Primordial non-Gaussian features from DBI Galileon inflation

Primordial non-Gaussian features from DBI Galileon inflation Sayantan Choudhury ∗ Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai - 400005, India and Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India Supratik Pal† arXiv:1210.4478v4 [hep-th] 20 May 2015 Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India We have studied primordial non-Gaussian features from a model of potential driven single field DBI Galileon inflation. We have computed the bispectrum from the three point correlation function considering all possible cross correlation between scalar and tensor modes from the proposed setup. Further, we have computed the trispectrum from four point correlation function considering the contribution from contact interaction, scalar and graviton exchange diagrams in the in-in picture. Finally we have obtained the non-Gaussian consistency conditions from the four point correlator, which results in partial violation of the Suyama-Yamaguchi four-point consistency relation. This further leads to the conclusion that sufficient primordial non-Gaussianities can be obtained from DBI Galileon inflation. Contents I. Introduction II. The Background Model 3 3 III. Tree Level Bispectrum Analysis A. Three scalar correlation B. One scalar two tensor correlation C. Two scalar one tensor correlation D. Three tensor correlation 5 5 7 9 11 IV. Tree Level Trispectrum Analysis from four scalar correlation A. Contact Interaction B. Scalar Exchange C. Graviton Exchange 12 12 14 16 V. Four point consistency conditions and violation of Suyama-Yamaguchi relation 18 VI. Summary and outlook 20 Acknowledgments 20 Appendix A. Functions appearing in three scalar correlation B. Functions appearing in one scalar two tensor correlation C. Functions appearing in two scalar one tensor correlation D. Functions appearing in three tensor correlation E. Functions appearing in four scalar correlator 1. Contact interaction 2. Scalar exchange 3. Graviton exchange 20 22 23 25 27 28 28 29 30 ∗ † Electronic address: sayantan@theory.tifr.res.in, sayanphysicsisi@gmail.com Electronic address: supratik@isical.ac.in 2 References 30 3 I. INTRODUCTION The physics of the early universe is a very rich area of theoretical physics, for there is a plethora of potential models that solve, at least partially, the well-known problems of the standard cosmological paradigm. Inflationary cosmology is the most successful branch which addressed all of these problems meticulously. This can however be explained by several class of models originated from a proper field theoretic or particle physics framework. But from observational point view a big issue may crop up in model discrimination and also in the removal of the degeneracy of cosmological parameters obtained from Cosmic Microwave Background (CMB) observations [1–3]. In this context the study of primordial non-Gaussian feature acts as a powerful computational tool to discriminate among inflationary models. In the very recent days the analysis of bispectrum and trispectrum derived from the study of primordial features of non-Gaussianity [4–20] from different models of inflation has thus become an intriguing aspect in the context of inflationary model building as well as studies of CMB physics. Galileon based inflationary models [21–23] and DBI inflationary models [24], [25] are both in vogue for quite some time now. Despite its successes, Galileon models generically give rise to unwanted degrees of freedom like ghosts, Laplacian and Tachyonic instabilities. Recently, a natural extension to these class of models has been brought forth by the present authors [26] in which DBI was clubbed together with Galileon. The framework, called DBI Galileon, consists of a D3 brane in the background of N =1,D=4 SUGRA derived from D4 brane in N =2,D=5 bulk SUGRA background. The interesting feature of this treaty is that those unwanted debris can be successfully thrown away keeping all the good features of Galileon intact. In the present paper, our prime objective is to investigate for some more interesting features of this rich structure of DBI Galileon [26], which ultimately results in sufficient nonGaussianity in this framework. Specifically, we explicitly calculate bispectrum and trispectrum from three and four point correlation functions by exploiting third and fourth order actions. The calculations reveal, along with the feature of large non-Gaussianity, some other interesting results like partial violation of the Suyama-Yamaguchi fourpoint consistency relation. Subsequently, we demonstrate that, in this framework, it is possible to have a parameter space for both non-Gaussianity and tensor-to-scalar ratio (r) consistent with combined constraint obtained from Planck+WMAP9+high-L+BICEP2 data [2, 3]. The plan of the paper is as follows. First we explore primordial non-Gaussian features from the third order action through the non-linear parameter fN L calculated from bispectrum (in equilateral limit configuration) including all possible scalar - tensor type of cross correlations in the different polarizing modes. Hence from the fourth order action we derive the expression for other two non-linear parameters gN L and τN L through trispectrum analysis considering the contribution from contact interaction, scalar and graviton exchange diagrams in the in-in picture. Finally, we explicitly derive the four point consistency relation from scalar and graviton exchange diagrams and also find a partial violation of standard Suyama-Yamaguchi relation [27, 28]. We also attempt to give some possible explanations for this violation. We end up with scanning the parameter space for non-Gaussianity and tensor-to-scalar ratio in the light of Planck+WMAP9+high-L+BICEP2 data. II. THE BACKGROUND MODEL For systematic development of the formalism, let us briefly review from our previous paper [26] how one can construct the effective 4D inflationary potential for DBI Galileon starting from N = 2, D = 5 SUGRA along with Gauss-Bonnet correction in the bulk geometry and D4 brane setup leads to an effective N = 1, D = 4 SUGRA in the D3 brane. Here the total five dimensional model is described by the following action (5) (5) (5) (5) (5) (5) ST otal = SEH + SGB + SDBI + SW Z + SBSUG (2.1) where h i p p   R α (5) (5) (5) 2 d5 x −g (5) R(5) − 2Λ5 , SGB = 2κ(5)2 d5 x −g (5) RABCD(5) RABCD − 4RAB(5) RAB + R(5) , 5 q R
(5) SDBI = − T24 d5 x exp(−Φ) − γ (5) + B (5) + 2πα′ F (5) ,   RP ′ (5) ˆ SW Z = − T24 n=0,2,4 Cn ∧ exp B̂2 + 2πα F2 |4 f orm  h n p ′ R FCD = 21 d5 x −g (5) ǫABCD ∂A ΦI ∂B ΦJ CIJ4TB4KL ∂C ΦK ∂D ΦL + πα CIJ 2  ′
io πα C0 ν4 C0 K L 2 ′2 B F B B ∂ Φ ∂ Φ + + 2π α T C F F − T ν + , + 8T IJ CD IJ KL C D 4 4 0 AB CD 4 0 2 Φ 4 h R 5 p (5) M53 R(5) 1 i 1 m̃ñq̃ i I I m̃ñ µ SBSUG = 2 d x −g(5) e(5) − 2 + 2 Ψ̄im̃ Γ ∇ñ Ψq̃ − SIJ Fm̃ñ F − 2 gαβ (Dm̃ φ )(Dm̃ φν ) + Fermionic + Chern − Simons + Pauli mass] (2.2) (5) SEH = 1 2κ25 R 4 ′ where T(4) is the D4 brane tension, α is the Regge Slope, exp(−Φ) is the closed string dilaton and C0 is the Axion. Here γ (5) , B (5) and F (5) represent the determinant of the 5D induced metric (γAB ) and the gauge fields (BAB , FAB ) respectively. Additionally here ν0 and ν4 represent the constants characterizing the interaction strength between D4¯ brane. In the present context 5-dimensional coordinates X A = (xα , y), where y parameterizes the extra dimension D4 compactified on the closed interval [−πR, +πR]. It is useful to introduce the 5D metric in conformal form
b20   ds24 + R2 β 2 dy 2 , ds24+1 = gAB dX A dX B = (2.3) 4 Λ b (5) 0 R2 exp(βy) + 24R 2 exp(−βy) and ds24 = gαβ dxα dxβ is FLRW counterpart. The parameter β determines the slope of the warp factor and R represents the compactification radius. Applying dimensional reduction technique via S1 /Z2 orbifolding symmetry and using the metric stated in equation(2.3) the total effective model for D3 DBI Galileon in background N =1, D=4 SUGRA is described by the following action [26]: Z h p ˆ S = d4 x −g (4) K̃(φ, X) − G̃(φ, X)✷(4) φ + l̃1 R(4) i (2.4)   (4) (4) 2 + l̃3 , + l̃4 C(1)Rαβγδ(4) Rαβγδ − 4I(2)Rαβ(4) Rαβ + A(6)R(4) where ˆ K̃(φ, X) = − f˜D̃ (φ) q  1 − 2QX f˜ − Q1 − C̃5 G̃(φ, X) − 2X M̃(T, T † ) − V (φ), √ † 2 ) D M̃ (T, T † ) = M(T,T , M (T, T † ) = (T2βR +T † ) , D̃ = 2κ2(4) , 2κ2(4)   g̃(φ)k1 C̃4 G̃(φ, X) = 2(1−2 , g̃(φ) = g̃0 + g̃2 φ2 , f˜(φ) ≃ (f˜ +f˜ φ12 +f˜ φ4 ) f˜(φ)Xk2 )) 0 2  4  h i α(4) α α(4) C(2) 1 l̃1 = 2κ2 1 + R2 β 2 (24I(2) − 24A(9) − 16A(10)) − κ2 R2 β 2 , l̃4 = 2κ(4) 2 , (4) (4) (4)  α (4) l̃3 = 2κ12 R4 β 4 (24C(4) − 144I(4) − 64A(5) + 144A(7) + 64A(8) + 192A(11)) − (4) . (2.5)  3M53 βb60 2 R5 I(1) 2κ2(4) MP L where α(4) , l̃1 , l̃3 , l̃4 are effective 4D couplings and κ(4) be the gravitational coupling strength. Here X represents the 4D kinetic term after dimensional reduction given by X := − 21 gµν ∂ µ φ∂ ν φ. In this context (T, T † ) are the four dimensional background SUGRA moduli fields which are constant after dimensional reduction. The one-loop corrected Coleman Weinberg potential is given by [26]:    2 X φ V (φ) = C2m 1 + D2m ln φ2m , (2.6) M m=−2,m6=−1 where D0 = 0 and the other constants are function of effective the brane tension for D3 brane and constant moduli in 4D. Hence using equation(2.4) the modified Friedman equation in presence of effective 4D Gauss-Bonnet coupling can be expressed as [26]: H4 = Λ(4) + 8πG(4) V (φ) Λ(4) + 8πG(4) ρφ ≈ , g̃1 g̃1 (2.7) where ρφ plays the role of energy density of the inflation in 4D effective theory, g̃1 represents the effective 4D GaussBonnet coupling dependent function on FLRW background which can be expressed in terms of the brane tension of D3 brane and Λ(4) is the 4D effective cosmological constant. It is important to note that in the 4D effective action as stated in Eq (2.4), the contribution of higher curvature effective Gauss-Bonnet like correction term is dominant compared to Ricci scalar. More precisely one can interpret this to be a non-perturbative solution of the effective field theory where the effective coupling parameter l̃4 >> l̃1 . Consequently the effective Friedmann equation in 4D takes a ¯ system. non-trivial form in the high energy regime, where energy density of the inflaton ρφ ≈ V (φ) >> g̃1 of D3 − D3 Here Eq (2.7) also implies that within our prescribed setup the non-perterbative regime of effective field theory cannot able to produce the well known solutions of GR in the low energy limiting situation where ρφ ≈ V (φ) << g̃1 . But in the perturbative regime of the effective theory the situation is completely different compared the non-perturbative case. In the regime where the effective coupling parameter l̃4 << l̃1 , it is possible to get back the known solution of GR. In literature it usually identified to be the low energy regime, where the inflaton energy density, ρφ ≈ V (φ) << g̃1 ¯ system. But in the high energy regime, where ρφ ≈ V (φ) >> g̃1 , it is not possible to realize the essence in D3 − D3 of the higher curvature terms through Fridemann equations, which will finally control the cosmological dynamics in a non-trivial manner. For more details see ref. [26], where the Friedmann equations are derived in detail. 5 III. TREE LEVEL BISPECTRUM ANALYSIS A. Three scalar correlation To calculate the scalar bispectrum for D3 DBI Galileon we consider here the third order action up to total derivatives. Using the uniform field gauge analysis the third order action for three scalar interaction can be written as:    Z C̄5 ∂ 2 ζ(∂ χ̃)2 Sζζζ = dtd3 x a3 C̄1 MP2 L ζ ζ̇ 2 + aC̄2 MP2 L ζ(∂ζ)2 + a3 C̄3 MP L ζ̇ 3 + a3 C̄4 ζ̇(∂i ζ)(∂i χ̃) + a3 2 M PL      C̄7  2 C̄8  2 δL2 aC̄6 ζ̇ 2 ∂ 2 ζ + ∂ ζ(∂ζ)2 − ζ∂i ∂j (∂i ζ)(∂j ζ) + a |1 , ∂ ζ∂i ζ∂i χ̃ − ζ∂i ∂j (∂i ζ)(∂j χ̃) + R a MP L δζ (3.1) where     δL2 d 2 2 3 (3.2) a YS ζ̇ − aYS cs ∂ ζ |1 = −2 δζ dt can be calculated from the second order action [26]   Z   c2 = dtd3 xa3 YS ζ̇ 2 − s2 (∂ζ)2 . S (4) a ζζ (3.3) Here C̄i (i = 1, 2, 3, ......, 8) are dimensionless co-efficients defined as: C̄1 C̄3 C̄4 C̄6 C̄8 " !  #   YS L1 H Y˙S 1 d d L1 = 2 3− 2 3+ , C̄ = 1 + + (aL {Y − t }) , 2 1 S 1 MP L cs HYS dt c2s a dt   YS YS L1 L1 (L1 a1 + a3 ) + a12 + (a9 + L1 a4 ) + 2 , = MP L  t1 cs         d A5 3HA5 MP2 L 3MP2 L MP2 L d A5 YS 1 + 2t1 − , C̄5 = , (1 − HL1 ) − =− 2t1 dt t21 t21 2t21 2 2 dt t21 2 2 2 2 2 2
