Anisotropic Non-Gaussianity from a Two-Form Field
aa r X i v : . [ a s t r o - ph . C O ] A p r KUNS-2444
Anisotropic Non-Gaussianity from a Two-Form Field
Junko Ohashi , Jiro Soda , and Shinji Tsujikawa Department of Physics, Faculty of Science, Tokyo University of Science,1-3, Kagurazaka, Shinjuku, Tokyo 162-8601, Japan and Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: October 16, 2018)We study an inflationary scenario with a two-form field to which an inflaton couples non-trivially.First, we show that anisotropic inflation can be realized as an attractor solution and that the two-form hair remains during inflation. A statistical anisotropy can be developed because of a cumulativeanisotropic interaction induced by the background two-form field. The power spectrum of curvatureperturbations has a prolate-type anisotropy, in contrast to the vector models having an oblate-type anisotropy. We also evaluate the bispectrum and trispectrum of curvature perturbations byemploying the in-in formalism based on the interacting Hamiltonians. We find that the non-linearestimators f NL and τ NL are correlated with the amplitude g ∗ of the statistical anisotropy in the powerspectrum. Unlike the vector models, both f NL and τ NL vanish in the squeezed limit. However, theestimator f NL can reach the order of 10 in the equilateral and enfolded limits. These results areconsistent with the latest bounds on f NL constrained by Planck. I. INTRODUCTION
The inflationary paradigm [1] can successfully account for the observed temperature fluctuations of the CosmicMicrowave Background (CMB) radiation and the distribution of large scale structures [2]. The basic outcome ofsimplest single-field slow-roll inflation models is the statistical isotropy, the Gaussian and (almost) scale-invariantpower spectrum [3]. These statistical properties can be tested by precise measurements of the CMB temperatureanisotropies.The observational data provided by the Wilkinson Microwave Anisotropy Probe (WMAP) [4] showed the evidenceof the scale dependence of the power spectrum, whose property has been used to discriminate between a host ofinflationary models. Interestingly, the WMAP data also indicated the deviation from the Gaussian perturbationsand they gave us a hint of the statistical anisotropy [5]. According to the recent results of Planck, the Gaussianperturbations are still consistent with the data, but the statistical anisotropy remains [6]. In the light of these newresults, it is worth investigating a possible statistical anisotropy based on concrete theoretical models.Naively, the statistical anisotropy implies that vector fields can play an important role during inflation. A mechanismfor creating the statistical anomaly at the end of inflation was proposed in Ref. [7] and extended in various ways [8].Moreover, a more concrete model has been proposed in the context of supergravity [9]. There, the anisotropicinflation is realized as an attractor and the vector hair survives during inflation. The latter vector hair gives riseto rich phenomenology [10–12] such as the statistical anisotropy and the cross correlation between the curvatureperturbations and the primordial gravitational waves (see Refs. [13, 14] for reviews). In particular, the latter wouldimply the correlation between the temperature fluctuation and the B-mode polarization [15].Remarkably, subsequent works revealed that vector fields can also induce the large non-Gaussianity [16]. In par-ticular, the non-Gaussianity of curvature perturbations has been investigated in the context of anisotropic inflation[17]-[22], which further emphasized the rich phenomenology of the anisotropic inflationary models [23]. As a result,the Planck constrained the anisotropic inflationary models with vector fields strongly. In Ref. [11], however, it waspointed out that not only vector fields but also two-form fields can potentially give rise to anisotropic inflation. In fact,it is well known that there are two-form fields in string theory [24]. Hence it is natural to explore this possibility fromthe theoretical point of view. One may suspect that there is no statistical anisotropy because the two-form field canbe represented by a pseudo scalar field, i.e., axion. However, there remains a possibility that a non-trivial polarizationof a two-form field induces the statistical anisotropy. In this case the anisotropy comes from an expectation value ofthe spatial derivative of the axion field. At first glance this seems to be odd, but it is a natural framework from thepicture of two-form fields. Hence, in this paper, we study this possibility in detail.We show that several interesting features are present in our model. Analogous to the results of Ref. [9], anisotropicinflation can be sustained by the background two-form field. Moreover, as suggested in Ref. [11], there exists a prolatetype anisotropy in the power spectrum of curvature perturbations. In contrast to vector models the non-Gaussianityvanishes in the squeezed limit, but the nonlinear estimator f NL of the equilateral and enfolded shapes can be as largeas the order of 10. Hence, our predictions are consistent with the Planck data and the future analysis may revealthese statistical anisotropies.The organization of our paper is as follows. In Sec. II, we introduce the two-form field model and explain howthe non-trivial hair remains during inflation. In Sec. III, we quantize the perturbations of two-form fields and derivetheir vacuum expectation values on super-Hubble scales. In Sec. IV, we evaluate various statistical quantities–suchas n -point correlation functions ( n = 2 , ,
4) of curvature perturbations. Sec. V is devoted to conclusions.
