On primordial trispectrum from exchanging scalar modes in general multiple field inflationary models
aa r X i v : . [ h e p - t h ] O c t Preprint typeset in JHEP style - HYPER VERSION
CAS-KITPC/ITP-194USTC/ICTS-10-09
On the primordial trispectrum from exchanging scalarmodes in general multiple field inflationary models
Xian Gao
Key Laboratory of Frontiers in Theoretical Physics,Institute of Theoretical Physics, Chinese Academy of SciencesNo.55, Zhong-Guan-Cun East Road, Hai-Dian District, Beijing 100080, P.R.ChinaE-mail: [email protected]
Chunshan Lin
The Interdisciplinary Center for Theoretical Study,University of Science and Technology of China, Hefei, Anhui 230026, P.R.ChinaandDepartment of Physics, McGill UniversityMontr´eal, QC, H3A 2T8, CanadaE-mail: [email protected] A BSTRACT : We make an complementary investigation of the primordial trispectrum from exchanging intermediate scalarmodes in multi-field inflationary models with generalized kinetic terms. Together with the calculation of irreducible con-tributions to the primordial trispectrum in Ref.[104], we give the full leading-order primordial trispectrum in generalizedmulti-field models.K
EYWORDS : Multi-field inflation, Non-gaussianity, Trispectrum. ontents
1. Introduction 12. Basic Setup 2
3. Non-linear perturbations 6
4. Conclusion 9A. Coefficients in the interactional Hamiltonian 10B. Basic Integrals 10
1. Introduction
One of the most exciting ideas of modern cosmology is inflation [1], which can solve the flatness, the horizon, and themonopole problem of the standard big bang cosmology. Such a period of cosmological inflation can be attained if theenergy density of the universe is dominated by the vacuum energy density associated with the potential of some scalarfield(s). Over the years, inflation has become so popular because of its prediction of nearly scale-invariant primordialdensity perturbation. In the inflationary scenario, the primordial fluctuations of quantum origin were generated and frozento seed wrinkles in the Cosmic Microwave Background(CMB) [2][3][4][5][6][7][8][9] and today’s Large-scale Structure(LSS) [10][11][12][13][14].Inflation is mostly a framework of theories rather than a single model or theory. From the observational point ofview, many inflationary models are “degenerate”. Measuring tensor modes in the CMB anisotropy and the spectral indexof the power spectrum of adiabatic perturbation are not adequate to efficiently discriminate among different inflationaryscenarios. Fortunately, we have another observable available, which proves to be valuable in providing us with additionalinformation beyond the power spectrum to discriminate models. It is the deviation from a purely Gaussian statistics amongCMB anisotropies [15][16], which arises from interaction(s) among perturbations, leading to non-vanishing higher-ordercorrelated functions. Due to its importance, constraining and predicting primordial non-Gaussianity has become one ofthe major efforts in modern cosmological community.The simplest single-field slow-roll inflation models, within the context of Einstein gravity and the standard initialadiabatic vacuum, is only able to generate negligible amount of non-Gaussianity [17], which is undetectable by currentobservations of the CMB or even LSS. In the theoretical aspect, there are several ways to approach large non-Gaussianity.A short list of these models and mechanisms includes k -inflation or models with general non-canonical kinetic terms[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 105, 30, 31, 32, 33, 34], multi-field inflation[35, 36, 37, 38, 39, 40, 41, 42, 43,44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65], the curvaton scenario [68, 69, 70, 71, 72, 73,74, 75, 76, 82, 83, 84, 85] , inhomogeneous “end-of-inflation” models such as hybrid/multibrid models [77, 78, 79, 81, 80],cosmic string [86, 87, 88], loops [89, 90, 91], modified initial vacuum [92, 93], ghost inflation [94, 95], quasi-single fieldmodel [96, 97], vector fields [98, 99, 100, 103, 101, 102] and so on.– 1 –ince much more observational data will be available in the near future from WMAP/PLANCK and LSS experiments,it is very necessary to study the four and higher-point correlation functions. In this paper, we make a complement to thecalculation of Ref. [104], in which we calculated the contributions to the primordial trispectrum in general multi-fieldinflation from the irreducible or so-called “contact” diagrams. A complete calculation of the trispectrum should alsoinclude the contributions from reducible or so-called “exchanging intermediate scalar modes” diagrams, as performed in[42, 105, 30, 45] in the investigation of the trispectrum in single-field and multi-field inflationary models, and in [28]where exchanging gravitons was considered. In this paper we show that, the contributions to the final trispectrum arisingfrom exchanging scalar modes has the same magnitude as those from the contact contributions, and thus is also veryimportant.The remainder of this paper is organized as follows. In Sec.2, we briefly review the background evolution and linearperturbations for our model. Readers who are interested in the details are encouraged to refer to [104]. In Sec.3, wecalculate the tri-spectrum which originating from correlating (or exchanging) scalar modes. The full trispectrum, whichincludes both contacting and correlating scalar contributions, is also discussed.
2. Basic Setup