  d M L M (1 − HL1 ) cs YS L1 MP L + PL − L31 , = L21 2MP2 L − L1 a4 , C̄7 = 1 P L 6 2t1 6 dt   L1 MP2 L c2s YS L1 MP2 L = MP L (HL1 − 1) + t1 t21 and the co-efficient of R= δL2 δζ |1 (3.4) involves spatial and time derivative in equation(3.1) is defined by the following expression: A5 L1  A5  (∂ζ)2 − ∂ −2 ∂i ∂j [(∂i ζ)(∂j ζ)] . (∂k ζ)(∂k χ̃) − ∂ −2 ∂i ∂j [(∂i ζ)(∂j χ̃)] + p1 ζ ζ̇ − 2 t1 2t1 a2 In this context R → 0 as k → 0 at large scale. Additionally ! L1 MP2 L MP2 L −2 L1 = , ζ̇), A = 2Y , A = − , χ̃ = ∂ (Y 3 S 5 S 2 HMP2 L − φ̇X
G̃X
  t1 4t1 t3 + 9t22 3 2Ht2 t21 − t4 t22 − 2t21 ṫ2 2 , YS = , c = , t = l̃ , t = 2H l̃ − 2 φ̇X G̃ 1 1 2 1 X s 2 3t22    t1 (4t1 t3 +  9t2 ) ˆ ˆ + 2X 2 K̃ 2 t3 = −9l̃1 H 2 + 3 X K̃ XX + 18H φ̇ 2X G̃X + X G̃XX , X   4X 3 ˆ ˆ ˆ − 4X 2 K̃ a1 = 3MP2 L H 2 − X K̃ X K̃XXX − 2H φ̇ 10X G̃X + 11X 2 G̃XX + 2X 3 G̃XXX XX − X 3 4X 3 14X 2 G̃φX + G̃φXX , + 2X G̃φ + 3 3 i h  2 a3 = −3a4 = −3 2MP2 L H − 2φ̇ 2X G̃X + X 2 G̃XX , a9 = − a12 = −2MP2 L . 3 (3.5) (3.6) It is important to mention here that for scalar and tensor modes ghosts and Laplacian instabilities can be avoided iff c2s > 0, Ys > 0. Throughout the paper we use the required parameters from [26] to compute the bispectrum and trispectrum. 6 Now following the prescription of in-in formalism in the interacting picture the three point correlation function for quasi-exponential limit, after some trivial algebra, look: hζ(k~1 )ζ(k~2 )ζ(k~3 )i = −i = 8 Z X 0    (j) dη a h0| ζ(k~1 )ζ(k~2 )ζ(k~3 ), Hint (η) j=1 −∞ (2π)3 δ (3) (k~1 + k~2 + k~3 )Bζζζ (k~1 , k~2 , k~3 ), ζζζ  |0i (3.7) where the total Hamiltonian in the interaction picture can be expressed in terms of the third order Lagrangian  R P8 (j) = − d3 x (L3 )ζζζ . Throughout this article we use the Bunch-Davies density as (Hint (η))ζζζ = j=1 Hint (η) ζζζ mode function as um (η, k) =
1 √  3  −ν (−kcm η) 2 m exp i[νm − 12 ] π2 2νm − 2 Γ(νm ) −kηcm (1) √ √ Hνm (−kηcm ) → Γ( 23 ) a 2Ym 2a Ym cm k (3.8) with m = (S[scalar], T [tensor]). Moreover following the momentum dependent ansatz given in [16],[29] the bispectrum Bζζζ (k~1 , k~2 , k~3 ) is defined as: (2π)4 Pζ2 6 Aζζζ (k~1 , k~2 , k~3 ) = fN L;1 Pζ2 Bζζζ (k~1 , k~2 , k~3 ) = Q3 3 5 i=1 ki (3.9) where the symbol ; 1 is used for three scalar correlation. Here Aζζζ (k~1 , k~2 , k~3 ) is the shape function for bispectrum and Pζ2 is used for normalization of E-mode polarization expressed in terms of the new combination of the cyclic permutations of two-point correlation functions given by Pζ2 = Pζ (k1 )Pζ (k2 ) + Pζ (k2 )Pζ (k3 ) + Pζ (k3 )Pζ (k1 ). (3.10) The Power spectra for scalar (Pζ (k)) and tensor modes (PT (k)) at the horizon crossing can be written as: ! !
2 p
2 p 2 2 V (φ) V (φ) 1 − ǫV − sSV 1 − ǫV − sTV 2νT −3 Γ(νT ) 2νs −3 Γ(νs ) √ √ , PT (k) = 2 . Pζ (k) = 2 8π 2 YS c3s g̃1 MP L 2π 2 YT c3T g̃1 MP L Γ( 23 ) Γ( 23 ) (3.11) Here for tensor modes we use (PT (k))ij;kl = |uh (η, k)|2 Nij;kl , PT (k) = (PT (k))ij;ij with the following helicity/spin P †(λ) dependent normalization factor: Nij;kl = λ eλij (~k)ekl (~k). In this context fN L represents the non linear parameter carrying the signature of primordial non-Gaussianities of the curvature perturbation in bispectrum. The explicit form of fN L characterizing the bispectrum can be expressed as: fN L;1 ( " # n −1  2 2 Γ(νs ) 3 k1 k2 k3 ζ 3 − ǫV 1 + YS I1 (ν̃) = C̄1 I1 (nζ − 1) − 2K 3 4 4c2s 1 + ǫV Γ( 23 ) 3 i=1 ki3   E3 3(1 − ǫV − sSV ) C̄4 C̄5 YS (3.12) + F3 I3 (nζ − 1) + 2 I3 (ν̃) + I5 (ν̃) I4 (ν̃) + 2YS  cs  8 4c2s  E6 3(1 − ǫV − sSV )2 C̄7 (1 − ǫV − sSV )2 C̄8 (1 − ǫV − sSV ) F6 I6 (nζ − 1) + 2 I6 (ν̃) + + I7 (ν̃) + I8 (ν̃) . 2 YS cs 2YS cs 8c2s 10 P3  where the functional form of the momentum dependent functions Ii (x)∀i are explicitly mentioned in the Appendix B.1. From the coefficients of Ii (ν̃) with i = 1, 3, 5, 7, 8 it seems that the non-Gaussian parameter fN L;1 is inverse proportional to the sound speed square for the scalar mode. But these co-efficients are not solely characterized by the sound speed for scalar mode since they depend on other factors like (1) effective Gauss-Bonnet coupling (α(4) ) and (2) higher order interaction between graviton and DBI Galileon in presence of quadratic correction of gravity in Einstein-Hilbert action. Additionally in this context counter terms which appears as the coefficients of Ii (nζ − 1) with i = 1, 3, 6 and I4 (ν̃) originated from the effective Gauss-Bonnet coupling (α(4) ) and higher order interaction between graviton (via Gauss-Bonnet correction) and DBI Galileon degrees of freedom in D3 brane in the background of four dimensional N =1 SUGRA multiplet play a very crucial role in this context. In α(4) 6= 0 limit such counter terms and dependence on the interaction between graviton and higher derivative DBI Galileon cannot be negligible in the slow-roll limit. Consequently, depending on the signature and the strength of the effective Gauss-Bonnet coupling 7 three situation arises: (1)the counter terms drives other terms, (2)the counter terms and other terms are tuned in such a way that the system is in equilibrium with respect to the sound speed and (3)the sound speed dominated terms win the war. Here the second situation is not physically interesting and the third situation leads to the trivial feature of DBI Galileon. Only the non-trivial features comes from the first situation in the context of single field DBI Galileon inflation. In equation(3.12) we have defined K = k1 + k2 + k3 , x = (nζ − 1, ν̃) and  S    sV − 2ǫV 2ǫV + sSV + δV ν̃ := , nζ − 1 = (3 − 2νs ) = − 1 − ǫV − sSV  1 − ǫV − sSV     T4 ρ4 φ̇X 2 G̃XX νs YS − ǫV + (1 + YS )ρ3 YS (1 + YS ) (3.13) + 2T3 , := T3 + 2 , = ρ3 + 2 , 1+2 F3 := − 1 + ǫV  1 + ǫV  H cs ΣG cs YS (1 + YS ) 2ρ4 (1 + YS )3 1 + YS 2(1 + YS )3 YS − ǫV E3 := − 2T4 − (1 − 2ρ4 ) , F6 := + ρ3 , E6 := 3 1 + ǫV 1 + ǫV (1 + ǫV ) 1 + YS (1 + ǫV )3 ċs with four new constants ρ3 , ρ4 , T3 , T4 . In the present context sSV = Hc is an extra slow-roll parameter appearing s due to the sound speed, cs 6= 1 as defined in [26]. For the numerical estimation we have further used the equilateral configuration (k1 = k2 = k3 = k and K = 3k) in which the non-linear parameter fN L can be simplified to the following form as: nζ −1   2   1 YS δV Γ(νs ) 1 YS2 2YS sSV 3 1 − − + − 54" c2s c2s c2s # c2s Γ( 23 )  2   E3 equil 3 equil 3 − ǫV 1 + YS 3(1 − ǫV − sSV ) equil equil F I (n − 1) + I × I1 (nζ − 1) − I (ν̃) (ν̃) + 3 3 ζ 1 4 4c2 1 + ǫV 2YS c2s 3     s YS YS 4ǫV − YS (3 − ǫV ) 1 YS I5equil (ν̃) + (3 − YS ) I4equil (ν̃) + 2 − 8 2 2 4c 4(1 + ǫ ) V s   3(1 − ǫV − sSV )2 E6 (1 − ǫV − sSV )2 (1 + YS )2 (YS − ǫV ) equil + F6 I6equil (nζ − 1) + 2 I6equil (ν̃) − I7 (ν̃) YS cs  2YS c2s (1 + ǫV )3 (1 + YS )(YS − ǫV )(1 − ǫV − sSV ) equil I8 (ν̃) . + 4c2s (1 + ǫV )2 (3.14) Now using the tensor-to-scalar ratio at the pivot scale k∗ : equil fN L;1 = 10 9k 3  r= 16.2 2(νT −νs ) Γ(νT ) Γ(νs ) 2  1 − ǫV − sTV 1 − ǫV − sSV 2 cs ǫ s  ! 3 2 1 − O(ǫT ) 2 (3.15) ⋆ the sound speed cs can be eliminated from the equation(3.14) also. equil ċT appearing due to the sound speed, cT 6= 1. See [26] for the details. The numerical value of fN Here sTV = Hc L;1 T equil in the equilateral limit is obtained from our set up as 4 < fN L;1 < 7 within the window for tensor-to-scalar ratio 0.213 < r < 0.250 [26]. This is extremely interesting result as it is different from other class of DBI models. The most equil impressing fact is that the upper bound of fN L;1 in the quasi-exponential limit are in good agreement with combined constraint obtained from Planck+WMAP9+high-L+BICEP2 [2, 3] data. B. One scalar two tensor correlation After applying the gauge fixing condition to uniform gauge the one scalar and two tensor interaction can be represented by the following third order action: ( Z 2 F̃2 F̃5 3 3 Sζhh = dtd x a F1 ζ h˙ij + + 2 ζhij,k hij,k + F̃3 ψ,k ḣij hij,k + F4 ζ̇ ḣ2ij + 2 ∂ 2 ζ ḣ2ij a a ) (3.16) F̃7 + F̃6 ψ,ij ḣik ḣjk + 2 ζ,ij ḣik ḣjk a 8 where the dimensionful coefficients Fi (i = 1, 2.....7) are defined as:    YT d L1 HL1 YT , F̃2 = Ys c2s , F̃3 = −2Ys , + F̃1 = 3YT 1 − c2T 3 "dt c2T !#    L1  2 Ys Y˙s σL1 HL1 YT ˆ 2 d F̃4 = 2 YT − K̃XX + 2σ 6 + , + 2Y −1− T cT YT c2T HYs dt YT c2T   2 4σYs cs , F̃7 = 4σYT L1 − 1 , F̃6 = − F̃5 = 2σYT L1 2 cT YT (3.17) where we use σ = φ̇XG5X . Now following the prescription of in-in formalism in the interaction picture three point one scalar two tensor correlation function can be expressed in the following form: "  # h 7 Z 0 i X (q) |0i dη a h0| ζ(~k1 )hij (~k2 )hkl (~k3 ), Hint (η) hζ(~k1 )hij (~k2 )hkl (~k3 )i = −i ij;kl ζhh (3.18) −∞ q=1 3 (3) ~ ~ ~ ~ ~ ~ = (2π) δ (k1 + k2 + k3 ) {Bζhh } (k1 , k2 , k3 ), ij;kl where the total Hamiltonian in the interaction picture in terms of the third order Lagrangian  can be expressed h   i i R 3 h P7 (q) density as [Hint (η)]ij;kl = q=1 Hint (η) = − d x (L3 )ζhh . Moreover the cross bispectrum ζhh ij;kl ζhh ij;kl {Bζhh }ij;kl (~k1 , ~k2 , ~k3 ) is defined as: u  (2π)4 Pu2 ~1 , k~2 , k~3 ) = 6 f {Bζhh }ij;kl (~k1 , ~k2 , ~k3 ) = Q3 (A ) ( k P2 ζhh N L;2 ij;kl ij;kl u 3 5 k i=1 i (3.19) where the symbol ; 2 stands for one scalar two tensor correlation. Here (Aζhh )ij;kl (k~1 , k~2 , k~3 ) is the shape function N for bispectrum and the polarization indices are u = 1(E − mode), 2(E B − mode), 3(B − mode). We adopt the following normalization depending on the polarization in which we are interested:   Pζ (k1 )Pζ (k2 ) + Pζ (k2 )Pζ (k3 ) + Pζ (k3 )Pζ (k1 ) :u=1(E-mode) N Pu2 = Pζ (k1 )Ph (k2 ) + Pζ (k2 )Ph (k3 ) + Pζ (k3 )Ph (k1 ) :u=2(E B mode) (3.20)  P (k )P (k ) + P (k )P (k ) + P (k )P (k ) :u=3(B-mode). h 1 h 2 h 2 h 3 h 3 h 1  u Consequently fN L;2 ij;kl represents the non-linear parameter which carries the signature of primordial non u Gaussianities of the one scalar two tensor interaction. The explicit form of fN L;2 ij;kl characterizing the bispectrum can be calculated as: 
2 4νT +2νs −9  
 1  
 2  3 OL u  K Cos νs − 21 π2 3 Cos νT − 21 π2 3 10QP u 2 − νT fN L;2 ij;kl = P3 s −3 4νT −6 c2ν cT (k1 )νs (k2 k3 )νT 3 i=1 ki3 s !
2
4 3 2 4 1 − ǫV − sSV 1 − ǫV − sTV V 2 (φ) Γ(νT ) 2νs +4νT −12 Γ(νs ) × 2 3 (3.21) Γ( 23 ) Γ( 32 ) YS YT2 c4T c3s g̃12 MP3 L h  u u u × 32F̃1 (∇1 )ij;kl + 4F̃2 (∇2 )ij;kl + 2 F̃3 (∇3 )ij;kl i u u u u + F̃4 (∇4 )ij;kl + F̃5 (∇5 )ij;kl + F̃6 (∇6 )ij;kl + F̃7 (∇7 )ij;kl N u with polarization index u = 1(E), 2(E B), 3(B). The functional form of the co-efficients (∇i )ij;kl ∀i are explicitly mentioned in the Appendix B.2. In this context we define K := cs k1 + cT (k2 + k3 ). The overall normalization factor for three types of polarization can be expressed as:  :u=1(E-mode) 8 N OL 128 :u=2(E B mode) QP = (3.22) u  2048 :u=3(B-mode). Further, to make the computation simpler without loosing any essential information we reduce the complete set in terms of the two-polarization (helicity) mode instead of four complicated tensor indices. For this purpose let us define a reduced physical quantity: Mλ †(λ) (3.23) (~k) = hij (~k)eij 9 in terms of which the one scalar two tensor correlation is defined as: Mλ2 Mλ3 (λ1 ;λ2 ) ~ ~ ~ hζ(~k1 ) (k1 , k2 , k3 ). (~k2 ) (~k3 )i = (2π)3 δ(~k1 + ~k2 + ~k3 )B(ζhh) (3.24) where the cross reduced bispectrum is defined as: u;(λ2 ;λ3 ) 2 6 (2π)4 Pu2 (λ2 ;λ3 ) (λ2 ;λ3 ) ~ ~ ~ Pu . fN L;2 A(ζhh) = (k1 , k2 , k3 ) = Q3 B(ζhh) 3 5 i=1 ki (3.25)  (λ2 ;λ3 ) Applying the basis transformation the explicit form of fN L;2 characterizing the crossed bispectrum can be written as:  u;(λ2 ;λ3 ) 10QP OL fN L;2 = P3u 3 3 i=1 ki 3 2 − νT
2   
 1  
 2  K 4νT +2νs −9 Cos νs − 12 π2 3 Cos νT − 12 π2 3 ν ν s −3 4νT −6 c2ν cT (k1 ) s (k2 k3 ) T s !