II. ANISOTROPIC BACKGROUND
We study the background dynamics of anisotropic inflation in the presence of two-form fields. The analysis is similarto the case of vector models studied in Ref. [9], but we repeat it for completeness. Here, we emphasize the shape ofanisotropy is different from that of vector models.Let us start with the action S = Z d x √− g " M p R − ∂ µ φ∂ µ φ − V ( φ ) − f ( φ ) H µνλ H µνλ , (1)where M p is the reduced Planck mass, R is a scalar curvature calculated from a metric g µν (with a determinant g ), φ is an inflaton field with a potential V ( φ ), and the field H µνλ has a non-trivial coupling f ( φ ) with the inflaton. Thefield H µνλ is related to a two-form field B µν , as H µνλ = ∂ µ B νλ + ∂ ν B λµ + ∂ λ B µν . (2)Without loss of generality, one can take the ( y, z ) plane in the direction of the two-form field. Then we can express B µν in the form 12 B µν dx µ ∧ dx ν = v ( t ) dy ∧ dz , (3)where v ( t ) is a function with respect to the cosmic time t .Since there exists a rotational symmetry in the ( y, z ) plane, it is convenient to parameterize the metric as follows: ds = −N ( t ) dt + e α ( t ) h e − σ ( t ) dx + e σ ( t ) ( dy + dz ) i , (4)where α describes the average expansion (the number of e-foldings) and σ represents the anisotropy of the Universe.Here, the lapse function N is introduced to derive the Hamiltonian constraint. With the above ansatz, the action (1)reduces to S = Z d x N e α (cid:20) M p ( − ˙ α + ˙ σ ) + 12 ˙ φ − N V ( φ ) + 12 f ( φ ) ˙ v e − α ( t ) − σ ( t ) (cid:21) , (5)where an overdot denotes a derivative with respect to t . After taking variations, we set the gauge N = 1.The equation of motion for the two-form field v is easily solved as˙ v = Af ( φ ) − e α +4 σ , (6)where A is a constant of integration. Taking the variation of the action with respect to N , α, σ, φ and using thesolution (6), we obtain the following background equations of motion˙ α = ˙ σ + 13 M p (cid:20)
12 ˙ φ + V ( φ ) + A f ( φ ) − e − α +4 σ (cid:21) , (7)¨ α = − α + 1 M p (cid:20) V ( φ ) + A f ( φ ) − e − α +4 σ (cid:21) , (8)¨ σ = − α ˙ σ − A M p f ( φ ) − e − α +4 σ , (9)¨ φ = − α ˙ φ − V ,φ + A f ( φ ) − f ,φ e − α +4 σ , (10)where the subscript in V ,φ and f ,φ denotes a derivative with respect to its argument φ .Let us introduce the energy density of the two-form field ρ v ≡ A f ( φ ) − e − α +4 σ , (11)and the shear Σ ≡ ˙ σ . We also define the expansion rate H ≡ ˙ α . First, we need to look at the shear to the expansionrate ratio Σ /H , which characterizes the anisotropy of the inflationary Universe. Notice that Eq. (9) reads˙Σ = − H Σ − ρ v M p . (12)In the regime where ˙Σ becomes negligible, the ratio Σ /H should converge to the valueΣ H = − ρ v V ( φ ) , (13)where we used Eq. (7) under the slow-roll approximation, i.e.,˙ α = H ≃ V ( φ )3 M p . (14)In order for the anisotropy to survive during inflation, ρ v must be almost constant. Employing the standard slow-roll approximation and assuming that the two-form field is sub-dominant in the inflaton equation of motion (10),one can show the coupling function f ( φ ) should be proportional to e − α to keep ρ v almost constant. In the slow-rollregime, the number of e-foldings α is related to φ , as dα = − V ( φ ) dφ/ ( M p V ,φ ). Then the functional form of f ( φ ) isdetermined as f ( φ ) = e − α = e R VM pV,φ dφ . (15)For the polynomial potential V ∝ φ n , for example, we have f = e φ nM p . The above case is, in a sense, a criticalone. What we want to consider is super-critical cases where the energy density of the two-form field increases. Forsimplicity, we parameterize f ( φ ) by f ( φ ) = e c R VM pV,φ dφ , (16)where c is a constant parameter. Let us consider the super-critical cases c >
1. From the definition (16), we canderive the following relation f ,φ f = c VM p V ,φ . (17)Then, the condition c > f ,φ f M p V ,φ V > . (18)Thus, any functional pairs of f and V satisfying the condition (18) in some range could produce the two-form hairduring inflation.On using Eq. (17), the inflation equation (10) reads¨ φ = − α ˙ φ − V ,φ (cid:20) − cǫ V ρ v V ( φ ) (cid:21) , (19)where the slow-roll parameter ǫ V is defined as ǫ V ≡ M p (cid:18) V ,φ V (cid:19) . (20)In this case, if the two-form field is initially small ( ρ v /V ( φ ) ≪ ǫ V /c ), then the conventional single-field slow-rollinflation is realized. During this stage f ∝ e − cα and the energy density of the two-form field grows as ρ v ∝ e