In this work we consider a general class of multi-field models containing N scalar fields coupled to Einstein gravity. Theaction takes the form S = Z d x √− g (cid:20) R P (cid:0) X IJ , φ I (cid:1)(cid:21) , (2.1)where φ I ( I = 1 , , · · · , N ) are scalar fields acting as inflaton fields, and X IJ ≡ − g µν ∂ µ φ I ∂ ν φ J , (2.2)is the kinetic term (matrix), g µν is the spacetime metric tensor with signature ( − , + , + , +) . “ I, J ”-indices are raised,lowered and contracted by the N -dimensional field-space metric G IJ = G IJ ( φ I ) . This form of the Lagrangian in-cludes multi-field k-inflation and multi-DBI models as special cases. For example, multi-field k-inflation has the scalar-field Lagrangian as P ( X, φ I ) , where X ≡ tr X IJ = G IJ X IJ , while in multi-field DBI models, P ( X IJ , φ I ) = − f ( φ I ) (cid:16) √D − (cid:17) − V ( φ I ) with D = 1 − f G IJ X IJ + 4 f X [ II X J ] J − f X [ II X JJ X K ] K + 16 f X [ II X JJ X KK X L ] L .We work in the ADM formalism of gravitation, in which the spacetime metric is written as ds = − N dt + h ij ( dx i + N i dt )( dx j + N j dt ) , (2.3)where N = N ( t, x ) is the lapse function, N i = N i ( t, x ) is the shift vector, and h ij is the spatial metric on constant timehypersurfaces. The ADM formalism is convenient because the equations of motion for N and N i are exactly the energyand momentum constraints which are easy to solve. Under the ADM formalism, the action (2.1) can be written as (up tototal derivative terms) S = Z dtd x √ hN (cid:18) R (3) + 12 N (cid:0) E ij E ij − E (cid:1)(cid:19) + Z dtd x √ hN P , (2.4)where h ≡ det h ij and the symmetric tensor E ij ≡ (cid:16) ˙ h ij − ∇ i N j − ∇ j N i (cid:17) , (2.5)with ∇ i the spatial covariant derivative defined with the spatial metric h ij and E ≡ tr E ij = h ij E ij . R (3) is the three-dimensional Ricci scalar which is computed from the spatial metric h ij . In the ADM formalism, spatial indices are raisedand lowered using h ij and h ij .In the ADM formalism, the kinetic matrix X IJ can be written as X IJ = − h ij ∂ i φ I ∂ j φ J + 12 N v I v J , (2.6)where v I ≡ ˙ φ I − N i ∇ i φ I . – 2 – .1.1 Equations of Motion The equations of motion for the scalar fields are ∇ µ (cid:0) P , h IJ i ∂ µ φ I (cid:1) + P ,J = 0 , (2.7)where ∇ µ is the four-dimensional covariant derivative. Here and in what follows, we denote P , h IJ i ≡ ∂P∂X IJ , P , h IJ ih KL i ≡ ∂ P∂X IJ ∂X KL , (2.8)as a shorthand notation.The equations of motion for N and N i are the Hamiltonian and momentum constraints respectively, R (3) + 2 P − N P , h IJ i v I v J − N (cid:0) E ij E ij − E (cid:1) = 0 , ∇ j (cid:18) N (cid:16) E ji − Eδ ji (cid:17)(cid:19) − P , h IJ i N v I ∇ i φ J = 0 . (2.9) In this work, we investigate scalar perturbations around a flat FRW background, the background spacetime metric takesthe form ds = − dt + a ( t ) δ ij dx i dx j , (2.10)where a ( t ) is the so-called scale-factor. The Friedmann equation and the continuity equation are H = ρ ≡ (cid:0) X IJ P , h IJ i − P (cid:1) , ˙ ρ = − H ( ρ + P ) . (2.11)In the above equations, all quantities are background values. From the above two equations we can also get anotherconvenient equation ˙ H = − X IJ P , h IJ i . (2.12)The background equations of motion for the scalar fields are P , h IJ i ¨ φ I + (cid:16) HP , h IJ i + ˙ P , h IJ i (cid:17) ˙ φ I − P ,J = 0 , (2.13)where P ,I denotes derivative of P with respect to φ I : P ,I ≡ ∂P∂φ I .In this work, we investigate cosmological perturbations during an exponential inflationary period. Thus, from (2.12)it is convenient to define a slow-roll parameter ǫ ≡ − ˙ HH = P , h IJ i ˙ φ I ˙ φ J H . (2.14) The scalar metric fluctuations about our background can be written as (see [106, 107] for nice reviews of the theory ofcosmological perturbations) δN = α ,δN i = ∂ i β ,δg ij = − a (cid:0) ψδ ij − ∂ i ∂ j E ) (2.15)where α, β, ψ and E are functions of space and time . The scalar field perturbations are denoted by δφ I ≡ Q I .Before proceeding, we would like to analyze the (scalar) dynamical degrees of freedom in our system. In the begin-ning we have N + 4 apparent scalar degrees of freedom. The diffeomorphism of Einstein gravity eliminates two of them , This form of ansatz corresponds to δg = 1 − N + N i N i and δg i = N i . See [106] for a detailed discussion on the gauge issue of cosmological perturbations. – 3 –eaving us N + 2 scalar degrees of freedom. Furthermore, two of these N + 2 degrees of freedom are non-dynamical.In the ADM formalism, these are just the fluctuations δN = α and δN i = ∂ i β . Thus, there are N propagating degreesof freedom in our system. As has been addressed, the diffeomorphism invariance allows us to choose convenient gaugesto eliminate two degrees of freedom. In single-field models, there are two convenient gauge choices: comoving gaugecorresponding to choosing δφ = E = 0 or spatially-flat gauge corresponding to ψ = E = 0 . In the multi-field case, thecomoving gauge loses its convenience since we cannot set δρ = 0 for every field in multi-field case. Thus, in this workwe use the spatially-flat gauge.In the spatially-flat gauge, propagating degrees of freedom for scalar perturbations are the inflaton field perturbations Q I ( t, x ) , while δN and δN i are non-dynamical constraints. In this work, we focus on scalar perturbations. In general, itis well-known that in the higher-order perturbation theories, scalar/vector/tensor perturbation modes are coupled together.However, from the point of view of the perturbation action approach, these couplings are equivalent to exchanging var-ious modes. In this work, we focus on interactions of scalar modes themselves, and neglect tensor perturbations. Theperturbations take the form φ I ( t, x ) = φ I ( t ) + Q I ( t, x ) ,h ij ≡ a δ ij N = 1 + α + α + · · · ,N i = ∂ i ( β + β + · · · ) + θ i + θ i + · · · , (2.16)where φ I ( t ) is the background value, and α n , β n , θ ni are of order O ( Q n ) .The next step is to solve the constraints α n , β n and θ ni in terms of Q I . Fortunately, in order to expand the action tothird-order in Q I , the solutions for the constraints up to the first-order are adequate. At the first-order in Q I , a particularsolution for equations (2.9) is: α = 12 H P , h IJ i ˙ φ I Q J ,β = a H ∂ − (cid:20)(cid:0) P , h IJ i + 2 X KL P , h IJ ih KL i (cid:1) (cid:18) X IJ H P , h KL i ˙ φ K Q L − ˙ φ I ˙ Q J (cid:19) − HP , h IJ i ˙ φ I Q J − P , h IJ i K Q K X IJ + P ,I Q I i ,θ i = 0 . (2.17)Here and in what follows, repeated lower indices are contracted using δ ij , and ∂ ≡ ∂ i ∂ i . ∂ − is a formal notation andshould be understood in fourier space. In multi-field model, we can decompose the perturbation into one instantaneous adiabatic sector and one instantaneousentropy sector. The “adiabatic direction” corresponds to the direction of the “background inflaton velocity” e I ≡ ˙ φ I q P , h JK i ˙ φ J ˙ φ K ≡ ˙ φ I ˙ σ , (2.18)where we define ˙ σ ≡ q P , h JK i ˙ φ J ˙ φ K , which is the generalization of the background inflaton velocity. Actually ˙ σ isessentially a shorthand notation and has nothing to do with any concrete field. Note that ˙ σ is related to the