3 2 4 S 2 T 4 2 (φ) 1 − ǫ − s V 1 − ǫ − s Γ(ν ) Γ(ν ) V V s T V V × 22νs +4νT −12 3 2 c4 c3 g̃ 2 M 3 Γ( 23 ) Γ( 23 ) Y Y S T T s 1 PL h  u;λ ;λ u;λ ;λ u;λ ;λ × 32F̃1 (∇1 ) 2 3 + 4F̃2 (∇2 ) 2 3 + 2 F̃3 (∇3 ) 2 3 i u;λ ;λ u;λ ;λ u;λ ;λ u;λ ;λ + F̃4 (∇4 ) 2 3 + F̃5 (∇5 ) 2 3 + F̃6 (∇6 ) 2 3 + F̃7 (∇7 ) 2 3 u;λ2 ;λ3 The functional form of the co-efficients (∇i ) pendix. In the equilateral limit we have iu;(λ2 ;λ3 ) h OL 10QP equil u fN = L;2 9k 3 3 2 − νT
2 (3.26) ∀i after basis transformation are explicitly mentioned in the Ap-    
 1  
 2 ((cs + 2cT )k)4νT +2νs −9 Cos νs − 21 π2 3 Cos νT − 12 π2 3 cs2νs −3 cT4νT −6 k νs +2νT !

3 2 4 S 2 T 4 2 (φ) 1 − ǫ − s 1 − ǫ − s V Γ(ν ) Γ(ν ) V V T s V V × 22νs +4νT −12 3 2 c4 c3 g̃ 2 M 3 Γ( 23 ) Γ( 23 ) Y Y h  S T T s 1 PL u;λ2 ;λ3 u;λ2 ;λ3 u;λ2 ;λ3 32F̃1 (∇1 )equil + 4F̃2 (∇2 )equil + 2 F̃3 (∇3 )equil u;λ2 ;λ3 u;λ2 ;λ3 u;λ2 ;λ3 2 ;λ3 + F̃4 (∇4 )equil + F̃5 (∇5 )u;λ + F̃6 (∇6 )equil + F̃7 (∇7 )equil equil i (3.27) where each coefficients and functions are evaluated in equilateral limit. C. Two scalar one tensor correlation After gauge fixing the interactions involving one tensor and two scalars are given by the following third order action:  Z Y1 Y4 Y2 Y5 3 3 Sζζh = dtd x a hij ζ,i ζ,j + 2 ḣij ζ,i ζ,j + Y3 ḣij ζ,i ψ,j + 2 ∂ 2 hij ζ,i ψ,j + 4 ∂ 2 hij ζ,i ζ,j (3.28) a2 a a a + Y6 ∂ 2 hij ψ,i ψ,j 10 where the dimensionful coefficients Yi (i = 1, 2.....6) are defined as: Y1 = Ys c2s , # " ! ˆ ˆ 
ẎT 1 HL1 K̃ 1 d  ˆ L1 K̃ XX XX 2 2 2 3+ L1 K̃XX − Ys cs − YT cT + L1 YT − + Y2 = 4 2 4 HYT 4 dt d σYs c2s + 2HL1 YT σ − YT (L1 σ) , + YT ! ! !# dt " ˆ ˆ ẎT d K̃ σ σ K̃ 3 XX L1 XX L1 − 3H + , + + + Y3 = Ys 2 dt 2 YT YT 2 YT # " ˆ 2
d σ (YT − K̃ XX cT )L1 − 2HσL1 + (L1 σ) + 2 YT c2T − Ys c2s , Y4 = Ys − 2 dt YT # " ˆ
YT2 L1 (YT − K̃XX c2T ) d σ 2 2 Y5 = + 2HL1 σ − (σL1 ) − 2 3YT cT − Ys cs , 2 2 dt YT    Ys2 σ 6Hσ d Y6 = , 1+ − 2YT 4YT YT dt YT2 (3.29) Following the prescription of in-in formalism in the interaction picture three point two scalar one tensor correlation function can be expressed in the following form: hζ(~k1 )ζ(~k2 )hkl (~k3 )i = −i = 7 Z X 0  i  h (q) dη a h0| ζ(~k1 )ζ(~k2 )hkl (~k3 ), Hint (η) q=1 −∞ (2π)3 δ (3) (~k1 kl ζζh + ~k2 + ~k3 ) {Bζζh }kl (~k1 , ~k2 , ~k3 ),  |0i (3.30) where the total Hamiltonian can be expressed in terms of the third order Lagrangian density as ([Hint (η)]kl )ζζh = i h R P7 h (q) i  Hint (η) = − d3 x (L3 )ζζh . Here the cross bispectrum {Bζζh }kl is defined as: q=1 kl ζζh kl u 6 (2π)4 Pu2 (Aζζh )kl = {Bζζh }kl = Q3 fN L;3 kl Pu2 , 3 5 i=1 ki (3.31) where (Aζζh )kl is the two scalar one tensor correlation shape function and the symbol ; 3 represents two scalar one  u tensor correlation. Consequently the non-linear parameter fN L;3 kl can be expressed as:   
 2  
 1  4ν +2ν −9 OL u  Cos νs − 12 π2 3 Cos νT − 21 π2 3 10LP Nij;kl K s T u fN L;3 kl = P3 s −6 2νT −3 c4ν cT (k1 k2 )νs k3νT 3 i=1 ki3 s ! 6 !

3 4 2   S 4 T 2 X 2 (φ) 1 − ǫ − s 1 − ǫ − s V Γ(ν ) Γ(ν ) V V T s V V ˆv Yv ∇ × 24νs +2νT −12 3 ij Γ( 23 ) Γ( 23 ) YS2 YT c6s c3T g̃12 MP3 L v=1 (3.32)   ˆ ∀v are explicitly mentioned in the Appendix B.3. In where the functional dependence of the co-efficients ∇v ij this context K := cs (k1 + k2 ) + cT k3 . For quasi-exponential limit the overall normalization factor for three types of polarization can be expressed as:  :u=1(E-mode) 1 N OL 16 :u=2(E B mode) (3.33) LP = u  256 :u=3(B-mode). As mentioned in the previous sub-section, performing basis transformation cross bispectrum for two scalars and one tensor can be expressed as: Mλ (ζζh) ~ ~ ~ hζ(~k1 )ζ(~k2 ) (~k3 )i = (2π)3 δ (3) (~k1 + ~k2 + ~k3 )Bλ (k1 , k2 , k3 ). (3.34) 11 where we have used the following parameterization: (ζζh) Bλ u;λ 2 6 (2π)4 Pu2 λ fN L;3 Pu . A(ζζh) = = Q3 3 5 i=1 ki (3.35)  u;λ The polarized non-Gaussian parameter for two scalar and one tensor mode fN L;3 can be rewritten as:   
 2  
 1  4ν +2ν −9 OL u  Cos νs − 21 π2 3 Cos νT − 12 π2 3 20LP δλλ′ K s T u fN L;3 λ = P3 s −6 2νT −3 c4ν cT (k1 k2 )νs k3νT 3 i=1 ki3 s ! ! 6
4
2 3 4 2 X   1 − ǫV − sSV 1 − ǫV − sTV V 2 (φ) Γ(νT ) 4νs +2νT −12 Γ(νs ) ˆv Yv ∇ × 2 3 λ′ Γ( 32 ) Γ( 23 ) YS2 YT c6s c3T g̃12 MP3 L v=1 (3.36)   ˆ where all the co-efficients ∇v ′ ∀v after basis transformation are explicitly written in the Appendix B.3. λ In the equilateral limit the expression for the non-Gaussian parameter (fN L ) reduces to the following form:    
 2  
 1 h iu OL 20LP δλλ′ ((2cs + cT )k)4νs +2νT −9 Cos νs − 21 π2 3 Cos νT − 21 π2 3 equil u fN L;3 = s −6 2νT −3 2νs +νT 9k 3 λ c4ν cT k s ! 6 !
4
2 3 4 2 X  equil 1 − ǫV − sSV 1 − ǫV − sTV V 2 (φ) Γ(ν ) Γ(ν ) T s 4νs +2νT −12 ˆ Yv ∇v ′ × 2 3 λ Γ( 23 ) Γ( 23 ) YS2 YT c6s c3T g̃12 MP3 L v=1 (3.37) D. Three tensor correlation The interactions involving three tensors are given by the following third order action:     Z 1 YT σ 3 3 ḣij ḣjk ḣki + 2 2 hik hjl − hij hkl hij,kl Shhh = dtd x a 12 4a cT 2 (3.38) Now following the prescription of in-in formalism in the interaction picture three point three tensor correlation function can be expressed in the following form: Z 0  h  i dη a h0| hi1 j1 (~k1 )hi2 j2 (~k2 )hi3 j3 (~k3 ), [Hint (η)]i1 j1 i2 j2 i3 j3 hhi1 j1 (~k1 )hi2 j2 (~k2 )hi3 j3 (~k3 )i = −i |0i (3.39) hhh −∞ = (2π)3 δ (3) (~k1 + ~k2 + ~k3 ) {Bhhh }i1 j1 i2 j2 i3 j3 (~k1 , ~k2 , ~k3 ),   = where the total Hamiltonian is expressed in terms of the third order Lagrangian density as [Hint (η)]i1 j1 i2 j2 i3 j3 hhh R 3 − d x [(L3 )hhh ]i1 j1 i2 j2 i3 j3 . In this context the bispectrum for three tensor correlation can be expressed as:  (2π)4 Pu2 hhh 6 fN L;4 Pu2 , {Bhhh }i1 j1 i2 j2 i3 j3 (~k1 , ~k2 , ~k3 ) = Q3 Ai1 j1 i2 j2 i3 j3 = 3 5 i=1 ki (3.40) where the symbol ; 4 represents three tensor correlation. Also, the non-Gaussian parameter is given by:  
3 3 Γ(νT ) 10WuP OL K 9−6νT Cos νT − 21 π2 3(νs +νT )−11 Γ(νs ) = 2 P 3 1 j1 i2 j2 i3 j3 (k1 k2 k3 )2νT Γ( 3 ) Γ( 23 ) 3 i=1 ki3 ! ! 3 2

(3.41) 3 3 3 X (p) 1 − ǫV − sSV 1 − ǫV − sTV V 2 (φ) ∆ 3 3 9 9 3 i1 j1 i2 j2 i3 j3 YS2 YT2 cs2 cT2 g̃12 MP3 L p=1 N where K = k1 +k2 +k3 and the polarization index u = 1(E −mode), 2(E B −mode), 3(B −mode). The functional (p) dependence of all the co-efficients ∆i1 j1 i2 j2 i3 j3 ∀p are summarized in Appendix B.4.   fN L;4 i 12 For quasi-exponential limit the overall normalization factor for three types of polarization can be expressed as:  :u=1(E-mode) 4 N :u=2(E B mode) (3.42) WuP OL = 64  1024 :u=3(B-mode). After performing basis transformation the relevant three point correlation function for three tensor interaction can be expressed in terms of bispectrum as: Mλ3 Mλ2 Mλ1 . (~k3 )i = (2π)3 δ (3) (~k1 + ~k2 + ~k3 )Bλhhh (~k2 ) (~k1 ) h 1 ,λ2 ,λ3 where (3.43) u 6 (2π)4 Pu2 λ1 ,λ2 ,λ3 fN L;4 λ ,λ ,λ Pu2 , A(ζζh) = = Bλhhh Q3 ,λ ,λ 1 2 3 3 1 2 3 5 i=1 ki (3.44) where the the non-linear parameter is given by:  fN L;4 u λ1 ,λ2 ,λ3 =  
3 3 10WuP OL K 9−6νT Cos νT − 21 π2 Γ(νT ) 3(νs +νT )−11 Γ(νs ) 2 P (k1 k2 k3 )2νT Γ( 23 ) Γ( 23 ) 3 3i=1 ki3 ! 3

3 3 3 X 1 − ǫV − sSV 1 − ǫV − sTV V 2 (φ) 3 3 9 9 3 YS2 YT2 cs2 cT2 g̃12 MP3 L (p) ∆λ1 λ2 λ3 p=1 ! (3.45) (p) Once again, all the helicity dependent co-efficients ∆λ1 λ2 λ3 ∀p after basis transformation are explicitly mentioned in the Appendix B.4. In the equilateral limit we have: h iu equil fN L;4 λ1 ,λ2 ,λ3  
3 3 Γ(νT ) 10WuP OL (3k)9−6νT Cos νT − 12 π2 3(νs +νT )−11 Γ(νs ) 2 = 9k 3 k 6νT Γ( 23 ) Γ( 3 ) ! 32 !