c − α .Therefore, the two-form field eventually becomes relevant to the inflaton dynamics described by Eq. (19). Nevertheless,the cosmic acceleration continues because ρ v /V ( φ ) does not exceed ǫ V /c . In fact, if ρ v /V ( φ ) exceeds ǫ V /c , the inflatonfield φ does not roll down, which makes ρ v = A f ( φ ) − e − α +4 σ / ρ v ≪ V ( φ ) is alwayssatisfied. In this way, there appears an attractor where inflation continues even when the two-form field affects theinflaton dynamics [9].Let us make the above statement more precise. Under the slow-roll approximation, the inflaton equation of motion(10) reads − α ˙ φ − V ,φ + A f − f ,φ e − α +4 σ ≃ . (21)Using Eqs. (14) and (21), we obtain dφdα = ˙ φ ˙ α = − M p V ,φ V + c A V ,φ e − α +4 σ − c R VM pV,φ dφ . (22)Neglecting the evolution of V , V ,φ and σ , this equation can be integrated to give e α − σ +2 c R VM pV,φ dφ = c A c − VM p V ,φ h e − c − α − σ i , (23)where Ω is a constant of integration. Substituting this back into the slow-roll equation (22), it follows that dφdα = − M p V ,φ V + c − c M p V ,φ V h e − c − α − σ i − . (24)At the initial stage of inflation ( α → −∞ ), the second term of Eq. (24) can be neglected relative to the first term.In the asymptotic future ( α → ∞ ), the term containing Ω disappears. This clearly shows a transition from theconventional slow-roll inflationary phase, where dφdα = − M p V ,φ V , (25)to what we refer to as the second inflationary phase, where the two-form field is relevant to the inflaton dynamics andthe inflaton gets 1 /c times slower as dφdα = − c M p V ,φ V . (26)In the second inflationary phase, we can employ Eq. (23) with discarding the Ω term to rewrite the energy densityof the two-form field as ρ v = A e − α +4 σ − c R VM pV,φ dφ = c − c ǫ V V ( φ ) , (27)which yields the anisotropy Σ H = − c − c ǫ V . (28)Moreover, from Eqs. (7) and (8), the slow-roll parameter ǫ ≡ − ˙ H/H is related to ǫ V as ǫ = − ¨ α ˙ α = − α (cid:18) −
12 ˙ φ − ρ v (cid:19) = 1 c ǫ V , (29)where we neglected the anisotropy and used Eqs. (14) and (26). Thus we arrive at the resultΣ H = − c − c ǫ. (30)Therefore, for a broad class of inflaton potentials and two-form kinetic functions, there exist anisotropic inflationarysolutions, with Σ /H proportional to ǫ . We also confirmed the existence of such anisotropic solutions by integratingEqs. (7)-(10) numerically. Note that the sign of Σ /H is opposite to that derived for the vector field [9].For c = 1, we need a separate treatment. In this case, integration of Eq. (22) gives e α − σ +2 R VM pV,φ dφ = 2 A VM p V ,φ ( α + α ) , (31)where α is an integration constant. Thus, we obtain the anisotropyΣ H = − α + α ) ǫ . (32)Notice that the anisotropy stemmed from the two-form field is a prolate type, while the anisotropy created by thevector field was an oblate type. This can be understood by the fact that the vector extending to the x -directionspeeds down the expansion in that direction, while the two-form field extending in the ( y, z ) plane speeds down theexpansion in the ( y, z ) direction.Before entering the study of perturbations, we should note the following important attractor mechanism [25]. Takinga look at the result (26), we find that the relation f = e c R VM pV,φ dφ ∝ e − α , (33)holds during the second anisotropic inflationary phase. Recall that this is the critical behavior. As we will see in thenext section, this attractor gives rise to the scale-invariant spectrum of the two-form field. III. PERTURBATIONS OF TWO-FORM FIELDS
From the phenomenological point of view the anisotropy of the expansion rate needs to be sufficiently small, so itis a good approximation to neglect the effect of the anisotropic expansion. However, we cannot ignore the effect ofthe two-form hair. Actually, in the next section, we will show several interesting results. In this section, we preparesome tools for the evaluation of n -point correlation functions of curvature perturbations.Let us consider the two-form field given by the action S int = − Z d x √− g f ( φ ) H µνλ H µνλ . (34)In the above action, there exists a gauge invariance under the gauge transformation δB µν = ∂ µ ξ ν − ∂ ν ξ µ . (35)Here, since we have the redundancy ξ µ → ξ µ + ∂ µ X , the parameter ξ can be restricted to be ∂ i ξ i = 0.For the derivation of the perturbation equations of the two-form field, we consider the isotropic background describedby the metric ds = a ( τ ) (cid:0) − dτ + δ ij dx i dx j (cid:1) , (36)where τ = R a − dt is the conformal time. From Eq. (2) it is convenient to perform the (3+1)-decomposition H ij = B ′ ij + ∂ i B j + ∂ j B i , (37) H ijk = ∂ i B jk + ∂ j B ki + ∂ k B ij , (38)where a prime represents a derivative with respect to τ . The interacting action (34) reads S int = − Z d x √− g f ( φ ) H ij H ij − Z d x √− g f ( φ ) H ijk H ijk . (39)Taking the variation of this action with respect to B i , we obtain ∂ i (cid:2) √− g f ( φ ) H ij (cid:3) = 0 . (40)Variation with respect to B ij leads to ∂∂τ (cid:2) √− g f ( φ ) H ij (cid:3) + ∂ k (cid:2) √− g f ( φ ) H ijk (cid:3) = 0 . (41)Then Eqs. (40) and (41) reduce to ∂ i (cid:2) B ′ ij + ∂ i B j − ∂ j B i (cid:3) = 0 , (42) − ∂∂τ (cid:20) a f ( φ ) (cid:8) B ′ ij + ∂ i B j + ∂ j B i (cid:9)(cid:21) + ∂ k (cid:20) a f ( φ ) { ∂ i B jk + ∂ j B ki + ∂ k B ij } (cid:21) = 0 . (43)Since we have the gauge degree of freedom δB ij = ∂ i ξ j − ∂ j ξ i , ∂ i ξ i = 0 , (44)choosing the parameter ∆ ξ j = − ∂ i B ij , (45)we can take the gauge ∂ i B ij = 0 . (46)Similarly, by taking into account the following gauge transformation δB i = ∂ i ξ − ξ ′ i , (47)we can set ∂ i B i = 0 . (48)From Eq. (42) it follows that B i = 0 . (49)Then, Eq. (43) reduces to − ∂∂τ (cid:20) a f ( φ ) B ′ ij (cid:21) + 1 a f ( φ ) ∂ k B ij = 0 . (50)Let us check the degrees of freedom. The components B ij have 3 degrees. There are 2 gauge conditions B ij,j = 0.Hence, we have one degree of freedom. This is the reason why we can map the two-form field to a scalar field.However, there is an important difference from the scalar field, namely, there exists a polarization in our case. Usingthe polarization tensor ǫ ij = − ǫ ji with k i ǫ ij = 0, we can expand the anti-symmetric tensor field as B ij ( τ, x ) = Z d k (2 π ) / (cid:2) a k χ ( τ, k ) ǫ ij e i k · x + c . c . (cid:3) , (51)where a k is the annihilation operator. We take the polarization tensor to be ǫ ij = ( k m /k ) ǫ mij , with the normalizationcondition ǫ ij ǫ ij = 2 . (52)The mode functions χ ( τ, k ) is known by solving Eq. (50). For convenience we introduce the following variable u ( τ, k ) = fa χ ( τ, k ) , (53)and parametrize the kinetic function as f = a p . This is always possible during inflation where both φ and a = e α aremonotonic functions with respect to time. Then, we obtain u ′′ + (cid:20) k + p (1 − p ) τ (cid:21) u = 0 . (54)Hence, for p = − p = 2, we obtain the scale-invariant spectrum for the two-form field. Although either choiceis possible, we set p = − p = −
1, we can deduce the mode functions as u = Ha √ k (1 + ikτ ) e − ikτ , (55)and χ = a u .Now, it is convenient to define E yz ≡ fa H yz = fa B ′ yz , (56) δE ij ≡ fa H ij = fa B ′ ij , (57)where E yz and δE ij correspond to the background value and the perturbation of the two-form field respectively. Notethat H ijk can be negligible on super-Hubble scales. We perform the Fourier transformation of the perturbation δE ij ,as δE ij = Z d k (2 π ) / e i k · x δ E ij ( τ, k ) . (58)On super-Hubble scales the Fourier modes are given by δ E ij ( τ, k ) = (cid:16) a k + a †− k (cid:17) E k ǫ ij , (59)where E k = fa χ ′ = ( a u ) ′ a ≃ H √ k . (60)In the last approximate equality of Eq. (60) we used the solution (55) in the limit τ → δE ij is given by h δE ij i = 1 π Z dk k |E k | ≃ H π Z IR dkk . (61)The Infrared (IR) modes are characterized by k i < k < k f , where k i and k f are the wavenumbers which crossedthe Hubble radius at the beginning and at the end of inflation respectively. Since the integral R IR dkk = ln( k f /k i ) isequivalent to the number of e-foldings N = ln( a f /a i ) on the de-Sitter background, it follows that h δE ij i = 9 H π N . (62)On super-Hubble scales the total two-form field is given by E classical ij = E yz + δE ij , (63)with the variance (62) of the perturbation δE ij . IV. STATISTICALLY ANISOTROPIC NON-GAUSSIANITY
In this section we estimate the statistical properties of our model, in particular, the scalar non-Gaussianity. Throughthis section, the anisotropy is assumed to be sufficiently small. We derive the interacting Hamiltonian by expandingthe action around the anisotropic background solution. We compute correlation functions according to the in-informalism by neglecting the anisotropic expansion of the Universe. The calculation is analogous to that carried outfor the vector field in Ref. [18]. This prescription can be justified by more rigorous calculations (see e.g., Ref. [17]).