slow-rollparameter ǫ as ˙ σ = 2 H ǫ .We introduce ( N − basis e In , ( n = 2 , · · · , N ) which are orthogonal with e I and also with each other. Theorthogonal condition can be defined as P , h IJ i e Im e Jn ≡ δ mn . (2.19)Thus the scalar-field perturbation Q I can be decomposed into instantaneous adiabatic/entropy basis: Q I ≡ e Im Q m , m = 1 , · · · N . (2.20)– 4 –p to now our discussion is rather general, without further restriction on the structure of P ( X IJ , φ I ) . In this work,we consider a general class of two-field models, with the following Lagrangian of the scalar fields : P ( X IJ , φ I ) = P ( X, Y, φ I ) , (2.21)with X ≡ X II = G IJ X IJ and Y ≡ X IJ X JI . This form of Lagrangian not only is the most general Lagrangian fortwo-field models and thus deserves detailed investigations, but also can make our discussions on the non-Gaussianities intwo-field models in a more general background.After performing the decomposition into instantaneous adiabatic/entropy modes, at the leading-order, the second-order action for the perturbations takes the form S (main) = Z dtd x a (cid:18) K mn ˙ Q m ˙ Q n − a δ mn ∂ i Q m ∂ i Q n (cid:19) , (2.22)with K mn ≡ δ mn + (cid:16) P , h MN i ˙ φ M ˙ φ N (cid:17) P , h IK ih JL i e I e Kn e J e Lm , = δ mn + (cid:18) c a − (cid:19) δ m δ n + (cid:18) c e − (cid:19) ( δ mn − δ m δ n ) , (2.23)where we introduce c a ≡ P ,X + 2 XP ,Y P ,X + 2 X ( P ,XX + 4 XP ,XY + 3 P ,Y + 4 X P ,Y Y ) ,c e ≡ P ,X P ,X + 2 XP ,Y , (2.24)which are the propagation speeds of adiabatic and entropy perturbations respectively. It is useful to note that K mn isdiagonal, K = 1 /c a , K = 1 /c e and K = K = 0 , as a consequence of the adiabatic/entropy decomposition. c a = c e is a generic feature in multi-field models; this can be seen explicitly from the definitions in (2.24), the speed ofsound for the adiabatic mode and the entropy mode(s) have different dependence on the P -derivatives .At this point, it is convenient to introduce two parameters: ξ ≡ X ( P ,XX + 2 P ,XY ) P ,X + 2 XP ,Y ,λ ≡ X P ,XX + 23 X P ,XXX + 2 (cid:18) Y P Y + 6 Y P ,Y Y + 83 Y P ,Y Y Y (cid:19) + 4 (cid:0) X Y P ,XXY + 2
XY P ,XY + 2 XY P ,XY Y (cid:1) , where all quantities are background values, and we have used Y = X . As we will see later, although the X , Y -dependences of P ( X, Y, φ I ) in general can be complicated, the non-linear structures of P affect the trispectra through theabove specific combinations of derivatives of P .After introducing new variables whose kinetic terms are canonically normalized ˜ Q σ ≡ ac a Q σ , ˜ Q s ≡ ac e Q s , (2.25)and changing into comoving time defined by dt = adη , the quadratic action takes the form S = Z dηd x h ˜ Q ′ σ + (cid:0) H + H ′ (cid:1) ˜ Q σ − c a ( ∂ ˜ Q σ ) + ˜ Q ′ s + (cid:0) H + H ′ (cid:1) ˜ Q s − c e ( ∂ ˜ Q s ) i . (2.26) This form of Lagrangian is motivated from that, for multi-field k -inflation models [55, 41], the Lagrangian is simply P ( X, φ I ) . In [43] a specialform of the Lagrangian ˜ P ( ˜ Y , φ I ) with ˜ Y ≡ X + b ( φ I )2 (cid:0) X − X IJ X IJ (cid:1) was chosen in the investigation of bispectrua in two-field models, whichis motivated by the multi-field DBI action. In this work, we use the more general form of the Lagrangian (2.21). In (2.22) we neglect the mass-square terms as M mn Q m Q n and the friction terms such as ∼ ˙ Q m Q n . In general these terms may becomeimportant, especially they may cause non-vanishing cross-correlations between adiabatic mode and entropy mode around horizon-crossing. See [63] fordetailed investigation of these cross-correlations for the same model in this paper, and [66, 67] for recent studies on multi-field perturbations. We use c a and c e rather than c σ and c s in order to avoid possible confusion, since in the literatures c s has special meaning, i.e. the speed of soundof perturbation in single-field models. This fact was first point out apparently in [59, 60] in the investigation of brane inflation models. See also [43, 37, 61, 62, 57, 41, 63] for extensiveinvestigations on general multi-field models with different c a and c e . – 5 –he action (2.22) or (2.26) describes a free theory. Performing a canonical quantization, we write ˜ Q σ ( k , η ) ≡ a k ˜ u k ( η ) + a †− k ˜ u ∗ k ( η ) , ˜ Q s ( k , η ) ≡ a k ˜ v k ( η ) + a †− k ˜ v ∗ k ( η ) , (2.27)where ˜ u k ( η ) and ˜ v k ( η ) are the mode functions, which satisfy the corresponding classical equations of motion ˜ u ′′ k + (cid:2) c a k − ( H + H ′ ) (cid:3) ˜ u k = 0 , ˜ v ′′ k + (cid:2) c e k − ( H + H ′ ) (cid:3) ˜ v k = 0 . (2.28)Finally, what we are interested in are the tree-level two-point functions for Q σ and Q s , defined as h Q σ ( k , η ) Q σ ( k , η ) i = (2 π ) δ ( k + k ) G k ( η , η ) , h Q s ( k , η ) Q s ( k , η ) i = (2 π ) δ ( k + k ) F k ( η , η ) , (2.29)with G k ( η , η ) ≡ u k ( η ) u ∗ k ( η ) , F k ( η , η ) ≡ v k ( η ) v ∗ k ( η ) , (2.30)where u k ( η ) and v k ( η ) are the mode functions for adiabatic perturbation and entropy perturbation respectively: u k ( η ) = i H √ c a k (1 + ic a kη ) e − ic a kη ,v k ( η ) = i H √ c e k (1 + ic e kη ) e − ic e kη . (2.31)The so-called “power spectra” for adiabatic and entropy perturbations are defined as P σ ( k ) ≡ G k ( η ∗ , η ∗ ) and P s ( k ) ≡ F k ( η ∗ , η ∗ ) , where η ∗ can be chosen as the time when the modes cross the sound-horizon, i.e. at c a k ≡ aH for adiabatic mode and c e k ≡ aH for entropy mode(s) . In the so-called comoving gauge, the perturbation Q σ is di-rectly related to the three-dimensional curvature of constant time space-like slices. This gives the gauge-invariant quantityreferred to as the “comoving curvature perturbation”: R ≡ H ˙ σ Q σ . (2.32)The entropy perturbation Q s is automatically gauge-invariant by construction. It is also convenient to introduce a renor-malized “isocurvature perturbation” defined as S ≡ H ˙ σ Q s . (2.33)In the cosmological context, it is also convenient to define the dimensionless power spectra for comoving curvatureperturbation and isocurvature perturbation respectively: P R∗ = H ˙ σ P σ ∗ ≡ H ˙ σ k π P σ ∗ ( k ) = 12 ǫc a (cid:18) H π (cid:19) , P S∗ = H ˙ σ P s ∗ ≡ H ˙ σ k π P s ∗ ( k ) = 12 ǫc e (cid:18) H π (cid:19) . (2.34)In the above results, all quantities are evaluated around the sound-horizon crossing.
