Cosmological singleton gravity theory and dS/LCFT correspondence

We study the evolution of cosmological perturbations generated during de Sitter inflation in the singleton gravity theory. This theory is composed of a dipole ghost pair in addition to tensor. We obtain the singleton power spectra which show that the de Sitter/logarithmic conformal field theory (dS/LCFT) correspondence works for computing the power spectra in the superhorizon limit. Also we compute the spectral indices for light singleton which contains a logarithmic correction.

Hence, it is quite interesting to compute the power spectrum of singleton (other than inflaton) generated during dS inflation because its equation is a degenerate fourth-order equation. In order to compute the power spectrum, one needs to choose the Bunch-Davies vacuum in the subhorizon limit of z → ∞. Therefore, one has to quantize the singleton canonically as we do for the inflaton. Also, it is important to see whether the dS/LCFT correspondence plays a crucial role in computing the power spectrum in the superhorizon limit of z → 0 [9]. As far as we know, there is no direct evidence for the dS/LCFT correspondence. We will show that the momentum LCFT-correlators σ a (k)σ b (−k) obtained from the extrapolation approach take the same form as the power spectra [P ab,0 (k, −1)] × k −3 . This shows that the dS/LCFT correspondence works well for obtaining the power spectra in the superhorizon limit.

Singleton gravity theory
Let us first consider the singleton gravity theory where a dipole pair φ 1 and φ 2 are coupled minimally to Einstein gravity. The action is given by where the first two terms are introduced to provide de Sitter background with Λ > 0 and the last three terms (S S ) represent the singleton theory composed of two scalars φ 1 and φ 2 [5][6][7]. Here we have κ = 8πG = 1/M 2 P , M P being the reduced Planck mass and m 2 is the degenerate mass-squared for the singleton. We stress that S SG denotes the action for the singleton gravity theory, whereas S S is the action for the singleton theory itself.
The Einstein equation takes the form G µν + κΛg µν = κT µν (2.2) with the energy-momentum tensor On the other hand, two scalar field equations are coupled to be which are combined to give a degenerate fourth-order equation This reveals the nature of the singleton theory as S S takes the following form upon using (2.4) to eliminate the auxiliary field φ 1 [27,28]: The solution of dS spacetime comes out when one chooses the vanishing scalars R = 4κΛ,φ 1 =φ 2 = 0. (2.7) Explicitly, curvature quantities are given bȳ with a constant Hubble parameter H 2 = κΛ/3. We choose the dS background explicitly by choosing a conformal time η where the conformal scale factor is (2.10) Here the latter denotes the scale factor with respect to cosmic time t. During the dS stage, a goes from small to a very large value like a f /a i ≃ 10 30 which implies that the conformal time η = −1/aH(z = −kη) runs from −∞(∞)[the infinite past] to 0 − (0) [the infinite future]. The two boundaries (∂dS ∞/0 ) of dS space are located at η = −∞ together with a point η = 0 − which make the boundary compact [26]. It is worth noting that the Bunch-Davies vacuum will be chosen at η = −∞, while the dual (L)CFT can be thought of as living on a spatial slice at η = 0 − . We choose the Newtonian gauge of B = E = 0 andĒ i = 0 for cosmological perturbation around the dS background (2.9). In this case, the cosmologically perturbed metric can be simplified to be with transverse-traceless tensor ∂ i h ij = h = 0. Also, one has the scalar perturbations In order to get the cosmological perturbed equations, one linearize the Einstein equation (2.2) directly around the dS We would like to mention briefly two metric scalars Ψ and Φ, and a vector Ψ i . The linearized Einstein equation requires Ψ = −Φ which was used to define the comoving curvature perturbation in the slow-roll inflation and thus, they are not physically propagating modes. In the dS inflation, there is no coupling between {Ψ, Φ} and {ϕ 1 , ϕ 2 } because of φ 1 =φ 2 = 0. The vector is also a non-propagating mode in the singleton gravity theory because it has no its kinetic term. The linearized scalar equations are given by (2.14) These are combined to provide a degenerate fourth-order scalar equation which is our main equation to be solved for cosmological purpose.

dS/LCFT correspondence in the superhorizon
First of all, we briefly review what are similarities and differences between AdS/CFT and dS/CFT dictionaries. The first version of the AdS/CFT dictionary was stated in terms of an equivalence between bulk and boundary partition functions in the presence of deformations: where on the bulk side φ 0 specifies the boundary conditions of bulk field φ propagating on M, whereas on the boundary CFT φ 0 denotes the sources of operators O on the boundary ∂M. Correlator of dual CFT can be computed by differentiating the partition function with respect to the sources and then, setting them to zero as This is called "differentiate" (GKPW) dictionary [29]. The second version consists of computing bulk-to-boundary propagators first and pulling CFT correlators to the boundary as This version was used in [30] and was referred to "extrapolate" (BDHM) dictionary [31].