3 3 3 X (p);equil 1 − ǫV − sSV 1 − ǫV − sTV V 2 (φ) ∆λ1 λ2 λ3 3 9 9 3 3 YS2 YT2 cs2 cT2 g̃12 MP3 L p=1 (3.46) Numerical values of all such non-Gaussian parameters from three point correlation for different polarizing modes are mentioned in the table(I). In this context PC and PV stands for the parity conserving and violating contribution for graviton degrees of freedom. IV. TREE LEVEL TRISPECTRUM ANALYSIS FROM FOUR SCALAR CORRELATION To derive the expression for scalar trispectrum for D3 DBI Galileon let us start from fourth order action up to total derivatives. Consequently the fourth order action in the uniform gauge can be expressed as: Sζζζζ = SCI + SSE + SGE (4.1) where SCI , SSE and SGE represent the contribution from the contact interaction, scalar exchange and graviton exchange appearing in the four point correlation. In the next subsections we will discuss the individual contributions separately. A. Contact Interaction Taking into account the contribution coming from contact interaction of effective DBI Galileon in the fourth order action in uniform gauge we get:   Z 3 (∂ζ)2 2 (∂ζ)4 4 3 a (4.2) Ū1 ζ̇ − ζ̇ Ū2 + Ū3 4 SCI = dtd x , 4 a2 a 13 where the co-efficients Ūi (i = 1, 2, 3) for effective DBI Galileon are defined as: ! i 1h h i i φ̇4 h ˆ ˆ ˆ 2 2 + , Ū1 = K̃4X − G̃4X φ̇2 + φ̇2 K̃ K̃ XXX − G̃XXX φ̇ XX − G̃XX φ̇ 6 2 i h  i  h 1 ˆ ˆ ˆ 2 2 , Ū3 = K̃XX − G̃XX φ̇2 . + K̃ Ū2 = φ̇2 K̃ XX − G̃XX φ̇ XXX − G̃XXX φ̇ 2 (4.3) ˆ where K̃(φ, X) and G̃(φ, X) are explicitly mentioned in equation(2.5). Using in-in procedure the four point correlation function for quasi exponential situation can be expressed in the following form: hζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 )iCI = −i 3 Z X j=1 3 (3) = (2π) δ 0 −∞  CI   (j) dη a h0| ζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 ), Hint (η) |0i ζζζζ CI where in the interaction picture the Hamiltonian can be written as: (Hint (η))ζζζζ = Here following the ansatz used in [8] the trispectrum 1 TζCI (k~1 , k~2 , k~3 , k~4 ) = Q4 3 i=1 ki where 3 = Pζ(1) TζCI (k~1 , k~2 , k~3 , k~4 ) P3 j=1  CI (j) . Hint (η) j<p,i6=j,p 3 = Pζ (kij )Pζ (kj )Pζ (kp ), Pζ(2) X ζζζζ for contact interaction is defined as:  3 3

−3 −3 −3 −3 (k1 k2 + k33 k43 ) k13 + k14 + (k13 k43 + k23 k33 ) k12 + k13
 n CI 3 −3 −3 τN L Pζ(1) + + (k13 k33 + k23 k43 ) k12 + k14 X (4.4) (k~1 + k~2 + k~3 + k~4 )TζCI (k~1 , k~2 , k~3 , k~4 ), 54 CI 3 25 gN L Pζ(2) Pζ (ki )Pζ (kj )Pζ (kp ) i<j<p o , (4.5) (4.6) such that 3 Pζ3 = Pζ(1) +
3456 1  3 3 −3 −3 (k1 k2 + k33 k43 ) k13 + k14 3 25 K̄

 3 −3 −3 −3 −3 + (k13 k43 + k23 k33 ) k12 + k13 + (k13 k33 + k23 k43 ) k12 + k14 Pζ(2) (4.7) CI CI and τN L and gN L are the two non linear parameters which carry the signatures of primordial non-Gaussianities of CI CI the curvature perturbation in trispectrum analysis. By knowing τN L the other parameter gN L can be calculated by making use of the following relation [? ]: CI gN L =
 CI

64  3 3 −3 −3 −3 −3 −3 −3 τN L , + (k13 k33 + k23 k43 ) k12 + k14 + (k13 k43 + k23 k33 ) k12 + k13 (k1 k2 + k33 k43 ) k13 + k14 3 K̄ (4.8) CI where K̄ = k1 + k2 + k3 + k4 . So, there is only one independent piece of information, namely τN L , that carries information about trispectrum obtained from contact interaction. To proceed further we denote the angle between k~i and k~j (with i 6= j) by Θij then Cos(Θ12 ) = Cos(Θ34 ) := Cos(Θ3 ), Cos(Θ23 ) = Cos(Θ14 ) := Cos(Θ1 ), Cos(Θ13 ) = Cos(Θ24 ) := Cos(Θ2 ) (4.9) subject to the constraint Cos(Θ1 )+Cos(Θ2 )+Cos(Θ3 ) = −1comes from the conservation of momentum. Additionally we have used q q k14 = k23 = |k~1 + k~4 | = |k~2 + k~3 | = k12 + k42 + 2k1 k4 Cos(Θ1 ) = k22 + k32 + 2k2 k3 Cos(Θ1 ), q q (4.10) k24 = k13 = |k~2 + k~4 | = |k~1 + k~3 | = k22 + k42 + 2k2 k4 Cos(Θ2 ) = k12 + k32 + 2k1 k3 Cos(Θ2 ), q q k34 = k12 = |k~3 + k~4 | = |k~1 + k~2 | = k32 + k42 + 2k3 k4 Cos(Θ3 ) = k12 + k22 + 2k1 k2 Cos(Θ3 ). CI The explicit form of τN L characterizing the trispectrum obtained from contact interaction can be expressed for our 14 model as: 28νs −6 π 6 Cos  νs − 1 2  π
2

 −3 −3 −3 −3 −3 (k13 k23 + k33 k43 ) k13 + k14 + (k13 k43 + k23 k33 ) k12 + k13 + (k13 k33 + k23 k43 ) k12 + k14  8 8Ū1 K̄ 8νs −5  (1 − ǫV − sSV )8 H 8 Γ(νs ) Γ(17 − 8νs )K̄ 8 G1 − iΓ(16 − 8νs )K̄ 7 G2 × 4 3 12 2ν 13 Γ( 2 ) YS cs (k1 k2 k3 k4 ) s + Γ(15 − 8νs )K̄ 6 G3 − iΓ(14 − 8νs )K̄ 5 G4 + Γ(13 − 8νs )K̄ 4 G5 − iΓ(12 − 8νs )K̄ 3 G6  Ū2 K̄ 8νs −3 h + Γ(11 − 8νs )K̄ 2 G7 − iΓ(10 − 8νs )K̄G8 + G9 + (k~3 .k~4 )Ī(3, 4; 1, 2) + (k~2 .k~4 )Ī(2, 4; 1, 3) 32 i + (k~2 .k~3 )Ī(2, 3; 1, 4) + (k~1 .k~4 )Ī(1, 4; 2, 3) + (k~1 .k~2 )Ī(1, 2; 3, 4) + (k~1 .k~3 )Ī(1, 3; 2, 4) i  Z̄ Γ(13 − 8ν ) Ū3 K̄ 8νs +12 h ~ ~ ~ ~ 1 s ~ ~ ~ ~ ~ ~ ~ ~ (k1 .k2 )(k3 .k4 ) + (k1 .k3 )(k2 .k4 ) + (k1 .k4 )(k2 .k3 ) + 13 8 (K̄)  Z̄4 Γ(16 − 8νs ) Z̄5 Γ(17 − 8νs ) Z̄2 Γ(14 − 8νs ) Z̄3 Γ(15 − 8νs ) − − + + 14 15 (K̄) (K̄) (K̄)16 (K̄)17 (4.11) where the functional dependence of the momentum dependent functions Gi ∀i, Zq ∀q and Ī(i, j; m, n) are given in Appendix B.5.A. It is important to mention here that the 4D effective coupling and the interaction between the higher order graviton and DBI Galileon plays a significant role in the slow-role regime. From equation(4.11) it evident CI that the non-Gaussian parameter τN L obtained from the contact interaction is inversely proportional to the 12th power of the sound speed for scalar mode. But depending on the signature and strength of the Gauss-Bonnet coupling the CI behavior of the τN L changes. Further, using the equilateral configuration (k1 = k2 = k3 = k4 = k and K̄ = 4k) and incorporating the contribution 2k ) the from the maximum shape of the trispectrum (Cos(Θ1 ) = Cos(Θ2 ) = Cos(Θ3 ) = − 31 and kij (f or i < j) = √ 3 non linear parameter can be expressed as: CI τN L = equil;CI τN = L
−3  
 224νs −5 π 6 Cos νs − 12 π2 Γ(νs ) 8 (1 − ǫV − sSV )8 H 8 8Ū1 h √ 65536Γ(17 − 8νs )k 8 Gequil 1 3 4 12 YS cs 13312k 5 Γ( 2 ) 9 3k 3 equil equil − 16384iΓ(16 − 8νs )k 7 G2 + 4096Γ(15 − 8νs )k 6 G3 − 1024iΓ(14 − 8νs )k 5 Gequil 4 + 256Γ(13 − 8νs )k 4 Gequil − 64iΓ(12 − 8νs )k 3 Gequil + 16Γ(11 − 8νs )k 2 Gequil 5 6 7 i Ū2 838861Ū3 Z̄1equil k 3 Γ(13 − 8νs ) equil equil − + G Ī + − 4iΓ(10 − 8νs )kGequil 9 8 1024k 12 67108864 !) Z̄2equil k 2 Γ(14 − 8νs ) Z̄3equil kΓ(15 − 8νs ) Z̄4equil Γ(16 − 8νs ) Z̄5equil Γ(17 − 8νs ) + . − − + 268435456 1073741824 4294967296 17179869184 (4.12) B. Scalar Exchange Within in-in picture formalism, to calculate the four-point correlation function resulting from a correlation established via the scalar exchange mode of effective DBI Galileon we start with the following action in the uniform gauge as: SSE = Z   (∂ζ)2 dtd3 x a3 Aζ̇ 3 − ζ̇ B , a2 (4.13) where the co-efficients (A, B) are defined as: A= ! i i i φ̇3 h φ̇ h ˆ φ̇ h ˆ ˆ 2 2 , B = − K̃ φ̇ K̃XX − G̃XX + K̃XXX − G̃XXX φ̇ − G̃ . XX XX 2 6 2 (4.14) 15 Using in-in procedure the four point correlation function for quasi-exponential limit can be expressed in the following form:  Z η 2 Z 0 2 X SE    SE  X (j) (p) dη dη̃ h0| ζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 ), Hint (η) , Hint (η̃) hζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 )iSE = −i |0i = j=1 p=1 −∞ (2π)3 δ (3) (k~1 + k~2 −∞ ζζζ ζζζ + k~3 + k~4 )TζSE (k~1 , k~2 , k~3 , k~4 ), (4.15) where in the interaction picture the Hamiltonian can be written in terms of the third order Lagrangian density R P2  (j) SE SE as: (Hint (η))ζζζ = = − d3 xLSE 3 . Hence following the ansatz used in [8] the trispectrum j=1 Hint (η) ζζζ TζSE (k~1 , k~2 , k~3 , k~4 ) is defined as: 1 TζSE (k~1 , k~2 , k~3 , k~4 ) = Q4 3 i=1 ki 

−3 −3 −3 −3 + (k13 k43 + k23 k33 ) k12 (k13 k23 + k33 k43 ) k13 + k13 + k14
 n SE 3 −3 −3 + (k13 k33 + k23 k43 ) k12 τN L Pζ(1) + + k14 54 SE 3 25 gN L Pζ(2) o , (4.16) SE SE where τN and g are the two non linear parameters which carry the signatures of primordial non-Gaussianities of L NL SE the curvature perturbation obtained from scalar exchange contribution in trispectrum analysis. By knowing τN the L SE other parameter gN L can be calculated by making use of the following relation [? ]: SE gN L =
 SE