3. Non-linear perturbations
In this section, We calculate the tri-spectrum which comes from correlating (or exchanging) scalar modes. The fulltrispectrum which includes both contacting and correlating scalar contributions is also discussed in this section. In general multi-field models, adiabatic/entropic modess with the same comoving wavenumber k exit their sound-horizons at different time, dueto their different speeds of sound, c a = c e . This fact will bring new interesting phenomenology in multi-field models. As was shown in [63], thecross-correlations between adiabatic/entropic modes would be enhanced by a small c e /c a ratio. – 6 – .1 Trispectra from Correlating Scalar Mode The third-order action for the model (2.1) has been derived in [43]: S (main) = Z dtd x a (cid:18)
12 Ξ mnl ˙ Q m ˙ Q n ˙ Q l − a Υ mnl ˙ Q m ∂ i Q n ∂ i Q l (cid:19) , (3.1)with Ξ mnl ≡ q P , h MN i ˙ φ M ˙ φ N (cid:20) P , h IK ih JL i e I e Km e Jn e Lℓ + 13 (cid:16) P , h MN i ˙ φ M ˙ φ N (cid:17) P , h IK ih JL ih P Q i e I e Km e J e Ln e P e Ql (cid:21) , Υ mnl ≡ q P , h MN i ˙ φ M ˙ φ N P , h IK ih JL i e I e Km e Jn e Ll . (3.2)In this article, we still work on the double-field model. It is a straightforward task to generalize our calculation to a moregeneral multi field model.Direct algebra gives the cubic-order interaction Hamiltonian: H I ( τ ) = Z dτ d x h − a σ Q ′ σ + a σ Q ′ σ ∂ i Q σ ∂ i Q σ − a c Q ′ σ Q ′ s + a s Q ′ σ ∂ i Q s ∂ i Q s + a c Q ′ s ∂ i Q σ ∂ i Q s i , (3.3)where the subscript “I” denotes the interactional picture and the five effective couplings Ξ σ etc. are given in Appendix A.The trispectrum is the four-point correlation function of perturbations. According to the in-in formalism [108], Thetrispectrum which comes from scalar exchanging can be formulated as h Q i ⊃ − ℜ "Z η −∞ + dη ′ Z η ′ −∞ + dη ′′ h I | Q I H I ( η ′ ) H I ( η ′′ ) | I i + Z η −∞ − dη ′ Z η −∞ + dη ′′ h I | H I ( η ′ ) Q I H I ( η ′′ ) | I i . (3.4)The calculation is straightforward but rather tedious. Here we simply collect the final results. The leading contributionfrom exchanging an intermediate scalar mode to the purely adiabatic four-point function (cid:10) Q σ (cid:11) is given by (see AppendixB for details) h Q σ ( τ, k ) Q σ ( τ, k ) Q σ ( τ, k ) Q σ ( τ, k ) i SE =(2 π ) δ ( X i k i ) n
92 Ξ σ c a I a ( c a k , c a k , c a k , c a k , c a k )+ 2Υ σ c a (cid:20) I (1) b ( c a k , c a k , c a k , c a k , c a k ) + I (2) b ( c a k , c a k , c a k , c a k , c a k ) + I (3) b ( c a k , c a k , c a k , c a k , c a k ) (cid:21) − σ Υ σ c a h I (1) c ( c a k , c a k , c a k , c a k , c a k ) + 2 I (2) c ( c a k , c a k , c a k , c a k , c a k ) i + 23 perms o , (3.5)where “23 perms” denotes the other 23 permutations among four external momenta k , · · · , k . In (3.5), the integrals I a ,– 7 – (1) b etc are defined in Appendix B. The mixed adiabatic/entropy four-point function (cid:10) Q σ Q s (cid:11) is given by h Q σ ( τ, k ) Q σ ( τ, k ) Q s ( τ, k ) Q s ( τ, k ) i SE + 5 perms =(2 π ) δ ( X i k i ) n σ Ξ c c a c e I a ( c a k , c a k , c e k , c e k , c a k ) + 2Ξ c c a c e I a ( c a k , c e k , c a k , c e k , c e k )+ Υ σ Υ s c a c e h I (1) b ( c a k , c a k , c e k , c e k , c a k ) + 2 I (2) b ( c e k , c e k , c a k , c a k , c a k ) i + Υ σ Υ c c a c e h I (2) b ( c a k , c a k , c e k , c e k , c a k ) + 2 I (3) b ( c a k , c a k , c e k , c e k , c a k ) i + 2Υ s c a c e I (3) b ( c a k , c e k , c a k , c e k , c e k )+ 2Υ s Υ c c a c e h I (2) b ( c a k , c e k , c a k , c e k , c e k ) + I (3) b ( c a k , c e k , c e k , c a k , c e k ) i + 12 Υ c c e c a h I (1) b ( c a k , c e k , c a k , c e k , c e k ) + 2 I (2) b ( c a k , c e k , c e k , c a k , c e k ) + I (3) b ( c e k , c a k , c e k , c a k , c e k ) i − σ Υ s c a c e I (1) c ( c a k , c a k , c e k , c e k , c a k ) − σ Υ c c a c e I (2) c ( c a k , c a k , c e k , c e k , c a k ) − Ξ c Υ σ c e c a I (1) c ( c e k , c e k , c a k , c a k , c a k ) − c Υ σ c e c a I (2) c ( c e k , c e k , c a k , c a k , c a k ) − c Υ s c e c a I (2) c ( c a k , c e k , c a k , c e k , c e k ) − c Υ c c a c e h I (1) c ( c a k , c e k , c a k , c e k , c e k ) + I (2) c ( c a k , c e k , c e k , c a k , c e k ) i + 23 perms o , (3.6)where in the first line “5 perms” denotes the other 5 possibilities of choosing two momenta for Q σ and two momenta for Q s out of the four external momenta. Note that in the permutations, the speeds of sound c a and c e are always associatedwith the given extra momenta. The purely entropic four-point function (cid:10) Q s (cid:11) is h Q s ( τ, k ) Q s ( τ, k ) Q s ( τ, k ) Q s ( τ, k ) i SE =(2 π ) δ ( X i k i ) n