A. Power spectrum
We first calculate the power spectrum of the comoving curvature perturbation ζ (see Refs. [26] for its definition). Inour case the curvature perturbation ζ can be written as the sum of the “unperturbed” field ζ (0) and the contribution δζ coming from the two-form field. We decompose the field ζ (0) into the Fourier components ζ (0) = Z d k (2 π ) / e i k · x ˆ ζ (0) k , ˆ ζ (0) k = ζ (0) k a k + ζ (0) ∗ k a †− k , (64)where the annihilation and creation operators satisfy the commutation relation[ a k , a † k ′ ] = δ (3) ( k − k ′ ) . (65)At leading order in slow-roll we have the following solution [27] ζ (0) k = H (1 + ikτ )2 √ ǫM p k / e − ikτ . (66)The total power spectrum P ζ is defined by the two-point correlation function of ζ , as h ˆ ζ k ˆ ζ k i = 2 π k δ (3) ( k + k ) P ζ ( k ) , (67)where ˆ ζ k is the Fourier component of ζ . The power spectrum can be written as the sum of the two contributions ζ (0) and δζ , as P ζ = P (0) ζ + δ P ζ . (68)Using the solution (66) long time after the Hubble radius crossing ( τ → P (0) ζ = H π ǫM p . (69)The next step is to derive the second contribution δ P ζ = δ h | ˆ ζ k ˆ ζ k | i from the two-form field. The interactingLagrangian following from Eq. (34) is L int = − a ∂ h f i ∂φ δφ ( H µνλ + δH µνλ ) (cid:0) H µνλ + δH µνλ (cid:1) , (70)where h i represents the background value. Since the function f is given by f = exp( R dφ/ √ ǫM p ), we can deducethe following relation ∂ h f i ∂φ δφ = 2 h f i ζ (0) , (71)where we used δφ = √ ǫM p ζ (0) . Note that there is no distinction between ǫ and ǫ V because we are considering thesituation c ∼
1. Thus, we obtain L int = 12 a − f (4 H yz δH yz + δH ij δH ij ) ζ (0) , = 12 a (4 E yz δE yz + δE ij δE ij ) ζ (0) , (72)where, in the second line, we employed the solution f = a − and Eqs. (56)-(57).The interacting Hamiltonian H int is related to L int , as H int = − R d x L int = H + H , where H and H followfrom the first and second terms of Eq. (72). Substituting Eqs. (58) and (64) into Eq. (72), we obtain H = − E yz H τ Z d k δ E yz ( τ, k )ˆ ζ (0) − k ( τ ) , (73) H = − H τ Z d k d p (2 π ) / δ E ij ( τ, k ) δ E ij ( τ, p )ˆ ζ (0) − k − p ( τ ) . (74)Using the in-in formalism [27], the two-point correction following from the interacting Hamiltonian H can be evaluatedas δ h | ˆ ζ k ˆ ζ k | i = − Z ττ min , dτ Z τ τ min , dτ h | hh ˆ ζ (0) k ( τ )ˆ ζ (0) k ( τ ) , H ( τ ) i , H ( τ ) i | i = E yz ǫ M p H Y i =1 Z τ − /k i dτ i τ i (cid:0) τ − τ i (cid:1) h | δ E yz ( τ , k ) δ E yz ( τ , k ) | i = 2 π k δ (3) ( k + k ) E yz N k cos θ k ,x π ǫ M p , (75)where θ k ,x is the angle between k and the x -axis. The two integrals in the first line of Eq. (75) have been evaluatedat the super-Hubble regime characterized by − k i τ <
1, from which τ min ,i = − /k i with i = 1 ,
2. In the second lineof Eq. (75) the upper bound τ of the second integral has been replaced by τ by dividing the factor 2! because of thesymmetry of the integrand. Note that we also used the following relation [18][ˆ ζ (0) k ( τ ) , ˆ ζ (0) k ′ ( τ ′ )] ≃ − iH ( τ − τ ′ )6 ǫM p δ (3) ( k + k ′ ) , (76)which is valid in the super-Hubble regime. One can show that the integral R τ − /k i dτ i ( τ − τ i ) /τ i is equivalent to N k i ≃ ln( − /k i τ ) under the approximation − k i τ ≪ N k i is the number of e-foldings before the end ofinflation at which the modes with the wavenumber k i left the Hubble radius. Since k = − k , we used the notation N k = N k ≡ N k .Using the power spectrum (69) and the slow-roll relation 3 M p H ≃ V , the two-form field gives rise to the correctionto the power spectrum δ P ζ = 6 E yz N k ǫV P (0) ζ cos θ k ,x . (77)The interacting Hamiltonian (74) has a contribution to δ P ζ with E yz replaced