64  3 3 −3 −3 −3 −3 −3 −3 τN L , + (k13 k33 + k23 k43 ) k12 + k14 + (k13 k43 + k23 k33 ) k12 + k13 (k1 k2 + k33 k43 ) k13 + k14 3 K̄ (4.17) SE where K̄ = k1 + k2 + k3 + k4 . The explicit form of τN L characterizing the scalar exchange trispectrum can be expressed for our model as: SE τN L 
 28νs −14 K̄ 10νs −15 Cos νs − 21 π2


 = 3 3 −3 −3 −3 −3 −3 −3 (k1 k2 + k33 k43 ) k13 + k14 + (k13 k43 + k23 k33 ) k12 + k13 + (k13 k33 + k23 k43 ) k12 + k14 8 Γ(νs ) (1 − ǫV − sSV )8 H 8  2 9A [Ξ1 (−k1 , −k2 , −k12 , k3 , k4 , k12 ) − Ξ1 (k1 , k2 , −k12 , k3 , k4 , k12 )] × Γ( 23h) YS5 c6s (k1 k2 k3 k4 )2νs + AB 3(k~3 .k~4 ) {Ξ3 (k1 , k2 , −k12 , k12 , k3 , k4 ) − Ξ3 (−k1 , −k2 , −k12 , k12 , k3 , k4 )} + 6(k~12 .k~4 ) {Ξ3 (k1 , k2 , −k12 , k3 , k4 , k12 ) − Ξ3 (−k1 , −k2 , −k12 , k3 , k4 , k12 )} + 3(k~1 .k~2 ) {Ξ4 (−k12 , k1 , k2 , k3 , k4 , k12 ) − Ξ4 (−k12 , −k1 , −k2 , k3 , k4 , k12 )} i − 6(k~12 .k~2 ) {Ξ4 (k1 , k2 , −k12 , k3 , k4 , k12 ) − Ξ4 (−k1 , −k2 , −k12 , k3 , k4 , k12 )} h − B2 (k~1 .k~2 )(k~3 .k~4 ) {Ξ2 (−k12 , k1 , k2 , k12 , k3 , k4 ) − Ξ2 (−k12 , −k1 , −k2 , k12 , k3 , k4 )} + 2(k~1 .k~2 )(k~12 .k~4 ) {Ξ2 (−k12 , k1 , k2 , k3 , k4 , k12 ) − Ξ2 (−k12 , −k1 , −k2 , k3 , k4 , k12 )} − 2(k~3 .k~4 )(k~12 .k~2 ) {Ξ2 (k1 , k2 , −k12 , k12 , k3 , k4 ) − Ξ2 (−k1 , −k2 , −k12 , k12 , k3 , k4 )} i − 4(k~12 .k~4 )(k~12 .k~2 ) {Ξ2 (k1 , k2 , −k12 , k3 , k4 , k12 ) − Ξ2 (−k1 , −k2 , −k12 , k3 , k4 , k12 )} + 23 permutations of (k1 , k2 , k3 , k4 )} (4.18) 16 where the momentum dependent functions Ξi ∀i are mentioned in the Appendix. Further, using the equilateral configuration the non-Gaussian parameter from scalar exchange contribution can be expressed as: equil;SE τN L 
 220νs −42 k 10νs −18 Cos νs − 21 π2 Γ(νs ) 8 (1 − ǫV − sSV )8 H 8 √ = 5 6 Γ( 23 )  YS cs   9 3  2k 2k 2k 2k − Ξ1 k, k, − √ , k, k, √ × 9A2 Ξ1 −k, −k, − √ , k, k, √ 3 3 3 3       2k 2k 2k 2k 2 + k AB 3 Ξ3 k, k, − √ , √ , k, k − Ξ3 −k, −k, − √ , √ , k, k 3 3 3 3     √ 2k 2k 2k 2k − Ξ3 −k, −k, − √ , k, k, √ + 4 3 Ξ3 k, k, − √ , k, k, √ 3 3 3  3    2k 2k 2k 2k − Ξ4 − √ , −k, −k, k, k, √ + 3 Ξ4 − √ , k, k, k, k, √ 3  3  3  3 √ 2k 2k 2k 2k − 4 3 Ξ4 k, k, − √ , k, k, √ − Ξ4 −k, −k, − √ , k, k, √ 3 3  3 3     2k 2k 2k 2k 4 2 Ξ2 − √ , k, k, √ , k, k − Ξ2 − √ , −k, −k, √ , k, k −k B 3 3  3 3     4 2k 2k 2k 2k +√ Ξ2 − √ , k, k, k, k, √ − Ξ2 − √ , −k, −k, k, k, √ 3  3 3 3 3   2k 2k 2k 2k 4 Ξ2 k, k, − √ , √ , k, k − Ξ2 −k, −k, − √ , √ , k, k −√ 3  3 3 3 3    2k 2k 2k 2k 16 Ξ2 k, k, − √ , k, k, √ − Ξ2 −k, −k, − √ , k, k, √ − 3 3 3 3 3 + 23 permutations} . C. (4.19) Graviton Exchange In this section we are interested to evaluate the contribution of four-point function of curvature perturbations from the exchange of graviton. This process involves a third-order interaction among scalar fluctuations and tensor perturbations. To proceed, we need here only the significant third order term in the action, which describes the graviton-scalar-scalar vertex in the uniform gauge as: Z 1 dt d3 x a2 Y1 hij ζ,i ζ,j , (4.20) SGE = 2 where Y1 = YS c2S . Using in-in procedure the four point correlation function both for quasi-exponential limit can be expressed in the following form: Z η Z η⋆ hh i i GE dη̃ h0| ζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 ), (Hint (η))GE dη hζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 )iGE = −i lim ζζζ , (Hint (η̃))ζζζ |0i ⋆ η →0 −∞ −∞ = (2π)3 δ (3) (k~1 + k~2 + k~3 + k~4 )TζGE (k~1 , k~2 , k~3 , k~4 ), (4.21) where in the interaction picture the Hamiltonian can be written in terms of the third order Lagrangian density as: R GE (Hint (η))ζζh = − d3 x LGE . 3 Here following the ansatz used in [8] the trispectrum TζGE (k~1 , k~2 , k~3 , k~4 ) obtained from the graviton exchange contribution is defined as:

 3 3 1 −3 −3 −3 −3 TζGE (k~1 , k~2 , k~3 , k~4 ) = Q4 + (k13 k43 + k23 k33 ) k12 + k13 (k1 k2 + k33 k43 ) k13 + k14 3 i=1 ki o
 n GE 3 −3 −3 54 GE 3 gN L Pζ(2) , τN L Pζ(1) + 25 + (k13 k33 + k23 k43 ) k12 + k14 (4.22) GE GE where τN L and gN L are the two non linear parameters which carry the signatures of primordial non-Gaussianities of GE GE the curvature perturbation in trispectrum analysis. By knowing τN L the other parameter gN L can be calculated by making use of the following relation [? ]: GE gN L =
 GE

64  3 3 −3 −3 −3 −3 −3 −3 τN L , + (k13 k33 + k23 k43 ) k12 + k14 + (k13 k43 + k23 k33 ) k12 + k13 (k1 k2 + k33 k43 ) k13 + k14 3 K̄ (4.23) 17 GE where K̄ = k1 + k2 + k3 + k4 . The explicit form of τN L characterizing the trispectrum obtained from the graviton exchange contribution can be expressed for our model as: GE τN L 
 
 28νs −31 π 2 (1 − ǫV − sSV )8 (1 − ǫV − sTV )2 K̄ 10νs −15 Sin8 νs − 12 π2 Sin2 νT − 21 π2


 = lim η ⋆ →0 (k 3 k 3 + k 3 k 3 ) k −3 + k −3 + (k 3 k 3 + k 3 k 3 ) k −3 + k −3 + (k 3 k 3 + k 3 k 3 ) k −3 + k −3 1 3 2 4 12 14 13 1 4 2 3 1 2 3 4 12 13 14         8 2   i j l m 10 X X X ka kb kc kd Γ(νs ) Γ(νT ) H λ ~ λ ~ × ǫ ( k )ǫ ( k ) , · ϑ (η ) abcd ⋆ ij ab lm cd 6 νT +3  Γ( 32 ) Γ( 23 ) YS4 YT2 c12 kab (ka kb kc kd )2νs s cT    i,j,l,m a<b λ=+[+,−],     c<d 23 perms. ×[+,−] (4.24) The momentum dependent functions ϑabcd (η⋆ ) are given in the Appendix. Here to write equation(4.24) we have used the fact that the exchange momentum dependent polarization tensor ǫλij (~kab ) is a symmetric tensor and also the fourpoint correlator obtained from the graviton exchange is invariant under the exchange of the subscripts of the momenta a ↔ b and c ↔ d. Additionally in equation(4.24) the sum is performed only over different indices a, b, c, d and we have extracted an overall symmetry factor of 4 which takes care about the exchanges a ↔ b and c ↔ d. Rewriting the sums appearing in equation(4.24) we get the following reduced formula for the non-Gaussian parameter: 
 
 28νs −31 π 2 (1 − ǫV − sSV )8 (1 − ǫV − sTV )2 K̄ 10νs −15 Sin8 νs − 21 π2 Sin2 νT − 12 π2 GE


 τN L =  3 3 −3 −3 −3 −3 −3 −3 (k1 k2 + k33 k43 ) k13 + k14 + (k13 k43 + k23 k33 ) k12 + k13 + (k13 k33 + k23 k43 ) k12 + k14  "  8 2  X   X Γ(νs ) H 10 Γ(νT ) k1i k2j k3l k4m λ ~ λ ~ × ϑ̂ + ϑ̂ · ǫ ( k )ǫ ( k ) 1234 3412 ij 12 lm 34 νT +3 6 Γ( 32 ) Γ( 23 ) YS4 YT2 c12 k12 (k1 k2 k3 k4 )2νs s cT   λ=+[+,−], i,j,l,m ×[+,−] k1i k3j k2l k4m νT +3 k13 (k1 k3 k2 k4 )2νs ki kj kl km + ǫλij (~k14 )ǫλlm (~k23 ) νT +3 1 4 2 3 k14 (k1 k4 k2 k3 )2νs + ǫλij (~k13 )ǫλlm (~k24 )   · ϑ̂1324 + ϑ̂2413   · ϑ̂1423 + ϑ̂2314 #) , (4.25) where we define limη⋆ →0 ϑabcd (η⋆ ) := ϑ̂abcd . There are divergent contributions in the limit η∗ → 0 appear with a logarithmic dependence on the momenta, but the additive cumulative contribution of Iabcd and Icdab give rise to a finite contribution at late times. P To represent Eq (4.25) in a simpler form, let us start with the polarization sum s ǫsij (~k12 )ǫslm (~k34 )k1i k2j k3l k4m in terms of the relative angles between the ~ka and ~k12 . The polarization tensors ǫsij can be rewritten as ǫ+ ei ⊗ ~ej − ~ēi ⊗ ~ēj , ǫ× ei ⊗ ~ēj + ~ēi ⊗ ~ej , ij = ~ ij = ~ (4.26) where ~e and ~ē are orthogonal unit vectors perpendicular to exchange momentum vector ~k12 . It is convenient to write ~ the momentum vector ~ka in a spherical polar coordinate system having {~e, ~ē, k̂12 ≡ ~k12 /k12 } as basis. In this coordinate ~ ~ system one can express the momentum vector as: ~ka = ka (sin θa cos φa , sin θa sin φa , cos θa ) , where cos θa ≡ k̂a · k̂12 ~ and cosφa ≡ k̂a · ~e. This implies × i j i j ǫ+ ij k1 k2 = k1 k2 sin θ1 sin θ2 cos(φ1 + φ2 ) , ǫij k1 k2 = k1 k2 sinθ1 sin θ2 sin(φ1 + φ2 ) , (4.27) × i j i j with an identical relation holding for ǫ+ ij k3 k4 and ǫij k3 k4 which will contribute to the polarization sum also. Since the projections of the momentum vectors ~k1 and ~k2 ( similarly for ~k3 and ~k4 ) on the plane orthogonal to exchange momentum vector ~k12 (~k34 ) have the same amplitude but opposite directions. Consequently we have two additional sets of constraint relationships given by: k2 sin θ2 = k1 sin θ1 Using these relations we get: X s and φ2 = φ1 + π, k4 sin θ4 = k3 sin θ3 and φ4 = φ3 + π. ǫsij (~kab )ǫslm (~kcd )kai kbj kcl kdm = ka2 kc2 sin2 θa sin2 θc cos 2Υab,cd , (4.28) (4.29) 18 where we define a new angular coordinate Υab,cd ≡ φa − φc with a = 1, (b, c) = 2, 3, 4, d = 3, 4 and b > a, d > c, a 6= b 6= c 6= d, which physically represents the angle between the projections of the two momentum vectors ~ka and ~kc on the plane orthogonal to ~k12 . Alternatively this can be interpreted as the angle between the two planes formed by the pair of momentum vectors {~k1 , ~k2 } and {~k3 , ~k4 }. Thus, the expression for the non-Gaussian parameter calculated from the graviton exchange contribution from the trispectrum can be simplified to the following expression: 

  28νs −31 π 2 (1 − ǫV − sSV )8 (1 − ǫV − sTV )2 K̄ 10νs −15 Sin8 νs − 12 π2 Sin2 νT − 12 π2 GE