Concerning correlation functions of a free massive scalar in AdS and dS, the following three statements appear importantly [32]: (a) In Euclidean AdS d+1 with ℓ 2 AdS = 1, either the differentiation of the partition function with respect to sources or extrapolation of the bulk operators to the boundary produce CFT correlators of an operator with dimension ∆ = d 2 + . (c) In Lorentzian dS d+1 with ℓ 2 dS = 1, functional derivatives of late-time Schrödinger wavefunction produce CFT correlators with dimension δ only. The dominant term in (b) was computed by Witten for a particular scalar [33], while a massless version of statement (c) was firstly made by Maldacena [26]. This implies that the dS/CFT "extrapolate" and "differentiate" dictionaries are inequivalent to each other. Particularly, the dimension of CFT operators associated to a massive scalar is different: for "extrapolate" dictionary in four dimensional dS space. Accordingly, following (c) to compute cosmological correlator of a massive scalar, it in momentum space is inversely proportional to CFT correlator with dimension ∆ + as which leads to the power spectrum for a massive scalar in the superhorizon limit. If one employs (c) to derive the dS/LCFT correspondence, the approach (c) may break down for deriving LCFT correlators because all LCFT correlators in AdS d+1 were derived based on the extrapolation approach (b) [5][6][7][8]. Hence, we wish to use the extrapolation approach (b) to derive the LCFT correlators from the bulk correlators. In this case, the cosmological correlator is directly proportional to the CFT correlator with different dimension as was shown in (3.3).
To develop the dS/LCFT correspondence [9], we first solve Eqs.(2.14) and (2.15) for the singleton gravity theory in the superhorizon limit of η → 0 − . Their solutions are given by The scaling of ϕ a,0 with a = 1, 2 is not conventional as they transform under The explicit connection between ϕ a,0 and σ a is encoded by [34] where the expectation value · · · is taken in the LCFT with the boundary fields ϕ a,0 as sources. Eq.(3.9) is a statement of the dS/LCFT correspondence. Here the bulk action is given by The bulk transformation (3.8) indicates that two operator σ a of conformal dimension w transform under dilations as where a dimension matrix ∆ b a is brought to the Jordan cell form as This implies that σ a transform under dilations of x → λx as In order to find the LCFT correlators σ a (x)σ b (y) , one might use the Ward identities for scale and special conformal transformations [9]. In this work, we wish to rederive them by using the extrapolation approach (b) (see Appendix for detail computations). The two-point functions of σ 1 and σ 2 are determined by Here w is a degenerate dimension of σ 1 and σ 2 . The coefficient A = w(2w−3) is determined by the normalization of σ 1 and σ 2 . However, D is arbitrary. The CFT vacuum |0 C is defined by three Virasoro operators L n |0 C = 0 for n = 0, ±1. The highest-weight state |σ a C = σ a (0)|0 C for two primary fields σ a of conformal weight h = w/2 is defined by This implies that for any pair of degenerate operators σ 1 and σ 2 (logarithmic pair), the Hamiltonian (L 0 ) becomes non-diagonalizable which shows us a crucial difference from an ordinary CFT. Actually, Eq.(3.19) represents the CFT version of the bulk transformation where CFT and LCFT represent their correlators in (3.17) and (3.18), respectively. In order to derive the relevant correlators in momentum space, one has to use the relation where we observe an inverse-relation of exponent 2w between |x|-space and k = |k|-space. Finally, the correlators in momentum space are easily evaluated as [9] where the prime ( ′ ) represents correlators without the (2π) 3 δ 3 (Σ i k i ) and A 0,w = 4w − 3 denotes derivatives of A 0 (w) = w(2w − 3) with respect to w. These correlators will be compared to the power spectra in the superhorizon limit of z → 0.