12 Ξ c c e c a I a ( c e k , c e k , c e k , c e k , c a k ) + 12 Υ s c e c a I (1) b ( c e k , c e k , c e k , c e k , c a k )+ Υ s Υ c c e c a I (2) b ( c e k , c e k , c e k , c e k , c a k ) + 12 Υ c c e I (3) b ( c e k , c e k , c e k , c e k , c a k ) − Ξ c Υ s c e c a I (1) c ( c e k , c e k , c e k , c e k , c a k ) − Ξ c Υ c c e c a I (2) c ( c e k , c e k , c e k , c e k , c a k ) + 23 perms o . (3.7) The scalar field perturbations Q σ and Q s themselves are not directly observable. What we are eventually interested in isthe curvature perturbation R . As has been investigated in [37, 38, 43], the comoving curvature perturbation R is relatedto the adiabatic and entropy perturbations of the scalar fields by R ≈ R ∗ + T RS S ∗ = (cid:18) H ˙ σ (cid:19) ∗ Q σ ∗ + T RS (cid:18) H ˙ σ (cid:19) ∗ Q s ∗ ≡ N σ Q σ ∗ + N s Q s ∗ . (3.8)Here T RS is the so-called transfer function from entropy perturbation to adiabatic perturbation . Note that N s ≡ T RS N σ is in general time-dependent. Thus contributions to the four-point correlation function for R are given by hR ( k ) R ( k ) R ( k ) R ( k ) i = N σ (cid:2) h Q σ ( k ) Q σ ( k ) Q σ ( k ) Q σ ( k ) i SE + h Q σ ( k ) Q σ ( k ) Q σ ( k ) Q σ ( k ) i C (cid:3) + N σ N s (cid:2) h Q σ ( k ) Q σ ( k ) Q s ( k ) Q s ( k ) i SE + h Q σ ( k ) Q σ ( k ) Q s ( k ) Q s ( k ) i C + (cid:3) + N s (cid:2) h Q s ( k ) Q s ( k ) Q s ( k ) Q s ( k ) i SE + h Q s ( k ) Q s ( k ) Q s ( k ) Q s ( k ) i C (cid:3) , (3.9)where the subscript SE denote the four-point functions which come from exchanging intermediate scalar modes, and thesubscript C denotes the four-point functions which come from contact diagrams. The four-point functions of exchanging As was pointed in [116, 117], as long as the fields roll slowly, these additional contributions after horizon-crossing are heavily suppressed. – 8 –calar modes for Q σ and Q s are given by (3.5), (3.6) and (3.7). The four-point function of the contact diagram is given inRef.[104]. In deriving (3.9), we have used the assumption that there is no cross-correlation between adiabatic and entropymodes, i.e. h Q σ Q s i ∗ ≡ , around horizon-crossing.It is convenient to define a so-called trispectrum hR ( k ) R ( k ) R ( k ) R ( k ) i ≡ (2 π ) δ ( X i =1 k i ) T ( k , k , k , k )= (2 π ) δ ( X i =1 k i )( T c ( k , k , k , k ) + T s ( k , k , k , k )) (3.10)where T c is given by eq.(5.29) of Ref.[104]. We can derive T s from eqs.(3.5)(3.6)(3.7).To investigate the size of Non-Gaussianity roughly, we choose regular tetrahedron limit, k = k = k = k = k = k , and take the approximation c a = c e ≡ c s ≪ . We define a real number t NL from the trispectrum to characterize itssize, T ( k , k , k , k ) | rth ≡ P R t NL . (3.11)We have t NL = t cNL + t sNl , (3.12)where t cNL comes from the contact diagram [104], t cNL = (cid:0) T RS (cid:1) − ( t + t + t ) , (3.13)where t = − c s (cid:0) c s λ − H ǫ (3 λ + 10Π) (cid:1) H ǫ − T RS c s (cid:0) H ǫ − c s λ (cid:1) c s H ǫ + T RS c s ,t = 13 (cid:0) − H ǫ + 3 c s λ (cid:1) c s H ǫ + T RS − H ǫ + 3 c s λ )128 c s H ǫ + T RS c s ,t = 5158192 c s + T RS c s . (3.14) t sNL comes from scalar exchange diagram, t sNL = t + t + t , (3.15)where t ≃ c s (cid:18) λH ǫ (cid:19) + (cid:18) . c − s + 0 . λH ǫ (cid:19) T RS + 0 . c − s T RS ,t ≃ . c − s + (cid:0) .
53 + 12 . ξ + 5 . ξ (cid:1) c − s T RS + (cid:0) .
43 + 1 . ξ + 1 . ξ (cid:1) c − s T RS ,t ≃ − . λH ǫ + (cid:18) . c − s + 14 . c − s ξ + 2 . λH ǫ + 2 . ξ λH ǫ (cid:19) T RS + (0 .
37 + 0 . ξ ) c − s T RS , (3.16)Here the contributions t , t , t come from diagram I a , I b and I c respectively (see Appendix B for details). Comparing t , t , t with t , t , t , we can see that the scalar exchange diagram makes a nontrivial contribution to the trispectrum.As for the contact contributions, the contributions to the trispectrum from exchanging scalar modes can be enhanced bysmall sound speed(s), large T RS , large ξ , and large λH ǫ .
4. Conclusion
In this note, we made a complementary calculation of the contributions to the trispectrum of primordial curvature pertur-bations from exchanging intermediate scalar modes in the context of generalized multi-field inflation, which completesthe calculation of our previous investigation [104]. We choose regular tetrahedron limit to estimate the size of non-Gaussianity. The calculation presented in this work, together with [104], can be employed as the starting point for furtheranalysis of the trispectrum of generalized multi-field inflatioyn models, such as the shapes, squeezed limit [109, 110, 111]and estimators [112, 113, 114, 115, 118] etc. We would like to come back to these issues in the near future.– 9 – cknowledgments
We would like to thank Miao Li, Yi Wang, Shinji Mukohyama for useful discussion, and thank Thorsten Battefeld forthe careful reading of the manuscript and suggestions on improvement. XG is deeply grateful to Prof. Miao Li for hisconsistent encouragement and support. This work was partly supported by the NSFC grant No.10535060/A050207, aNSFC group grant No.10821504 and Ministry of Science and Technology 973 program under grant No.2007CB815401,and in part by Perimeter Institute for Theoretical Physics. CL is supported by Chinese Scholarship Council. CL wouldlike to thank the hospitality of Perimeter Institute, where the paper was finalized when CL visited PI.
A. Coefficients in the interactional Hamiltonian
The variously introduced coefficients in (3.3) are given by Ξ σ = 4 λ ˙ σ , Ξ c = H √ ǫ √ XP ,X (cid:18) c a − (cid:19) , Υ σ = 1 H √ ǫ (cid:18) c a − (cid:19) , Υ s = ˙ σξXP ,X Υ c = √ H √ ǫ (cid:18) c e − (cid:19) . (A.1) B. Basic Integrals
The full expressions for the four-point functions are rather complicated. In this work, at the leading-order, all contributionsto the four-point functions can be grouped into six basic integrals, which we denote as I a , I (1) b , I (2) b , I (3) b , I (1) c and I (1) c ,and their “conjugate” which we define as below (see Fig. B). k k k k k τ τ I a I (1) b I (2) b I (3) b I (1) c I (2) c Figure 1:
Diagrammatic representation of the six basic integrals: I a , I (1) b , I (2) b , I (3) b , I (1) c and I (1) c . Allthe momenta configurations and τ , τ are the same as in I a . A red dot denotes the temporal derivative, ablue dot denote the spatial derivative or momentum in Fourier space, where a blue line between two dotsrepresents the dot product. From Fig.B, it is straightforward to read the expressions for these integrals, we find I a ( k , k , k , k , k ) ≡ − H ℜ (cid:20)Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ , τ ) (cid:21) + 14 H Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ , τ ) ∂ G k ( τ , τ ) ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ , τ ) , (B.1)– 10 –here and in what follows ∂ , ≡ ddτ , , ∂ ≡ d dτ dτ and in this appendix we denote G k ( τ , τ ) = u k ( η ) u ∗ k ( η ) with u k ( η ) = i H √ k (1 + ikη ) e − ikη . I (1) b ( k , k , k , k , k ) ≡ − H ( k · k ) ( k · k ) ℜ (cid:20)Z τ −∞ dτ Z τ −∞ dτ τ τ G k ( τ, τ ) G k ( τ, τ ) G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) (cid:21) + 14 H ( k · k ) ( k · k ) Z τ −∞ dτ Z τ −∞ dτ τ τ G k ( τ , τ ) G k ( τ , τ ) G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) , (B.2) I (2) b ( k , k , k , k , k ) ≡ − H ( k · k ) ( k · k ) ℜ (cid:20)Z τ −∞ dτ Z τ −∞ dτ τ τ G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) (cid:21) + 14 H ( k · k ) ( k · k ) Z τ −∞ dτ Z τ −∞ dτ τ τ G k ( τ , τ ) G k ( τ , τ ) ∂ G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) , (B.3) I (3) b ( k , k , k , k , k ) ≡ H ( k · k ) ( k · k ) ℜ (cid:20)Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ, τ ) G k ( τ, τ ) G k ( τ , τ ) (cid:21) − H ( k · k ) ( k · k ) Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ , τ ) G k ( τ , τ ) ∂ G k ( τ, τ ) G k ( τ, τ ) G k ( τ , τ ) , (B.4)and I (1) c ( k , k , k , k , k ) ≡ H ( k · k ) ℜ Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) − H ( k · k ) Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ , τ ) ∂ G k ( τ , τ ) G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) , (B.5) I (2) c ( k , k , k , k , k ) ≡ H ( k · k ) ℜ Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) − H ( k · k ) Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ , τ ) ∂ G k ( τ , τ ) ∂ G k ( τ, τ ) G k ( τ, τ ) ∂ G k ( τ , τ ) . (B.6)It is useful to introduce the “conjugate” contributions, defined as follows. Up to the second-order in perturbationtheory, there are two interaction vertices and thus two temporal integrals with respect to τ and τ respectively. Wecall two contributions (diagrams) are conjugate to each other with exchanging τ ↔ τ while keeping all the momentarelations. Having known the expression for a diagram, it is easy to write down the integral expression for its conjugate,e.g. ˜ I a ( k , k , k , k , k ) ≡ − H ℜ (cid:20)Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ , τ ) (cid:21) + 14 H (cid:20)Z τ −∞ dτ Z τ −∞ dτ τ τ ∂ G k ( τ , τ ) ∂ G k ( τ , τ ) ∂ G k ( τ, τ ) ∂ G k ( τ, τ ) ∂ G k ( τ , τ ) (cid:21) ∗ , (B.7)where ∗ denotes complex conjugate. It is analogous for the other conjugate integrals, which we do not write here forsimplicity. Moreover, we introduce the combination of a contribution and its conjugate, e.g. I a ( k , k , k , k , k ) ≡ [ I a + ˜ I a ] ( k , k , k , k , k ) . (B.8)Before we evaluate the integrals, it is useful to make it clear about the smallest set of integrals we need. There are twocases. For left-right asymmetric diagrams, e.g. I (2) b (or ˜ I (2) b ), we always encounter the combination I (2) b ≡ I (2) b + ˜ I (2) b – 11 –ather than ˜ I (2) b itself. While for the left-right symmetric diagrams, e.g. I a , ˜ I a is simply exchanging simultaneously k ↔ k , k ↔ k . Thus, after the 6 permutations (which specify two momenta associated with τ and other twomomenta associated with τ ) among the four extra momenta k , · · · , k , the final contribution to the correlation functionfrom I a is equal to I a / . Thus, what we really need is the following six basic integrals: I a , I (1) b , I (2) b , I (3) b , I (1) c and I (2) c .Now we collect the final results for these integrals, in the limit of τ → . We find I a ( k , k , k , k , k ) ≡ H k K Y i =1 k i ! " A ( K , K , k ) + K ( K + k ) ( K + k ) , (B.9)with A ( s , s , r ) ≡ s + ( s + 3 r ) (4 s + K ) + 6 r ( s + r ) + ( s ↔ s ) , (B.10)where here and in what follows we denote K ij ≡ k i + k j and K ≡ k + k + k + k . I (1) b ( k , k , k , k , k ) = ( k · k ) ( k · k ) H k K Y i =1 k i ! × (cid:2) Γ ( K , K ; k k , k k ; J ) + (12 ↔
34) + K F ( K , k , k k ) F ( K , k , k k ) (cid:3) . (B.11)with J ij ≡ K ij − k , and Γ( s , s ; q , q , t ) ≡ K − t ) (cid:8) K + K ( − t + s + 2 s )+ K [ t ( t − s ) + 3 ( − t + s ) s + 2 q + 6 q ]+ K (cid:2) t ( s + s ) + 8 s q + 12 s q − t (5 s s + 4 q + 6 q ) (cid:3) + K (cid:2) t ( ts s + ( t − s ) q ) + 2 (cid:0) t − ts + 20 q (cid:1) q (cid:3) +6 Kt [ ts q + ( ts − q ) q ] + 24 t q q (cid:9) , (B.12)and F ( s, t, q ) ≡ s + 2 q + 3 st + t ( s + t ) . (B.13) I (2) b ( k , k , k , k , k ) ≡ ( k · k ) ( k · k ) (cid:18) k k (cid:19) H k K Y i =1 k i ! × (cid:2) Γ ( K , K ; k k , k k ; J ) + Γ (cid:0) ¯ K , K ; − k k , k k ; J (cid:1) + K F ( K , k , k k ) F ( K , k , k k ) (cid:3) , (B.14)where ¯ K ij ≡ k i − k j . I (3) b ( k , k , k , k , k ) ≡ ( − k · k ) ( k · k ) k k k H K Y i =1 k i ! × (cid:2) Γ (cid:0) ¯ K , K ; − k k , k k ; J (cid:1) + Γ (cid:0) ¯ K , K ; − k k , k k ; J (cid:1) + K F ( K , k , k k ) F ( K , k , k k ) (cid:3) . (B.15)And I (1) c ( k , k , k , k , k ) ≡ ( k · k ) H K k k k k k " C ( K , k k , J ) + ¯ C ( K , k k , J ) + K F ( K , k , k k )( K + k ) , (B.16)– 12 – (2) c ( k , k , k , k , k ) ≡ ( k · k ) H K k k k k k C ( K , k k , J ) + ¯ C (cid:0) ¯ K , − k k , J (cid:1) + K F ( K , k , k k ) (cid:16) K + ˜ k (cid:17) , (B.17)with C ( s, q, t ) ≡ K ( K − t ) [ − t ( K + 3 s ) + K ( K + 4 s )] + 2 (cid:0) K − Kt + 6 t (cid:1) q ( K − t ) , ¯ C ( s, q, t ) ≡ K (cid:0) K ( K + 2 s ) + t ( K + 3 s ) − Kt (3 K + 8 s ) (cid:1) + 2 (cid:0) K − Kt + 6 t (cid:1) q ( K − t ) . (B.18) References [1] A. Guth, Phys. Rev. D 23 (1981) 347.