 τN L =  3 3 −3 −3 −3 −3 −3 −3 (k1 k2 + k33 k43 ) k13 + (k13 k43 + k23 k33 ) k12 + (k13 k33 + k23 k43 ) k12 + k14 + k13 + k14 8 2  k 2 k 2 [1 − (~kˆ · ~kˆ )2 ][1 − (~kˆ · ~kˆ )2 ]   Γ(νs ) Γ(νT ) H 10 1 12 3 12 1 3 × cos 2Υ · ϑ̂ + ϑ̂ 12,34 1234 3412 νT +3 6 Γ( 32 ) Γ( 23 ) YS4 YT2 c12 k12 (k1 k2 k3 k4 )2νs s cT  ˆ ˆ ˆ ˆ   k 2 k 2 [1 − (~k1 · ~k13 )2 ][1 − (~k2 · ~k13 )2 ] + 1 2 cos 2Υ · ϑ̂ + ϑ̂ 1324 2413 13,24 νT +3 k13 (k1 k3 k2 k4 )2νs  ˆ ˆ ˆ ˆ   k12 k22 [1 − (~k1 · ~k14 )2 ][1 − (~k2 · ~k14 )2 ] cos 2Υ14,23 · ϑ̂1423 + ϑ̂2314 , + νT +3  k14 (k1 k4 k2 k3 )2νs (4.30) Further, incorporating the contribution from the maximum shape of the trispectrum one can show that the graviton exchange contribution does not contribute anything in the equilateral limit. Now summing up all the significant contributions of four-point four scalar correlation coming from contact interaction, scalar exchange and graviton equil equil exchange interaction the numerical value of τN L in the equilateral limit is obtained from our set up as 48 < τN L < 97 in quasi-exponential limit within the window for tensor-to-scalar ratio 0.213 < r < 0.250 which is significantly large from other class of DBI models and consistent with combined constraint obtained from Planck+WMAP9+highL+BICEP2 [2, 3] data. V. FOUR POINT CONSISTENCY CONDITIONS AND VIOLATION OF SUYAMA-YAMAGUCHI RELATION In the counter-collinear limit collecting the contribution from the scalar exchange diagram we derive the following expression for the four point consistency condition: (nζ − 1)2 Pζ (k12 )Pζ (k1 ) [Pζ (k3 ) + · · · ] hζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 )iSE ≈ (2π)3 δ 3 (~k1 + ~k2 + ~k3 + ~k4 ) 4 (5.1) which can be interpreted as the scalar exchange contribution arising from the product of two back-to-back bispectra in the squeezed limit. Additionally, we consider the contribution from the graviton exchange diagram from which we derive another expression for the four point consistency condition: 3 3 ~ ~ ~ ~ hζ(k~1 )ζ(k~2 )ζ(k~3 )ζ(k~4 )iGE ≈ 9cs ǫs (2π)  δ (k1 + k2 + k3 + k4 )Pζ (k12 )Pζ (k1 )  ×  X X λ=+[+,−], i,j,l,m ×[+,−]   ki kj kl km ǫλij (~k12 )ǫλlm (~k34 ) 1 12 32 3 Pζ (k3 ) + · · ·   k1 k3 (5.2) Here using k12 → 0, θ1 , θ3 → π the polarization sum appearing in Eq (5.2) can be simplified to the following expression as: X X λ=+[+,−], i,j,l,m ×[+,−] ki kj kl km ǫλij (~k12 )ǫλlm (~k34 ) 1 12 32 3 = cos 2Υ12,34 . k1 k3 (5.3) Further substituting Eq (5.3) in Eq (5.2) and using Eq (3.15) the four-point correlation function from the graviton exchange contribution in the counter-collinear limit (k12 << k1 ≈ k2 , k3 ≈ k4 ) reduces to the following expression:    2 2 3 1 − ǫV − sSV Γ(νs ) GE 2(νs −νT −4) 2 ~ ~ ~ ~ hζ(k1 )ζ(k2 )ζ(k3 )ζ(k4 )i = 9.2 r⋆ . 1 + O(ǫT ) . (5.4) 2 1 − ǫV − sTV ⋆ Γ(νT ) ⋆ 3 3 ~ ~ ~ ~ × (2π) δ (k1 + k2 + k3 + k4 )Pζ (k12 )Pζ (k1 ) [cos 2Υ12,34 Pζ (k3 ) + · · · ] 19 To check the validity of well known Suyama-Yamguchi consistency relation we start with the in-in picture where the four-point correlator can be written as: X hζ 2 (~x)ζ 2 (0)i~k = |hn~k |ζ 2 (0)i|2 (5.5) n where n is a label for individual states or particle number within the momentum eigen space. Here the sum is written over positive definite terms. On the other hand in this context one of the contributions is the square of the squeezed limit of the three-point correlation function of the scalar contribution. This implies: hζ 2 (~x)ζ 2 (0)i~k = |hζ(~k)|ζ 2 (0)i|2 X |hñ~k |ζ 2 (0)i|2 . + Pζ (k) (5.6) ñ As the second term in Eq (5.6) is always positive definite we conclude that: hζ 2 (~x)ζ 2 (0)i~k ≥ |hζ(~k)|ζ 2 (0)i|2 . Further Pζ (k) using this result in quasi-exponential limit we get: 2 lim q→0 Z k~1 d3 k1 (2π)3 Z k~3 d3 k3 hζ(k~1 )ζ(~ q − k~1 )ζ(k~3 )ζ(−~q − k~3 )i ≥ lim | {z } q→0 | {z } (2π)3 k~2 k~4 R 3 d k2 ~k )ζ(k~2 )ζ(−~q − k~2 (2π)3 hζ(|{z} | {z k~1 k~3 Pζ (q) Hence using Eq (5.7) finally we get:   Z Z 36 d3 k3 d3 k1 2 ≥0 P (k )P (k ) τ − (f ) lim ζ 1 ζ 2 N L N L q→0 k~ (2π)3 k~ (2π)3 25 3 1 resulting in a generic outcome of DBI Galileon inflation, viz, 36  ˆ 2 τ̂N L ≥ fN L 25 k~2 )i } (5.7) (5.8) (5.9) where τ̂N L and fˆN L are used to represent soft limits of the three and four point correlation functions. This relation 2  36 directly confirms the partial violation of standard Suyama-Yamaguchi relation [27, 28, 30] τ̂N L = 25 fˆN L . These non-trivial features allow us to go beyond the no-go theorem in the present context. Some other aspects of the violation of well known consistency relations in the context of single field inflation has been studied in [31, 32]. Let us now investigate for some possible explanations of the partial violation of standard Suyama-Yamaguchi relation. The standard relations and limits of Non-Gaussianity are usually derived under the following assumptions: • The background is Einsteinian gravity, • Inflation is driven by a single scalar field, • The scalar field action is canonical, • Perfect slow roll conditions hold good throughout, • The vacuum is Bunch-Davies. Of course, most of the results derived using these assumptions are true to a great extent, it is not obvious that they will still hold good if one or more assumptions are relaxed. Only when one deals with a context where he/she has to relax one or more assumptions, one can investigate for the consequences and conclude if the relations are still valid or not. In the present scenario, a non-Einstein framework forms the background along with a non-canonical action appearing in the matter sector for DBI Galilon. The contributions of them arise through the first two terms of Eq (2.4) which will further effect Eq (5.1) and Eq (5.4). On top of that, we have higher derivative contributions for DBI Galileon matter sector, for contact interaction, scalar and graviton exchange contributions are coupled with the higher curvature contributions through highly non-linear terms as appearing in the perturbative action as mentioned in Eq (4.2,4.13,4.20), which directly affects the interaction vertex factors as well as the propagators of the setup, 20 resulting in deviation from standard results. We suspect that these non-standard inputs might have reflected in the violation of the no-go theorem. Having said this, we do admit that this can at best serve as a qualitative explanation of the violation. A huge amount of work needs to be done before one can comment conclusively on deviation from which assumption still respects the relation and deviation from which one leads to violation, and, in case it does, to what extent. This is a highly non-trivial task which one can only hope to attempt in future. VI. SUMMARY AND OUTLOOK In this article we have explored primordial non-Gaussian features of DBI Galileon inflation in D3 brane. We have derived the expressions for three and four point correlation functions in terms of the non-linear parameters fN L and τN L for equilateral type of non-Gaussian configurations in the nontrivial polarization modes. This resulted in a significantly large value for non-Gaussianity from this setup. We could also find a parameter space for both nonGaussianity and tensor-to-scalar ratio (r) consistent with combined constraint obtained from Planck+WMAP9+highL+BICEP2 data. The detectable features of primordial non-Gaussianity lead to the conclusion that this type of models can directly be verified by upcoming data. Moreover, the calculations reveal some other interesting results like partial violation of the Suyama-Yamaguchi four-point consistency relation. Some issues which can be addressed in the context of non-Gaussianity for DBI Galileon are studies of mass spectrum of primordial black hole formation [33], [34] as a tool for constraining non-Gaussianity at small scales; effect of the presence of one loop and two loop radiative corrections in the presence of all possible scalar and tensor mode fluctuations in the bispectrum and trispectrum; study of different shapes in equilateral, local, orthogonal, squeezed limit configuration for the tree, one and two loop level of non-Gaussianity and calculation of other higher order n-point correlation functions to find out the proper consistency relations between all higher order non-Gaussian parameters as well as the analysis of CMB bispectrum and trispectrum in the presence of Galileon in SUGRA background. Given the promise the results of the present paper shows, these open issues worth exploring in future as they may give rise to interesting results. Acknowledgments SC thanks Council of Scientific and Industrial Research, India for financial support through Senior Research Fellowship (Grant No. 09/093(0132)/2010). Appendix In this section we mention all the momentum dependent functions appearing in the context of bispectrum and trispectrum analysis coming from all scalar-tensor three point correlations and four point scalar correlation. [fNL;A ]u;(λ1 λ2 λ3 ) (E- mode) [fN L;1 ]1;(000) (PC) [fN L;2 ]1;(0++) (PV) [fN L;2 ]1;(0−−) (PV) [fN L;2 ]1;(0+−) (PV) [fN L;2 ]1;(0−+) (PV) [fN L;3 ]1;(00+) (PC) [fN L;3 ]1;(00−) (PC) [fN L;4 ]1;(+++) (PV) [fN L;4 ]1;(−−−) (PV) [fN L;4 ]1;(++−) (PV) [fN L;4 ]1;(+−−) (PV) [fN L;4 ]1;(−+−) (PV) [fN L;4 ]1;(−++) (PV) [fN L;4 ]1;(−−+) (PV) ×10−3 4000 - 7000 3.2 - 6.7 1.4 - 5.7 2.6 - 9.6 1.7 - 6.9 121 - 432 549 - 878 0.23 - 0.97 0.06 - 0.41 0.23 - 0.93 0.01 - 0.35 0.04 - 0.39 0.03 - 0.56 0.09 - 0.34 u;(λ1 λ2 λ3 ) [fNL;A N ] (E B- mode) [fN L;1 ]2;(000) (PC) [fN L;2 ]2;(0++) (PV) [fN L;2 ]2;(0−−) (PV) [fN L;2 ]2;(0+−) (PV) [fN L;2 ]2;(0−+) (PV) [fN L;3 ]2;(00+) (PC) [fN L;3 ]2;(00−) (PC) [fN L;4 ]2;(+++) (PV) [fN L;4 ]2;(−−−) (PV) [fN L;4 ]2;(++−) (PV) [fN L;4 ]2;(+−−) (PV) [fN L;4 ]2;(−+−) (PV) [fN L;4 ]2;(−++) (PV) [fN L;4 ]2;(−−+) (PV) ×10−3 0 2.1 - 4.5 2.1 - 8.9 2.9 - 11.0 3.5 - 7.4 78 - 349 304 - 883 0.08 - 0.32 0.09 - 0.67 0.18 - 0.67 0.07 - 0.44 0.02 - 0.32 0.1 - 0.43 0.07 - 0.41 [fNL;A ]u;(λ1 λ2 λ3 ) (B- mode) [fN L;1 ]3;(000) (PC) [fN L;2 ]3;(0++) (PV) [fN L;2 ]3;(0−−) (PV) [fN L;2 ]3;(0+−) (PV) [fN L;2 ]3;(0−+) (PV) [fN L;3 ]3;(00+) (PC) [fN L;3 ]3;(00−) (PC) [fN L;4 ]3;(+++) (PV) [fN L;4 ]3;(−−−) (PV) [fN L;4 ]3;(++−) (PV) [fN L;4 ]3;(+−−) (PV) [fN L;4 ]3;(−+−) (PV) [fN L;4 ]3;(−++) (PV) [fN L;4 ]3;(−−+) (PV) ×10−4 0 2.8 - 8.7 2.7 - 7.2 2.7 - 8.4 1.8 - 10.6 45 - 221 189 - 588 0.02 - 0.34 0.23 - 1.7 0.03 - 0.53 0.02 - 0.42 0.09 - 0.51 0.17 - 0.63 0.05 - 0.44 non-Gaussian ([fN L;A ]u;(λ1 λ2 λ3 ) ) parameters related to the primordial bispectrum for A=1 (threeNscalar), 2(one scalar and two tensor), 3(two scalar and one tensor), 4(three tensor) with polarization index u = 1(E − mode), 2(E B − mode), 3(B − mode) including all helicity degrees of freedom represented by λ1 , λ2 and λ3 estimated from our model. In this context “+”,“-” stands for two projections of helicity for graviton degrees of freedom and “0” represents helicity for scalar mode. Here PC and PV stands for the parity conserving and violating contributions appearing in the tree level primordial bispectrum analysis. TABLE I: Different 21 22 A. Functions appearing in three scalar correlation The functions appearing in the context of three scalar correlation can be expressed as: i h  
P P 1+x 2 3 2 2 I1 (x) = Cos x − 21 π2 Γ(1 + x) 2+x i6=j ki kj , i>j ki kj − K 2 K    i h 
 P (k1 k2 k3 )3 Γ(3 + x) 1 π 1+x 1 K 1 π , I2 (x) = Cos x − 2 2 Γ(1 + x) 1−x − K i>j ki kj − K 2 k1 k2 k3 , I3 (x) = Cos x − K3 2 2 2 n ~ ~ 2  
 2   ~ ~ (k1 .k2 )k3 (k .k )k I4 (x) = Cos x − 12 π2 (3 + x)Γ(1 + x) − Γ(2 + x) kK3 + 2 K3 1 (3 + x)Γ(1 + x) − Γ(2 + x) kK1 K o (k~ .k~ )k2  + 3 K1 2 (3 + x)Γ(1 + x) − Γ(2 + x) kK2 , 
n (k~1 .k~2 )k2   (k~ .k~ )k2    3 Γ(1 + x) + Γ(2 + x) kK3 + 2 K3 1 Γ(1 + x) + Γ(2 + x) kK1 + I5 (x) = Cos x − 12 π2 K o (k~3 .k~1 )k22  k2 , Γ(1 + x) + Γ(2 + x) K K 
 (k1 k2 k3 )2 1 π (6+x)Γ(3+x) I6 (x) = , Cos x − 2 2 K3 12 o 
 n (k~1 .k~2 )k2 
 (k~2 .k~3 )k12 (k~3 .k~1 )k12 k1 k2 k3 1 π 2+x 2 k3 +k3 k1 3 I7 (x) = Cos x − 2 2 2 Γ(1 + x) + Γ(2 + x) k1 k2 +kK + + , + (3 + x) 2 K3 K i K ~ ~ 2 K n ~ ~ 2h 2 
 (k .k )k k (k1 .k2 )k3 (3 + x)Γ(1 + x) + (3 + x)Γ(2 + x) kK3 − Γ(3 + x) K32 + 2 K3 1 [(3 + x)Γ(1 + x) I8 (x) = Cos x − 12 π2 K i io h 2 2 (k~ .k~ )k k k2 + (3 + x)Γ(2 + x) kK1 − Γ(3 + x) K12 + 3 K1 2 (3 + x)Γ(1 + x) + (3 + x)Γ(2 + x) kK1 − Γ(3 + x) K12 . (6.1) In the equilateral configuration these functions are related through the following expression: I1equil (x) = I5equil (x) =  I4equil (x) equil 3I equil (x) equil x  equil I (x). , I2 (x) = 8 , I6 (x) = 1 + 2 2(1 − x) 2 3 (6.2) Additionally in the squeezed limit these functions are reduced to the following expressions:           1 π 2 + x (1 + x) 2k1 k1 1+x 1 π sq sq 3 I1 = Cos x − − − − k3 , k1 Γ(1 + x) , I2 = Cos x − Γ(1 + x) 2 2 2 1−x 2 4  2   2 2   2 3 k1 k3 1 π 1 π (6 + x)Γ(3 + x) (k1 k3 ) Γ(3 + x) , I6sq = Cos x − Cos x − , I3sq = 8 2   2 2 8 2 2 12     1 π 1 k3 k1 k32 I4sq = Cos x − (3 + x)Γ(1 + x) − Γ(2 + x) , + (k~1 .k~3 )k1 (3 + x)Γ(1 + x) − Γ(2 + x) 2 2 ( 2 2k1 2 )        k3 k1 k32 (k~1 .k~3 )k12 k1 1 π Γ(1 + x) + Γ(2 + x) , + Γ(1 + x) + Γ(2 + x) I5sq = Cos x − 2 2 2 2k k1 K      1    1 π 2+x 1 k3 k1 k32 Γ(1 + x) + Γ(2 + x) I7sq = Cos x − + (3 + x) + (k~1 .k~3 )k1 , 2 2   2 4 8k1 2    k3 k1 k32 k32 1 π sq (3 + x)Γ(1 + x) + (3 + x)Γ(2 + x) − Γ(3 + x) 2 I8 = Cos x − 2 2 2 2k1 4k1  o 1 ~ ~ + (k1 .k3 )k1 (3 + x)Γ(1 + x) + (3 + x)Γ(2 + x) 2 − Γ(3 + x) 14 . (6.3) 23 B. Functions appearing in one scalar two tensor correlation The functional dependence of the co-efficients appearing in the context of one scalar two tensor correlation can be expressed as: h iu h iu h iu   6  Jp (k~2 , k~1 , k~3 ) Jp (k~3 , k~2 , k~1 )  Jp (k~1 , k~2 , k~3 )  X Γ(7 + p − 4νT − 2νs ) ij;kl ij;kl ij;kl u (∇1 )ij;kl = , + + p 2p νs ν ν 7 14 s s νT νT νT − 2ν − s   k k k (k k ) (k k ) (k k ) 3 3 4νT − 3 − 3 2 3 1 3 2 1 1 2 3  cs cT K 7+p−4νT −2νs p=1  i h (k~2 .k~3 ) (k~3 .k~1 ) (k~1 .k~2 ) 4 + + νs νs νs X ν ν ν Γ(9 + p − 4νT − 2νs ) k (k2 k3 ) T k2 (k1 k3 ) T k3 (k2 k3 ) T u u Hp Nij,kl , (∇2 )ij;kl = 1
2p 2 2 2νs − p 3 9+p−4νT −2νs 3 −3 4νT − 3 −6 c − ν T c c K p=1 T s 2 T i            h u k~1 .k~2 Y123 + k~1 .k~3 Y132 + k~2 .k~3 Y213 + k~2 .k~1 Y231 + k~3 .k~2 Y312 + k~3 .k~1 Y321 Nij,kl u (∇3 )ij;kl = ,
2 2 3 cT 2 − νT  h i 3 u (∇4 )uij;kl = , − νs J˜123 + J˜132 + J˜213 + J˜231 + J˜312 + J˜321 Nij,kl 2 iu h (∇5 )uij;kl = C˜123 + C˜132 + C˜213 + C˜231 + C˜312 + C˜321 , ij,kl i h u , (∇6 )uij;kl = Ŵ123 + Ŵ132 + Ŵ213 + Ŵ231 + Ŵ312 + Ŵ321 Nij,kl  u     u u Nmn,m (6.4) (∇7 )ij;kl = k1m k1m′ X̄123 + X̄132 + k2m k2m′ X̄231 + X̄213 + k3m k3m′ X̄312 + X̄321 Nij,kl ′ . n with h iu J1 (k~a , k~b , k~c ) h iij;kl u J3 (k~a , k~b , k~c ) h iuij;kl J4 (k~a , k~b , k~c ) h iij;kl u J5 (k~a , k~b , k~c ) ij;kl iu h iu
2 h −1 −2 , J2 (k~a , k~b , k~c ) = J1 (k~a , k~b , k~c ) Kcs 3 cT 3 , ij;kl ij;kl
 2
 3 3 2 u (k + k + k k ) + k (k + k ) − ν − ν = Nij,kl a b a b c T T a b 2 2
  3 u = Nij,kl kb kc (kb + kc ) + ka (kb2 + kc2 + kb kc ) , 2 − νT iu  2 2  h u u = Nij,kl kb kc + ka kb kc (kb + kc ) , J6 (k~a , k~b , k~c ) = iNij,kl ka kb2 kc2 , u = Nij,kl 3 2 − νT ij;kl (includes 3 permutations of a, b, c), (6.5) 24 Yabc 1 = νs ka (kb kc )νT ( H 1 = H2 = L 1 , H 3 = 4 X Hp a2 Ys c2s t1 a2 Ys c2s =− t1 2p 2νs − p 3 −3 4νT − 3 −6 cT K 9+p−4νT −2νs + cs p=1 H1 (ka kb + kb kc + kc ka ) −2 −4 2  Γ(9 + p − 4νT − 2νs ) cs 3 cT 3 K 2 5 X Γ(8 + q − 4νT − 2νs ) Aabc q 16 2νs − q3 − 83 4νT − 2q 3 − 3 cs cT K 8+q−4νT −2νs q=1 ) (with a, b, c = 1, 2, 3 with a 6= b 6= c), ka kb kc , H4 = − −1 −2 3 , cs cT K
3 −1 −2 abc = − ka kb + kb kc + kc ka + ka2 Aabc − νs , Aabc = cs 3 cT 3 KAabc 1 , 1 , A3 2 2      2 2 a Ys cs 2 3 3 = ka kb kc , − νs ka kb kc − νs + ka2 (kb + kc ) , Aabc Aabc 5 4 2 2 t1 h iu 6 Jp (k~a , k~b , k~c ) iu h X Γ(7 + p − 4νT − 2νs ) ij;kl C˜abc = (includes 6 permuations of a, b, c), 2p νs 7 14 ν 2νs − p k 7+p−4νT −2νs ij;kl 3 − 3 4νT − 3 − 3 a (kb kc ) T c c K p=1 s T ) ( 7 1 Γ(6 + p − 4νT − 2νs ) a2 Ys c2s X abc Γ(8 − 4νT − 2νs ) 2 Ŵabc = νs a + , ka 2ν − 8 4ν − 16 ka (kb kc )νT t1 p=1 p c2νs − p3 −2 c4νT − 2p 3 −4 cs s 3 c T 3 K 8−4νT −2νs K 6+p−4νT −2νs Aabc = 1 s T 7 X T aabc Γ(7 + p − 4νT − 2νs ) p , X̄abc = p νs 14 ν T ka (kb kc ) c2νs − 3 − 73 c4νT − 2p 7+p−4νT −2νs 3 − 3 K p=1 s T  2   7 X aabc 3 Γ(6 + p − 4νT − 2νs ) 3 −1 −2 p abc abc J˜abc = , a = = aabc cs 3 cT 3 K, − ν − ν T s , a2 p 2p 1 1 νs ν T ka (kb kc ) c2νs − 3 −2 c4νT − 3 −4 K 6+p−4νT −2νs 2 2 p=1 s T    2    3 3 3 aabc = ka2 , − νT − νs ka (kb + kc ) + kb2 + kc2 + kb kc + − νT 3 2 2     2   3 3 3 2 2 abc 2 a4 = ka (kb + kc ) − νT + ka (kb + kc + kb kc ) + kb kc (kb + kc ) − νT − νs , 2  2     2    3 3 3 3 2 2 2 2 2 abc − νT k (k + k + kb kc ) + − νs kb kc + − νT − νs ka kb kc (kb + kc ) , a5 = 2  a b c 2 2  2  3 3 = −ka2 kb2 kc2 , − νT kb kc + − νs ka (kb + kc ) , aabc aabc = ka kb kc 7 6 2 2 (6.6) After using the basis transformation mentioned in equation(3.23) the reduced form of the above mentioned coefficients can be expressed in the following form: (∇1 ) u;λ2 ;λ3 = h iu;λ2 ;λ3 6   Jp (k~1 , k~2 , k~3 ) X k1νs (k2 k3 )νT  p=1  (∇2 ) u;λ2 ;λ3 (∇3 ) u;λ2 ;λ3 (∇4 ) u;λ2 ;λ3 (∇5 ) u;λ2 ;λ3 (∇6 ) u;λ2 ;λ3 (∇7 ) u;λ2 ;λ3 2 = h + h iu;λ2 ;λ3 Jp (k~2 , k~1 , k~3 ) k2νs (k1 k3 )νT × ~ ~ (k1 .k2 ) k1νs (k2 k3 )νT + ~ ~ (k2 .k3 ) k2νs (k1 k3 )νT + ~ ~ (k3 .k1 ) k3νs (k1 k2 )νT iu;λ2 ;λ3 4 X + h iu;λ2 ;λ3    Jp (k~3 , k~2 , k~1 ) k3νs (k2 k1 )νT   Γ(7 + p − 4νT − 2νs ) 2ν − p − 7 4ν − 2p − 14 cs s 3 3 cT T 3 3 K 7+p−4νT −2νs , Γ(9 + p − 4νT − 2νs ) , − νT p=1 iu;λ2 ;λ3            h 2 k~1 .k~2 Y123 + k~1 .k~3 Y132 + k~2 .k~3 Y213 + k~2 .k~1 Y231 + k~3 .k~2 Y312 + k~3 .k~1 Y321 , =
2 2 3 cT 2 − νT  h i 3 =2 − νs J˜123 + J˜132 + J˜213 + J˜231 + J˜312 + J˜321 δ λ2 λ3 , 2 iu;λ2 ;λ3 h ˜ , = C123 + C˜132 + C˜213 + C˜231 + C˜312 + C˜321 i h = 2 Ŵ123 + Ŵ132 + Ŵ213 + Ŵ231 + Ŵ312 + Ŵ321 δ λ2 λ3 , i h    = Z1u;λ2 ;λ3 X̄123 + X̄132 + Z2u;λ2 ;λ3 X̄231 + X̄213 + Z3u;λ2 ;λ3 X̄312 + X̄321 . 3 2
2 c2T Hp 2ν − p −3 4ν − 2p −6 cs s 3 cT T 3 K 9+p−4νT −2νs (6.7) 25 with h iu;λ2 ;λ3 J1 (k~a , k~b , k~c ) h iu;λ2 ;λ3 J3 (k~a , k~b , k~c ) h iu;λ2 ;λ3 J4 (k~a , k~b , k~c ) h iu;λ2 ;λ3 J5 (k~a , k~b , k~c ) iu;λ2 ;λ3 1 2 iu;λ2 ;λ3 h
2 h − − cs 3 cT 3 K, = J1 (k~a , k~b , k~c ) , J2 (k~a , k~b , k~c )