Singleton propagation in dS spacetime
In order to compute the power spectrum, we have to know the solution to singleton equations Eqs.(2.14) and (2.15) in the whole range of η(z). For this purpose, the scalars ϕ i can be expanded in Fourier modes φ i k (η) The first equation of (2.14) leads to which can be further transformed into with the index The solution to (4.5) is given by the Hankel function H ν . Accordingly, one has the solution to (4.2) with C undetermined constant. In the subhorizon limit of z → ∞, Eq.(4.2) reduces to which leads the positive-frequency solution with the normalization 1/ √ 2k This is a typical mode solution of a massless scalar propagating on dS spacetime. Inspired by (4.9) and asymptotic form of H (4.10) In the superhorizon limit of z → 0, Eq.(4.2) takes the form On the other hand, plugging (4.1) into (2.15) leads to the degenerate fourth-order differential equation which seems difficult to be solved directly. However, we may solve Eq.(4.14) in the two limits of subhorizon and superhorizon. In the subhorizon limit of z → ∞, Eq.(4.14) takes the form whose direct solution is given by with two coefficientsc 1 andc 2 . The c.c. of φ 2,d k,∞ is a solution to (4.15) too. Here Ei(2iz) is the exponential integral function defined by [35] Ei(2iz) = Ci(2z) + iSi(2z) + i π 2 , (4.17) where the cosine-integral and sine-integral functions are given by We note that Ei(2iz) satisfies the fourth-order equation However, we wish to point out that the direct solution (4.16) is not suitable for choosing the Bunch-Davies vacuum to give quantum fluctuations. In order to find an appropriate solution, we note that (∇ 2 − m 2 )ϕ 2 = µ 2 ϕ 1 in (2.14) reduces to in the subhorizon limit whose solution is We note that φ 2 k,∞ (z) is included as the first term of (4.16) [as a solution to the fourth-order equation (4.15)].
On the other hand, Eq.(4.14) takes the form in the superhorizon limit of z → 0 as whose solution is given by This also satisfies for µ 2 = (3 − 2w)H 2 which is the superhorizon limit of Eq.(2.14). The presence of "ln z" implies that (4.23) is a solution to the fourth-order equation (4.22) Finally, the trick used in [6] implies that one may solve (4.14) directly by differentiating (∇ 2 − m 2 )ϕ 1 = 0 with respect to m 2 . The explicit steps are given by which provides a way to obtain φ 2 k (z) from φ 1 k (z) as We note that (4.14) can be obtained by acting (∇ 2 − m 2 ) on (4.27). Explicitly, d with the digamma function ψ(x) = ∂ ln[Γ(x)]/∂x. Here we observe the appearance of ln[z]term. It turns out that φ 2 k (z) takes the form when considering J ±ν → Γ(±ν + 1) −1 (z/2) ±ν in the superhorizon limit of z → 0 as which recovers (4.23). We mention that ∂ ∂ν J −ν in (4.29) is dominant because it behaves as z −ν ln[z] in the superhorizon limit of z → 0. However, we do not recover its asymptotic form (4.21) in the subhorizon limit of z → ∞. Hence, it is not easy to obtain a full solution φ 2 k (z) to (4.14) by the trick used in [6]. Fortunately, its superhorizon-limit solution (4.23) could be found by this trick.

Power spectra
The power spectrum is defined by the two-point function which could be computed when one chooses the Bunch-Davies (BD) vacuum state |0 BD in the subhorizon limit (∂dS ∞ ) of η → −∞(z → ∞) [14]. The defining relation is given by where F represents singleton and tensor and k = √ k · k is the comoving wave number. Quantum fluctuations were created on all length scales with wave number k. Cosmologically relevant fluctuations start their lives inside the Hubble radius which defines the subhorizon: k ≫ aH. On later, the comoving Hubble radius 1/(aH) shrinks during inflation while keeping the wavenumber k constant. Eventually, all fluctuations exit the comoving Hubble radius, they reside on the superhorizon region of k ≪ aH after horizon crossing.
In general, one may compute the power spectrum of scalar and tensor by taking the BD vacuum. In the dS inflation, we choose the subhorizon limit of z → ∞ to define the BD vacuum. This implies that in the infinite past of η → −∞(z → ∞), all observable modes had time-independent frequencies ω = k and the Mukhanov-Sasaki equation reduces to F ′′ k,∞ + k 2 F k,∞ ≈ 0 whose positive solution is given by F k,∞ = e −ikη / √ 2k = e iz / √ 2k. This defines a preferable set of mode functions and a unique physical vacuum, the BD vacuum |0 BD .
On the other hand, we choose the superhorizon region of z ≪ 1 to get a finite form of the power spectrum which stays alive after decaying. For example, fluctuations of a massless scalar (∇ 2 δφ = 0) and tensor (∇ 2 h ij = 0) with different normalization originate on subhorizon scales and they propagate for a long time on superhorizon scales. This can be checked by computing their power spectra given by In the limit of z → 0, they are finite as Accordingly, it would be very interesting to check what happens when one computes the power spectra for the dipole pair (singleton) generated from during the dS inflation in the framework of the singleton gravity theory.