[2] A. D. Miller et al. , Astrophys. J. , L1 (1999) [arXiv:astro-ph/9906421].[3] P. de Bernardis et al. [Boomerang Collaboration], Nature , 955 (2000) [arXiv:astro-ph/0004404].[4] S. Hanany et al. , Astrophys. J. , L5 (2000) [arXiv:astro-ph/0005123].[5] N. W. Halverson et al. , Astrophys. J. , 38 (2002) [arXiv:astro-ph/0104489].[6] B. S. Mason et al. , Astrophys. J. , 540 (2003) [arXiv:astro-ph/0205384].[7] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. , 330 (2009) [arXiv:0803.0547 [astro-ph]].[8] E. Komatsu et al. , arXiv:1001.4538 [astro-ph.CO].[9] D. Larson et al. , arXiv:1001.4635 [astro-ph.CO].[10] V. Mukhanov, and G. Chibisov, JETP 33, 549 (1981).[11] A. H. Guth, and S.-Y. Pi, Phys. Rev. Lett. 49, 1110 (1982).[12] S. W. Hawking, Phys. Lett. B115, 295 (1982).[13] A. A. Starobinsky, Phys. Lett. B117, 175 (1982).[14] J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D28, 679 (1983).[15] E. Komatsu et al., arXiv:0902.4759 [astro-ph.CO].[16] N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Phys. Rept. , 103 (2004) [arXiv:astro-ph/0406398].[17] J. M. Maldacena, JHEP , 013 (2003) [arXiv:astro-ph/0210603].[18] D. Seery and J. E. Lidsey, JCAP , 003 (2005) [arXiv:astro-ph/0503692].[19] X. Chen, M. x. Huang, S. Kachru and G. Shiu, JCAP , 002 (2007) [arXiv:hep-th/0605045].[20] X. Chen, M. x. Huang and G. Shiu, Phys. Rev. D , 121301 (2006) [arXiv:hep-th/0610235].[21] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, JHEP , 014 (2008) [arXiv:0709.0293 [hep-th]].[22] X. Chen, R. Easther and E. A. Lim, JCAP , 023 (2007) [arXiv:astro-ph/0611645].[23] X. Chen, R. Easther and E. A. Lim, JCAP , 010 (2008) [arXiv:0801.3295 [astro-ph]].[24] D. Seery, J. E. Lidsey and M. S. Sloth, JCAP , 027 (2007) [arXiv:astro-ph/0610210].[25] D. Seery and J. E. Lidsey, JCAP , 008 (2007) [arXiv:astro-ph/0611034].[26] C. T. Byrnes, M. Sasaki and D. Wands, Phys. Rev. D , 123519 (2006) [arXiv:astro-ph/0611075].[27] F. Arroja and K. Koyama, Phys. Rev. D , 083517 (2008) [arXiv:0802.1167 [hep-th]].[28] D. Seery, M. S. Sloth and F. Vernizzi, JCAP , 018 (2009) [arXiv:0811.3934 [astro-ph]].[29] X. Chen, B. Hu, M. x. Huang, G. Shiu and Y. Wang, JCAP , 008 (2009) [arXiv:0905.3494 [astro-ph.CO]].[30] F. Arroja, S. Mizuno, K. Koyama and T. Tanaka, Phys. Rev. D , 043527 (2009) [arXiv:0905.3641 [hep-th]].[31] C. Armendariz-Picon, T. Damour and V. F. Mukhanov, Phys. Lett. B , 209 (1999) [arXiv:hep-th/9904075].[32] J. Garriga and V. F. Mukhanov, Phys. Lett. B , 219 (1999) [arXiv:hep-th/9904176]. – 13 –
33] N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, JCAP (2010) 008 [arXiv:1004.0893 [astro-ph.CO]].[34] N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, arXiv:1006.5411 [astro-ph.CO].[35] N. Bartolo, S. Matarrese and A. Riotto, Phys. Rev. D , 103505 (2002) [arXiv:hep-ph/0112261].[36] D. Seery and J. E. Lidsey, JCAP , 011 (2005) [arXiv:astro-ph/0506056].[37] D. Langlois, S. Renaux-Petel, D. A. Steer and T. Tanaka, Phys. Rev. D , 063523 (2008) [arXiv:0806.0336 [hep-th]].[38] D. Langlois, S. Renaux-Petel, D. A. Steer and T. Tanaka, Phys. Rev. Lett. , 061301 (2008) [arXiv:0804.3139 [hep-th]].[39] D. Langlois, S. Renaux-Petel and D. A. Steer, JCAP , 021 (2009) [arXiv:0902.2941 [hep-th]].[40] S. Renaux-Petel, JCAP (2009) 012 [arXiv:0907.2476 [hep-th]].[41] X. Gao, JCAP , 029 (2008) [arXiv:0804.1055 [astro-ph]].[42] X. Gao and B. Hu, JCAP (2009) 012 [arXiv:0903.1920 [astro-ph.CO]].[43] F. Arroja, S. Mizuno and K. Koyama, JCAP , 015 (2008) [arXiv:0806.0619 [astro-ph]].[44] E. Kawakami, M. Kawasaki, K. Nakayama and F. Takahashi, JCAP , 002 (2009) [arXiv:0905.1552 [astro-ph.CO]].[45] S. Mizuno, F. Arroja, K. Koyama and T. Tanaka, Phys. Rev. D , 023530 (2009) [arXiv:0905.4557 [hep-th]].[46] S. Mizuno, F. Arroja and K. Koyama, Phys. Rev. D (2009) 083517 [arXiv:0907.2439 [hep-th]].[47] J. L. Lehners and S. Renaux-Petel, Phys. Rev. D , 063503 (2009) [arXiv:0906.0530 [hep-th]].[48] F. Bernardeau and J. P. Uzan, Phys. Rev. D , 103506 (2002) [arXiv:hep-ph/0207295].[49] D. Wands, N. Bartolo, S. Matarrese and A. Riotto, Phys. Rev. D , 043520 (2002) [arXiv:astro-ph/0205253].[50] C. T. Byrnes, K. Y. Choi and L. M. H. Hall, JCAP , 008 (2008) [arXiv:0807.1101 [astro-ph]].[51] C. T. Byrnes, K. Y. Choi and L. M. H. Hall, JCAP , 017 (2009) [arXiv:0812.0807 [astro-ph]].[52] D. Battefeld and T. Battefeld, JCAP (2009) 010 [arXiv:0908.4269 [hep-th]].[53] D. Langlois, F. Vernizzi and D. Wands, JCAP , 004 (2008) [arXiv:0809.4646 [astro-ph]].[54] M. Kawasaki, K. Nakayama, T. Sekiguchi, T. Suyama and F. Takahashi, JCAP , 019 (2008) [arXiv:0808.0009 [astro-ph]].[55] D. Langlois and S. Renaux-Petel, JCAP , 017 (2008) [arXiv:0801.1085 [hep-th]].[56] G. I. Rigopoulos, E. P. S. Shellard and B. J. W. van Tent, Phys. Rev. D , 083521 (2006) [arXiv:astro-ph/0504508].