by the IR solution δE ij with theexpectation value (62). Taking into account this contribution, the total correction to the power spectrum can bederived by replacing E yz in Eq. (77) for E classical ij defined in Eq. (63). Using the notation E c ≡ | E classical ij | , the totalpower spectrum of the curvature perturbation reads P ζ = P (0) ζ (cid:0) IN k cos θ k ,x (cid:1) , where I ≡ E c ǫV . (78)In contrast to the vector case where the anisotropy is oblate [10, 11], we now have the prolate anisotropy.The statistics of the WMAP anisotropies uses the parametrization P ζ = P (0) ζ (1 + g ∗ cos θ k , V ), where V is a“privileged” direction close to the ecliptic poles [28, 29]. The WMAP data provides the constraint g ∗ = 0 . ± . g ∗ = 12 IN k , (79)from which we obtain I = 2 . × − (cid:16) g ∗ . (cid:17) (cid:18) N k (cid:19) . (80)Note that the quantity Iǫ = E c / (2 V ) characterizes the ratio of the energy densities of the two-form field ( E c /
2) andinflaton ( V ), which is much smaller than 1 from Eq. (80). B. Bispectrum
The three-point correlation of ζ can be evaluated by using the in-in formalism along the same line of Ref. [18]. Thetree-level contribution coming from the interacting Hamiltonian (73) is given by δ h | ˆ ζ k ˆ ζ k ˆ ζ k | i = i Z τ − /k dτ Z τ − /k dτ Z τ − /k dτ h | hhh ˆ ζ (0) k ˆ ζ (0) k ˆ ζ (0) k ( τ ) , H ( τ ) i , H ( τ ) i , H ( τ ) i | i + 2 perm . = E yz ǫ M p H Y i =1 Z τ − /k i dτ i τ i (cid:0) τ − τ i (cid:1) Z d p (2 π ) / ×h | δ E ij ( τ , k − p ) δ E ij ( τ , p ) δ E yz ( τ , k ) δ E yz ( τ , k ) + δE ij ( τ , k − p ) δ E ij ( τ , p ) δ E yz ( τ , k ) δ E yz ( τ , k ) + δE ij ( τ , k − p ) δ E ij ( τ , p ) δ E yz ( τ , k ) δ E yz ( τ , k ) | i = 3 E yz H √ π / ǫ M p δ (3) ( k + k + k ) N k N k N k (cid:20) cos θ k , k cos θ k ,x cos θ k ,x k k + 2 perm. (cid:21) , (81)0where “2 perm.” represents two terms obtained by the permutation. In the last line of Eq. (81) we used the relation ǫ ij ( k ) ǫ ij ( k ) = 2 cos θ k , k . The loop contribution following from the product of the three interacting Hamiltonians H provides the bispectrum in which E yz of Eq. (81) is replaced by the IR solution δE ij with the variance (62). Thenthe total anisotropic bispectrum B ζ , defined by δ h | ˆ ζ k ˆ ζ k ˆ ζ k | i = B ζ δ (3) ( k + k + k ), reads B ζ = 72 √ π / I ( P (0) ζ ) N k N k N k (cid:20) cos θ k , k cos θ k ,x cos θ k ,x k k + 2 perm. (cid:21) , (82)where I = E c / (2 ǫV ) is given by Eq. (80).We define the non-linear parameter f NL according to the relation B ζ = 310 (2 π ) / f NL ( P ζ ) X i =1 k i / Y i =1 k i , (83)by which we have f NL = 60 I ( P (0) ζ ) ( P ζ ) N k N k N k r + r [ r cos θ k , k cos θ k ,x cos θ k ,x + cos θ k , k cos θ k ,x cos θ k ,x + r cos θ k , k cos θ k ,x cos θ k ,x ] , (84)where r ≡ k k , r ≡ k k . (85)In the following we employ the approximations ( P ζ ) ≃ ( P (0) ζ ) and N k ≃ N k ≃ N k ≡ N CMB . We also fix r = 1and define the angle β = π − θ in the range 0 < β < π (i.e., 0 < r < r = 2(1 − cos β ) . (86)The local, equilateral, and enfolded shapes correspond to (i) β → r →
0, (ii) β = π/ r = 1, and (iii) β → π , r →
2, respectively.Let us consider the situation in which the angle between k and the x -axis is given by γ . On using Eq. (80), thenon-linear parameter (84) reduces to f NL ≃ . (cid:16) g ∗ . (cid:17) (cid:18) N CMB (cid:19) F ( r , γ ) , (87)where F ( r , γ ) ≡
12 + r (cid:20) r cos β cos γ (cos β cos γ + sin β sin γ ) + 12 (cos β cos γ + sin β sin γ − cos γ ) (cid:21) . (88)From Eq. (86) we can express β in terms of r , as cos β = 1 − r / β = r p − r /