To compute the power spectrum, we have to know the commutation relations and the Wronskian conditions. The canonical conjugate momenta are given by The canonical quantization is accomplished by imposing equal-time commutation relations: The two operatorsφ 1 andφ 2 are expanded in terms of Fourier modes as [27,28,36] with N andÑ the normalization constants. Plugging (5.7) and (5.8) into (5.6) determines the relation of normalization constants as NÑ = 1/2k and commutation relations between which reflects the quantization of singleton. Here, the commutation relation of [ĉ 2 (k),ĉ † 2 (k ′ )] is implemented by the following Wronskian condition with (4.9) andc 2 = −iH/(2 √ 2k 3 ) in (4.21): It is important to note that the commutation relations (5.9) were used to derive the power spectra of conformal gravity [37]. On the other hand, if one uses the solution φ 1 k,∞ (4.9) and φ 2,d k,∞ (4.16), the Wronskian condition leads to which cannot be independent of z unlessc 1 =c * 1 = 0, This explains why the direct solution φ 2,d k,∞ (4.16) is not suitable for choosing the Bunch-Davies vacuum in the subhorizon limit. At this stage, we wish to mention when do the fluctuations of singleton become classical. The commutators in (5.6) commute on the superhorizon region of z < 1 after horizon crossing.
We are ready to compute the power spectrum of the dipole pair defined by Here we choose the BD vacuum |0 BD by imposingĉ a (k)|0 BD = 0. On the other hand, the cosmological correlator defined in momentum space are related to the power spectra as [14] φ a Since the singleton theory is quite different from the two-free scalar theory, we explain what the BD vacuum is. For this purpose, we remind the reader that the Gupta-Bleuler condition of B + (x)|phys = 0 where B is a conjugate momentum of scalar photon A 0 was introduced to extract the physical states of transverse photons A 1 and A 2 by confining scalar photon A 0 and longitudinal photon A 3 as members of quartet [38,39]. For this purpose, we note that the dipole pair (ϕ 1 , ϕ 2 ) is turned into the zero-norm state by making use of the BRST transformation in Minkowski spacetime [40]. We suggest that if the dS/LCFT correspondence works, the boundary logarithmic operator σ 2 is related to the negative-norm state of ϕ 2 . In order to remove the negative-norm state, we impose the subsidiary condition as ϕ + 1 (x)|phys = 0 where ϕ + 1 (x) is the positive-frequency part of the field operator. Then, the physical space (|phys ) will not include any ϕ 2 -particle state. This corresponds to the dipole mechanism to cancel the negative-norm state. Here, the subsidiary condition of ϕ + 1 (x)|phys = 0 is translated intoĉ 1 (k)|phys = 0 which shares a property of the BD vacuum |0 BD defined byĉ 1 (k)|0 BD = 0, in addition toĉ 2 (k)|0 BD = 0.
The tensor power spectrum for ϕ 1 is given as P 11 = 0 (5.14) when one used the unconventional commutation relation [ĉ 1 (k),ĉ † 1 (k ′ )] = 0. On the other hand, it turns out that the power spectrum of ϕ 2 is defined by In the superhorizon limit of z → 0, the power spectrum takes the form (5. 17) which implies that P 22 approaches zero as z → 0. In the massless case of m 2 = 0 (ν = 3/2, w = 0), P 22 leads to the power spectrum P δφ = (H/2π) 2 in (5.2) for a massless scalar. It is important to note that in the superhorizon limit of z → 0, P 22 is given by (5.18) which implies that P 22 approaches zero as z → 0. In deriving (5.18), ξ denotes a real quantity given by φ 1 k = −iξz w and φ 2 k ∼ ξz w ln[z]. We mention that the remaining power spectra P 12 and P 21 take the same form as P (1) 22 where we fixed N = 1/ √ 2k. Finally, we obtain the power spectra of singleton in the superhorizon limit of z → 0 .
Its explicit form is given by .