[57] X. d. Ji and T. Wang, Phys. Rev. D , 103525 (2009) [arXiv:0903.0379 [hep-th]].[58] S. Pi and T. Wang, Phys. Rev. D , 043503 (2009) [arXiv:0905.3470 [astro-ph.CO]].[59] D. A. Easson, R. Gregory, D. F. Mota, G. Tasinato and I. Zavala, JCAP , 010 (2008) [arXiv:0709.2666 [hep-th]].[60] M. x. Huang, G. Shiu and B. Underwood, Phys. Rev. D , 023511 (2008) [arXiv:0709.3299 [hep-th]].[61] Y. F. Cai and W. Xue, Phys. Lett. B , 395 (2009) [arXiv:0809.4134 [hep-th]].[62] Y. F. Cai and H. Y. Xia, Phys. Lett. B , 226 (2009) [arXiv:0904.0062 [hep-th]].[63] X. Gao, JCAP (2010) 019 [arXiv:0908.4035 [hep-th]].[64] T. Wang, arXiv:1008.3198 [astro-ph.CO].[65] A. S. Sakharov and M. Y. Khlopov, Phys. Atom. Nucl. (1993) 412 [Yad. Fiz. (1993) 220].[66] D. I. Kaiser and A. T. Todhunter, Phys. Rev. D (2010) 124037 [arXiv:1004.3805 [astro-ph.CO]].[67] C. M. Peterson and M. Tegmark, arXiv:1005.4056 [astro-ph.CO].[68] M. Sasaki, J. Valiviita and D. Wands, Phys. Rev. D , 103003 (2006) [arXiv:astro-ph/0607627].[69] K. A. Malik and D. H. Lyth, JCAP , 008 (2006) [arXiv:astro-ph/0604387].[70] Q. G. Huang and Y. Wang, JCAP , 025 (2008) [arXiv:0808.1168 [hep-th]].[71] Q. G. Huang, JCAP , 017 (2008) [arXiv:0807.1567 [hep-th]].[72] Q. G. Huang, Phys. Lett. B , 260 (2008) [arXiv:0801.0467 [hep-th]].[73] S. Li, Y. F. Cai and Y. S. Piao, Phys. Lett. B , 423 (2009) [arXiv:0806.2363 [hep-ph]].[74] K. Ichikawa, T. Suyama, T. Takahashi and M. Yamaguchi, Phys. Rev. D , 023513 (2008) [arXiv:0802.4138 [astro-ph]].[75] T. Kobayashi and S. Mukohyama, JCAP , 032 (2009) [arXiv:0905.2835 [hep-th]]. – 14 –
76] M. Li, C. Lin, T. Wang and Y. Wang, Phys. Rev. D , 063526 (2009) [arXiv:0805.1299 [astro-ph]].[77] L. Alabidi, JCAP , 015 (2006) [arXiv:astro-ph/0604611].[78] M. Sasaki, Prog. Theor. Phys. , 159 (2008) [arXiv:0805.0974 [astro-ph]].[79] A. Naruko and M. Sasaki, Prog. Theor. Phys. , 193 (2009) [arXiv:0807.0180 [astro-ph]].[80] Q. G. Huang, JCAP , 005 (2009) [arXiv:0903.1542 [hep-th]].[81] Q. G. Huang, JCAP , 035 (2009) [arXiv:0904.2649 [hep-th]].[82] J. O. Gong, C. Lin and Y. Wang, JCAP , 004 (2010) [arXiv:0912.2796 [astro-ph.CO]].[83] C. Lin and Y. Wang, JCAP (2010) 011 [arXiv:1004.0461 [astro-ph.CO]].[84] J. Zhang, Y. F. Cai and Y. S. Piao, JCAP (2010) 001 [arXiv:0912.0791 [hep-th]].[85] Y. F. Cai and Y. Wang, arXiv:1005.0127 [hep-th].[86] D. M. Regan and E. P. S. Shellard, arXiv:0911.2491 [astro-ph.CO].[87] M. Hindmarsh, C. Ringeval and T. Suyama, Phys. Rev. D (2009) 083501 [arXiv:0908.0432 [astro-ph.CO]].[88] M. Hindmarsh, C. Ringeval and T. Suyama, Phys. Rev. D (2010) 063505 [arXiv:0911.1241 [astro-ph.CO]].[89] J. Kumar, L. Leblond and A. Rajaraman, JCAP (2010) 024 [arXiv:0909.2040 [astro-ph.CO]].[90] H. R. S. Cogollo, Y. Rodriguez and C. A. Valenzuela-Toledo, JCAP (2008) 029 [arXiv:0806.1546 [astro-ph]].[91] Y. Rodriguez and C. A. Valenzuela-Toledo, Phys. Rev. D (2010) 023531 [arXiv:0811.4092 [astro-ph]].[92] P. D. Meerburg, J. P. van der Schaar and P. S. Corasaniti, JCAP (2009) 018 [arXiv:0901.4044 [hep-th]].[93] P. D. Meerburg, J. P. van der Schaar and M. G. Jackson, JCAP (2010) 001 [arXiv:0910.4986 [hep-th]].[94] Q. G. Huang, JCAP (2010) 025 [arXiv:1004.0808 [astro-ph.CO]].[95] K. Izumi and S. Mukohyama, JCAP (2010) 016 [arXiv:1004.1776 [hep-th]].[96] X. Chen and Y. Wang, Phys. Rev. D (2010) 063511 [arXiv:0909.0496 [astro-ph.CO]].[97] X. Chen and Y. Wang, JCAP , 027 (2010) [arXiv:0911.3380 [hep-th]].[98] C. A. Valenzuela-Toledo and Y. Rodriguez, Phys. Lett. B (2010) 120 [arXiv:0910.4208 [astro-ph.CO]].[99] N. Bartolo, E. Dimastrogiovanni, S. Matarrese and A. Riotto, JCAP (2009) 028 [arXiv:0909.5621 [astro-ph.CO]].[100] N. Bartolo, E. Dimastrogiovanni, S. Matarrese and A. Riotto, JCAP (2009) 015 [arXiv:0906.4944 [astro-ph.CO]].[101] C. A. Valenzuela-Toledo, Y. Rodriguez and D. H. Lyth, Phys. Rev. D (2009) 103519 [arXiv:0909.4064 [astro-ph.CO]].[102] E. Dimastrogiovanni, N. Bartolo, S. Matarrese and A. Riotto, arXiv:1001.4049 [astro-ph.CO].[103] M. Karciauskas, K. Dimopoulos and D. H. Lyth, Phys. Rev. D (2009) 023509 [arXiv:0812.0264 [astro-ph]].[104] X. Gao, M. Li and C. Lin, JCAP , 007 (2009) [arXiv:0906.1345 [astro-ph.CO]].[105] X. Chen, B. Hu, M. x. Huang, G. Shiu and Y. Wang, JCAP , 008 (2009) [arXiv:0905.3494 [astro-ph.CO]].[106] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. , 203 (1992).[107] R. H. Brandenberger, Lect. Notes Phys. , 127 (2004) [arXiv:hep-th/0306071].[108] S. Weinberg, Phys. Rev. D , 043514 (2005) [arXiv:hep-th/0506236].[109] P. Creminelli and M. Zaldarriaga, JCAP (2004) 006 [arXiv:astro-ph/0407059].[110] J. Ganc and E. Komatsu, arXiv:1006.5457 [astro-ph.CO].[111] S. Renaux-Petel, arXiv:1008.0260 [astro-ph.CO].[112] J. R. Fergusson, M. Liguori and E. P. S. Shellard, Phys. Rev. D (2010) 023502 [arXiv:0912.5516 [astro-ph.CO]].[113] D. Munshi, A. Heavens, A. Cooray, J. Smidt, P. Coles and P. Serra, arXiv:0910.3693 [astro-ph.CO].[114] S. Mizuno and K. Koyama, arXiv:1007.1462 [hep-th].[115] D. M. Regan and E. P. S. Shellard, Phys. Rev. D (2010) 023520 [arXiv:1004.2915 [astro-ph.CO]].[116] T. Battefeld and R. Easther, JCAP (2007) 020 [arXiv:astro-ph/0610296].[117] D. Battefeld and T. Battefeld, JCAP (2007) 012 [arXiv:hep-th/0703012].[118] J. Smidt, A. Amblard, C. T. Byrnes, A. Cooray, A. Heavens and D. Munshi, Phys. Rev. D (2010) 123007 [arXiv:1004.1409[astro-ph.CO]].(2010) 123007 [arXiv:1004.1409[astro-ph.CO]].