4, so that f NL is afunction of r for a given value of γ . The non-linear parameters for the local, equilateral, and enfolded shapes aregiven, respectively, by f localNL = 7 . (cid:16) g ∗ . (cid:17) (cid:18) N CMB (cid:19) β sin γ , (89) f equilNL = 3 . (cid:16) g ∗ . (cid:17) (cid:18) N CMB (cid:19) , (90) f enfoldedNL = 29 . (cid:16) g ∗ . (cid:17) (cid:18) N CMB (cid:19) cos γ , (91)where, in the local case, we expanded f localNL around β = 0. The local non-linear parameter (89) vanishes in thesqueezed limit β →
0, which is a distinctive feature of our model. The reason why this happens is that, unlike thevector models, f NL is proportional to the inner product of two vectors k i and k j . In Eq. (84) the squeezed limitcorresponds to the case in which the angles θ k , k and θ k , k approach π/ r → f N L r cos γ =0cos γ =1/4cos γ =1/2cos γ =3/4cos γ =1 0 5 10 15 20 25 30 0 0.5 1 1.5 2 f N L r cos γ =0cos γ =-1/4cos γ =-1/2cos γ =-3/4cos γ =-1 FIG. 1: The non-linear estimator f NL versus r = k /k for a number of different values of cos γ with g ∗ = 0 . N CMB = 60.The left and right panels show the plots for the angles 0 ≤ γ ≤ π/ π/ ≤ γ ≤ π , respectively. The local, equilateral, andenfolded limits correspond to r = 0, r = 1, and r = 2, respectively. For γ close to π/
2, the equilateral non-linear parameteris largest. For γ close to 0 or π , f NL has a maximum at r = 2. From Eq. (90) the equilateral non-linear parameter does not depend on the angle γ . For g ∗ = 0 . N CMB = 60, f equilNL is as large as 10. The enfolded non-linear parameter (91) depends on γ . For cos γ = 1, g ∗ = 0 . N CMB = 60, f enfoldedNL is as large as 30.In Fig. 1 we plot the non-linear parameter (87) versus r (0 < r <
2) for g ∗ = 0 . N CMB = 60. The leftpanel and right panel correspond to positive and negative values of cos γ , respectively. In the local limit ( r → f NL vanishes for any value of γ . For the angle γ close to π/ f NL has a maximum at the equilateralconfiguration ( r = 1). With the increase of | cos γ | , however, the enfolded estimator gets larger. In particular, for γ close to 0 or π , f NL has a maximum at r = 2. C. Trispectrum
The four-point correlation function of ζ corresponding to the tree-level contribution is given by δ h | ˆ ζ k ˆ ζ k ˆ ζ k ˆ ζ k | i = Z τ − /k dτ Z τ − /k dτ Z τ − /k dτ Z τ − /k dτ ×h | hhhh ˆ ζ (0) k ˆ ζ (0) k ˆ ζ (0) k ˆ ζ (0) k ( τ ) , H ( τ ) i , H ( τ ) i , H ( τ ) i , H ( τ ) i | i + 5 perm . = 12 · E yz H ǫ M p Y i =1 Z τ − /k i dτ i τ i (cid:0) τ − τ i (cid:1) Z d pd p ′ (2 π ) ×h | δ E ij ( τ , k − p ) δ E ij ( τ , p ) δ E ij ( τ , k − p ′ ) δ E ij ( τ , p ′ ) δ E yz ( τ , k ) δ E yz ( τ , k )+ 11 perms . | i = − · (2 π ) E yz H ǫ M p δ (3) ( k + k + k + k ) N k N k N k N k × (cid:20) k k k cos θ k , k cos θ k , k cos θ k ,x cos θ k ,x + 11 perm. (cid:21) , (92)where k ij = k i + k j . The loop contribution, which follows from the product of the four interacting Hamiltonians H ,gives rise to the four-point correlation function where E yz in Eq. (92) is replaced by the IR solution δE ij with the2variance (62). Defining the total anisotropic trispectrum T ζ , as δ h | ˆ ζ k ˆ ζ k ˆ ζ k ˆ ζ k | i = T ζ δ (3) ( k + k + k + k ), weobtain T ζ = − π I ( P (0) ζ ) N k N k N k N k (cid:20) k k k cos θ k , k cos θ k , k cos θ k ,x cos θ k ,x + 11 perm. (cid:21) . (93)We introduce the non-linear estimator τ NL according to the relation T ζ = (2 π ) ( P ζ ) τ NL (cid:18) k k k + 11 perm. (cid:19) . (94)In the squeezed limit characterized by k →
0, the non-linear estimator reduces to τ localNL = − IN k N k [cos θ k , k cos θ k , k cos θ k ,x cos θ k ,x + cos θ k , k cos θ k , k cos θ k ,x cos θ k ,x + cos θ k , k cos θ k , k cos θ k ,x cos θ k ,x + cos θ k , k cos θ k , k cos θ k ,x cos θ k ,x ] , (95)where we used the approximation ( P ζ ) ≃ ( P (0) ζ ) . Since the angles between the vectors k i ( i = 1 , , ,