. (5.22) Interestingly, k −3 P ab,0 (k, −1) has the same form as the momentum correlators of LCFT (3.24). This may show how the dS/LCFT correspondence works for deriving the power spectra in the superhorizon limit. For a light singleton with m 2 ≪ H 2 , one has w ≃ m 2 3H 2 . Hence, these power spectra are given by whose spectral indices are given by . (5.24) played an important role to derive the power spectra in the superhorizon limit. On the other hand, the CFT vacuum |0 C was defined by imposing the Virasoro operators L n |0 C = 0 for n = 0, ±1. The highest-weight state |Φ C = Φ(0)|0 C for any primary field Φ of conformal weight h is defined by L 0 |Φ C = h|Φ C and L n |Φ C = 0 for n > 0. Consequently, we have derived the power spectra and spectral indices of singleton in the superhorizon limit by using two boundary conditions at the infinite past (η = −∞) and infinite future (η = 0 − ) where the BD vacuum was taken on the former time, while the CFT vacuum was employed on the latter time. The dS/LCFT correspondence was firstly realized as the computation of singleton power spectra. Since the LCFT as dual to the singleton suffers from the non-unitarity (for example, P 22,0 | m 2 H 2 ≪1 (k, −ǫ) < 0 for ǫk < 0.607), a truncation mechanism will be introduced to cure the non-unitarity in dS spacetime [8,40,43]. However, there remains nothing (σ 11 = 0) for the rank-2 LCFT dual to singleton after truncating (3.20). If one considers three-coupled scalar theory instead of singleton, its dual correlators will be not a 2 × 2 matrix (3.20) but a 3 × 3 matrix of The truncation process be carried out by throwing all terms which generate the third column and row of (6.1). Actually, this corresponds to finding a unitary CFT. We point out that a unitary CFT (σ 22 ) obtained after truncation is nothing but an ordinary CFT. Finally, let us ask how could this scenario account for cosmological observables like the amplitude of the power spectrum and the tensor-to-scalar ratio in the cosmic microwave background. In this work, we have chosen the dS inflation withφ 1 =φ 2 = 0 instead of the slow-roll (dS-like) inflation for simplicity. If we choose the slow-roll inflation, then the Einstein equation takes the form of G µν = T µν /M 2 P which provides the energy density ρ =φ 1φ2 + (m 2 φ 1 φ 2 + µ 2 φ 2 1 /2) and the pressure p =φ 1φ2 − (m 2 φ 1 φ 2 + µ 2 φ 2 1 /2). The first and second Friedmann equations are given by . Also, their scalar equations are given byφ 1 + 3Hφ 1 + m 2 φ 1 = 0 andφ 1 + 3Hφ 1 + m 2 φ 2 = −µ 2 φ 1 which are combined to give ( d 2 dt 2 + 3H d dt + m 2 ) 2 φ 2 = 0. However, it requires a formidable task to perform its cosmological perturbations around the slow-roll inflation instead of the dS inflation. Hence, we wish to remain "cosmological perturbations of singleton" as a future work by answering to the question how could this theory account for the observed cosmological parameters in the cosmic microwave background.
On the other hand, one may consider the holographic inflation and thus, the dS/CFT correspondence determines the tensor central charge. If one accepts holographic inflation such that the dS inflation era of our universe is approximately described by a dual CFT 3 living on the spatial slice at the end of inflation, the BICEP2 results might determine the central charge c T = 1.2 × 10 9 of the CFT 3 [44]. This is because every CFT 3 has a transverse-traceless tensor T ij with two DOF which satisfies T ij (x)T kl (0) = c T |x| 6 I ij,kl (x). Since a single complex scalar ψ represents two polarization modes of the graviton, its tensor correlator in momentum space is defined by ψ k ψ k ′ = (2π) 3 δ 3 (k + k ′ ) 2π 2 k 3 P T 2 which determines the tenor power spectrum P T = 2 Ht P π 2 = P h,0 in (5.4). This was determined to be 5 × 10 −10 by BICEP2 [12]. Also, its improvement of energy-momentum tensor was reported in [45] by including a curvature coupling of ζφ 2 R. As a result, if one uses the critical gravity including curvature squared terms to describe the holographic inflation, the dS/LCFT picture for tensor modes would play a role in determining other cosmological observables.
Appendix: LCFT correlators from "extrapolate" dictionary In this appendix, we derive the LCFT correlators by making use of the extrapolation approach (b) in the superhorizon limit. For this purpose, we consider the Green's function for a massive scalar propagating on dS spacetime . Taking a transformation form of hypergeometric function we obtain the asymptotic form for △ − = w lim η,η ′ →0 (ηη ′ ) −w G 0 (η, x; η ′ , y) ∝ 1 |x − y| 2w , (6.4) which corresponds to LCFT correlators e O 1 (x)O 2 (y) e = e O 2 (x)O 1 (y) e . Furthermore, the Green's function G 1 is derived by taking derivative with respect to w as where F denotes F = H 2 Γ(3 − w)Γ(w) 2 F 1 (w, w − 1, 2; 1 − 4/ξ)/(16π). It turns out that its asymptotic form is given by where ζ 1 is some constant and (6.6) corresponds to e O 2 (x)O 2 (y) e .