4) and k approach π/ k →
0, the estimator τ localNL vanishes in this limit.We also consider the regular tetrahedron limit, i.e., k = k = k = k = k = k ≡ k (see e.g., figure 2 of Ref. [31]for illustration). For this configuration, the angle between k and k is π/ k = √ k . We also focus on thecase in which the direction of k is the same as that of the x -axis. Then the trispectrum (93) reads T equil ζ ≃ − √ π I ( P (0) ζ ) N k k , (96)by which the non-linear estimator can be derived as τ equilNL ≃ − . × (cid:16) g ∗ . (cid:17) (cid:18) N k (cid:19) . (97)Unlike the local shape, | τ equilNL | can be of the order of 10 -10 . V. CONCLUSIONS
For the models in which the inflaton field φ couples to an anti-symmetric tensor B µν , we showed that anisotropicinflation occurs for the coupling f ( φ ) given by Eq. (15). In this case there is an attractor solution along which theratio of the anisotropic shear Σ to the Hubble parameter H is proportional to the slow-roll parameter ǫ . Even for thesuper-critical case in which the coupling f ( φ ) is generalized to Eq. (16) with c >
1, there is the regime of anisotropicinflation where Σ /H is nearly constant with f proportional to a − . The anisotropy induced by the two-form fieldcorresponds to the prolate type (the expansion of the Universe slows down in the ( y, z ) plane), in contrast to theoblate type stemming from the vector field.The presence of the two-form field coupled to inflaton gives rise to modifications to statistical quantities observedin CMB temperature fluctuations. From the action (34) we derived the interacting Hamiltonians (73) and (74)between curvature perturbations and the two-form field. We evaluated the n -point correlation functions ( n = 2 , , g ∗ = 12 IN k parametrizes the strength of anisotropy. Even if theenergy density of the two-form field is very much smaller than that of inflaton, the parameter g ∗ can be of the orderof 0.1, as suggested by the WMAP data [30].In Eqs. (82) and (84) we derived the three-point correlation function B ζ (bispectrum) and the non-linear estimator f NL , which exhibit a number of interesting properties. By considering the triangle of three momenta ( k + k + k = 0)with k = k , we showed that f NL can be expressed by functions of r = k /k and the angle γ between k and the x -axis. In the local, equilateral, and enfolded limits, the non-linear estimators are simplify given by Eqs. (89), (90),and (91), respectively. We found that f localNL vanishes in the squeezed limit ( r → f equilNL and f enfoldedNL can beof the order of 10 (see Fig. 1). These results are consistent with the recent constraints by Planck, i.e., f localNL = 2 . ± . f equilNL = − ±
75 (68 % CL) [6].The four-point correlation function T ζ (trispectrum) has been also computed in Eq. (93). Defining the non-linearestimator τ NL as Eq. (94), we found that τ NL vanishes in the squeezed limit ( k → | τ NL | can be of the order of 10 -10 . This is an interesting property by which ourscenario can be distinguished from the vector case as well as other models with large non-Gaussianities.It will be of interest to understand the physics of a dipole-type anisotropy suggested by the Planck data [6]. Althoughthere is a phenomenological description of this type of anisotropy [32], no physically well-motivated models are presentto our best knowledge. Recent attempt to explain the dipole-type anisotropy by a contrived geometrical set up isintriguing [33], but it still lacks a consistent dynamical picture. Since our framework is natural and consistent, itwould be great if our mechanism is generalized to explain the origin of the dipole-type anisotropy observed by Planck. Acknowledgments
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