 = 2λ2 λ3 32 − νT (ka2 + kb2 + ka kb ) + 23 − νT ka (kb + kc )
  = 2 32 − νT λ32 kb kc (kb + kc ) + λ33 ka (kb2 + kc2 + kb kc ) , iu;λ2 ;λ3   h = 2λ22 λ23 kb2 kc2 + ka kb kc (kb + kc ) , J6 (k~a , k~b , k~c ) = 2iλ23 λ22 ka kb2 kc2 , (includes 3 permutations of a, b, c), h iu;λ2 ;λ3 6 iu;λ2 ;λ3 X h Jp (k~a , k~b , k~c ) Γ(7 + p − 4νT − 2νs ) (includes 6 permuations of a, b, c), = C˜abc p νs 7 ν T 2ν − − 4ν − 2p − 14 ka (kb kc ) cs s 3 3 cT T 3 3 K 7+p−4νT −2νs p=1 −1 −2 u  cs 3 cT 3 K u;λ2 ;λ3 = Za (ka − kb − kc )(ka + kb − kc )(ka − kb + kc ) ka2 − (λ2 kb + λ3 kc )2 . 2 2 2 32ka kb kc (6.8) =2 3 2 − νT C. Functions appearing in two scalar one tensor correlation The functional dependence of the co-efficients appearing in the context of two scalar one tensor correlation can be expressed as:     (k2i k3j + k3i k2j ) (k1i k3j + k3i k1j ) (k1i k2j + k2i k1j ) ˆ Õ, = ∇1 + + k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs ij       3 (k2i k3j P123 + k3i k2j P132 ) (k1i k3j P213 + k3i k1j P231 ) (k1i k2j P312 + k2i k1j P321 ) ˆ = cs ∇2 , − νT + + 2 k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs ij     (k2i k3j R123 + k3i k2j R132 ) (k1i k3j R213 + k3i k1j R231 ) (k1i k2j R312 + k2i k1j R321 ) ˆ3 ,, + + = cs ∇ k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs ij          k2i k3j R̃123 + k3i k2j R̃132 k1i k3j R̃213 + k3i k1j R̃231 k1i k2j R̃312 + k2i k1j R̃321 ˆ4 , = k12 ∇ + k22 + k32 k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs ij     (k2i k3j + k3i k2j ) 2 (k1i k3j + k3i k1j ) 2 (k1i k2j + k2i k1j ) ˆ5 Õ, + k + k = k12 ∇ 2 3 k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs ij          k1i k3j L̃213 + k3i k1j L̃231 k1i k2j L̃312 + k2i k1j L̃321 k2i k3j L̃123 + k3i k2j L̃132 ˆ6  .(6.9) = k12 ∇ + k22 + k32 k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs ij with Õ =  4 X  Pabc = p=1 5 X Op mabc = 3  4ν − 2p −6 2ν − p −3 cs s 3 cT T 3 K 9+p−4νs −2νT mabc p p=1 3 2 Γ(9 + p − 4νs − 2νT ) Γ(8 + p − 4νT − 2νs )    16 2ν − p − 8 4νs − 2p 3 − 3 cT T 3 3 K 8+p−4νs −2νT cs
−2 −1 , O1 = 1, O2 = ics 3 cT 3 K, O3 = −(ka kb + kb kc + kc ka ), O4 = −ika kb kc ,  − νT (ka kb + kb kc + kc ka ) + ka2 , mabc 4  3 −2 −1 = cs 3 cT 3 Kmabc − νT , mabc 1 , 2 2    3 2 = ka2 kb kc , = − νT ka kb kc + ka (kb + kc ) , mabc 5 2 , mabc = 1  (6.10) 26 Rabc = L1  3 − νT 2 X 5 Γ(8 + p − 2νT − 4νs ) Āabc p 16 4νs − 2p 2ν − p − 8 3 − 3 cs cT T 3 3 K 8+p−2νT −4νs p=1 + ( a2 Ys t1 3 2 − νT 6 h iu X Jˆq (k~a , k~b , k~c ) × q=1
kc2 3 2 − νs
Γ(7 + p − 2νT − 4νs ) 14 2ν − p − 7 4νs − 2p 3 − 3 cT T 3 3 K 7+p−2νT −4νs cs    (6.11)   h iu h iu h iu  3 3 −2 −1 Jˆ1 (k~a , k~b , k~c ) = − νs − νT , Jˆ2 (k~a , k~b , k~c ) = ics 3 cT 3 K Jˆ1 (k~a , k~b , k~c ) , 2 2       h iu 3 3 3 3 Jˆ3 (k~a , k~b , k~c ) = − − νs ka2 + − νT kb2 + − νs − νT {ka kb + kb kc + kc ka } , 2 2    2  2  h iu 3 3 3 2 2 Jˆ4 (k~a , k~b , k~c ) = − − i − νs ka kc + − νT kc ka + − νs ka2 kb 2 2 2   h iu  3 , + − νT kc2 kb + ka kb kc Jˆ1 (k~a , k~b , k~c )     h2  h iu  i u 3 3 Jˆ5 (k~a , k~b , k~c ) = ka2 kc2 + − νs ka2 kb kc + − νT ka kb kc2 , Jˆ6 (k~a , k~b , k~c ) = ika2 kb kc2 , 2 2       3 3 − 32 − 31 abc abc abc abc − νT , Ā2 = ics cT K Ā1 , Ā3 = − − νT ka kb + kb kc + kc ka + ka2 , Ā1 = 2    2 3 2 abc 2 abc Ā4 = ka kb kc − νT + ka (kb + kc ) , Ā5 = ka kb kc , 2   a2 Ys ka2 3 R̃abc = ka2 L1 Õ + − ν s Pabc , t1 kb2 2
5 L1 a2 Ys 32 − νs X abc Γ(8 + p − 4νs − 2νT ) Labc = L21 Õ − nq 16 4νs − 2p 2ν − p − 8 t1 3 − 3 cs cT T 3 3 K 8+p−4νs −2νT p=1
2 6 a4 Ys2 23 − νs X abc Γ(7 + p − 4νs − 2νT ) , dr + 14 7 4νs − 2p 2νT − p t21 kb2 kc2 7+p−4νs −2νT 3 − 3 3−3 c c K r=1 s T    3 1 1 − 23 − 31 abc abc abc n1 = − νs + 2 , n2 = ics cT Kn1 , 2 2 kb  ka        3 kc 1 kb 1 3 abc 2 n3 = − 2 + , + + + 2 − νs − νs ka (kb + kc ) 2 ka2 kb 2  kb kc  kc kb 3 , nabc = ka (kc + kb ), + − νs nabc = −i (kc + kb ) + ka 2 + 5 4 2 kb kc  2 3 −2 −1 abc = −(kb2 + kc2 + kb kc + ka kb + ka kc ), dabc = ika kb2 kc2 dabc = = ics 3 cT 3 Kdabc − νs , dabc 1 , d3 6 1 2 2         2 3 3 2 2 abc 2 2 2 4 2 2 k k + k k dabc = −i k k + , d = k k + k . k k + k (k + k + k k ) − ν − ν b c a a b b c b c s s b c 4 b c 5 b c c 2 2 (6.12) , 27 After using the basis transformation mentioned in equation(3.23) we get:   ′ ′′    ′ ′′   ′ ′′ ′′ ′′ ′ ′ ′′ ′   k1λ k3λ + k3λ k1λ k1λ k2λ + k2λ k1λ k2λ k3λ + k3λ k2λ ˆ1  Õδλ′ λ′′ , = ∇ + + k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs λ′     ′ ′′    ′ ′′   ′ ′′ ′ ′ ′′ ′′ ′ ′′ ˆ2 ∇ k1λ k3λ P213 + k3λ k1λ P231 k1λ k2λ P312 + k2λ k1λ P321 k2λ k3λ P123 + k3λ k2λ P132 ′ λ
 δλ′ λ′′ , = + + k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs cs 32 − νT   ′ ′′    ′ ′′   ′ ′′ ′′ ′′ ′′ ′ ′′ ′   k1λ k3λ R213 + k3λ k1λ R231 k1λ k2λ R312 + k2λ k1λ R321 k2λ k3λ R123 + k3λ6 k2λ R132 ˆ3  δλ′ λ′′ , , + + = cs  ∇ k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs λ′  ′ ′′  ′ ′′      ′ ′′ ′ ′′ ′′ ′′ ′′   k2λ k3λ R̃123 + k3λ k2λ R̃132 k1λ k3λ R̃213 + k3λ k1λ R̃231 k1λ k2λ R̃312 + k2 k1λ R̃321 ˆ4  δλ′ λ′′ , = k12 ∇ + k22 + k32 k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs λ′  ′ ′′  ′ ′′   ′ ′′ ′′  ′′   ′ ′ ′′  ′′   k1λ k3λ + k3λ k1λ k1λ k2λ + k2λ k1λ k2λ k3λ + k3λ k2λ ˆ5  Õδλ′ λ′′ , + k22 + k32 = k12 ∇ k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs λ′  ′ ′′  ′ ′′      ′ ′′ ′′ ′′ ′ ′ ′′ ′′   k1λ k3λ L̃213 + k3λ k1λ L̃231 k1λ k2λ L̃312 + k2λ k1λ L̃321 k2λ k3λ L̃123 + k3λ k2λ L̃132 ˆ6  δλ′ λ′′ , = k12 + k22 + k32 ∇ k1νT (k2 k3 )νs k2νT (k1 k3 )νs k3νT (k1 k2 )νs λ′ (6.13) where kiλ = ki where i = 1, 2, 3. Most surprisingly, the above coefficients are independent of λ due to no parity violation. D. Functions appearing in three tensor correlation The functional dependence of the co-efficients appearing in the context of three tensor correlation can be expressed as:  3 3 σ (1) ∆i1 j1 i2 j2 i3 j3 = Ni1 j1 ;ij Ni2 j2 ;jk Ni3 j3 ;ki c3s − νT [M123 + M132 + M213 + M231 + M312 + M321 ] , 12 2 YT (2) ∆i1 j1 i2 j2 i3 j3 = 2 Ni1 j1 ;ik Ni2 j2 ;jl Ni3 j3 ;ij [k3k k3l + k2k k2l + k1k k1l ] Q, 2cT YT (3) ∆i1 j1 i2 j2 i3 j3 = − 2 Ni1 j1 ;i3 j3 Ni2 j2 ;kl [k3k k3l + k2k k2l + k1k k1l ] Q (6.14) 2cT with Q= babc 3 babc 4 babc 5 babc 6 4 X 3  7 X Γ(9 + p − 6νT ) 3 abc Γ(6 + p − 6νT ) abc , babc b = babc , M = , b = − ν abc T p 2 1 1 K, 9+p−6νT 6+p−6νT K K 2 p=1 p=1 2    3 ka (kb + kc ) + kb2 + kc2 + kb kc + ka2 , − νT = 2 "   2 #   3 3 2 2 2 , − νT + ka (kb + kc + kb kc ) + kb kc (kb + kc ) − νT = ka (kb + kc ) 2 2 # "   2  2 2 3 3 2 2 2 ka (kb + kc + kb kc ) + kb kc + ka kb kc (kb + kc ) , − νT − νT = 2 2   3 = −ka2 kb2 kc2 . − νT ka kb kc [kb kc + ka (kb + kc )] , babc = 7 2 Op (6.15) 28 After using the basis transformation mentioned in equation(3.23) the helicity dependent functions are given by:  3 h ′ ′′ ′′′ i ′ ′′ ′′′ ′ ′′ ′′′ ′ ′′ ′′′ ′ ′′ ′′′ ′ ′′ ′′′ 3 σ (1) λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , + M321 + M231 + M213 + M312 + M132 M123 δλ1 λ′ δλ2 λ′′ δλ3 λ′′′ c3s − νT ∆λ1 λ2 λ3 = 12 2 h ′ ′′ ′′′ ′′ i ′ ′′ ′ YT (2) ∆λ1 λ2 λ3 = 2 δλ1 λ′ δλ2 λ′′ δλ3 λ′′′ k3λ k3λ + k2λ k2λ + k1λ k1λ Qλ , 2cT h ′′′ ′′ ′′ i ′′′ ′′ ′′′ YT (3) ∆λ1 λ2 λ3 = − 2 δλ′′ λ′ δλ2 λ1 δλ3 λ′′′ k3λ k3λ + k2λ k2λ + k1λ k1λ Q, (6.16) 2cT with 4 ′′′ Γ(9 + p − 6ν )
λ′ λ′′ λ′′′ Γ(6 + p − 6νT ) λ′′′ X T , Q = Opλ , 6+p−6ν 9+p−6νT T K K p=1 p=1  3
′ ′′ ′′′
λ′ λ′′ λ′′′
λ′ λ′′ λ′′′ 3 abc λ λ λ = b1 K, = babc , babc − νT 1 2 2  2 h ′ ′ ′′ ′′′ i ′ ′′′ ′′ ′′′ ′′ ′′′ ′′
λ λ λ 3 λ 2 λ 2 λ λ λ 2 λ λ λ ) , + (k k ) + k ) + (k ) + (k + k (k = babc − ν k T a c b c b c b 3 a "2    2 # n ′ ′′′ ′′ ′ ′′′ o ′′ ′′′ ′′ ′′′ ′′ ′′′ ′′
′ ′′ ′′′ 3 3 λ λ 2 λ abc λ λ λ λ λ λ λ λ 2 λ λ λ 2 λ = (ka ) (kb + kc ) b4 , − νT + ka ((kb ) + (kc ) + kb kc ) + kb kc (kb + kc ) − νT 2 2 " # n ′ 2 ′ ′′ ′′′ o 3 ′′′ ′′ ′′′ ′′ ′′′ ′′′ ′′ ′′
λ′ λ′′ λ′′′ 3 2 λ λ λ 2 λ λ λ 2 λ 2 + = babc ) k ) + (k k ) + k ) + (k ) ((k (k kaλ kbλ kcλ (kbλ + kcλ ) , − ν − ν T T 5 c b c b c b a 2 2   ′ ′′ ′′′ h ′′ ′′′ ′ ′′ ′′′ ′ ′′ ′′′ ′′′ i ′′′ ′′ ′′ ′ ′

3 λ λ λ λ λ λ = −(kaλ kbλ kcλ )2 , babc = − νT kaλ kbλ kcλ kbλ kcλ + kaλ (kbλ + kcλ ) , babc 7 6 2 ′′′ ′′′ ′′′ ′′′ ′′′ ′′′ ′′′ ′′′ ′′′ O1λ = 1, O2λ = iK λ , O3λ = −(kaλ kb + kbλ kc + kcλ ka ), O4λ = −ikaλ kb kc , (6.17) ′ ′′ ′′′ ′ ′′ ′′′ λ λ λ where k1 = λ k1 , k2 = λ k2 and k3 = λ k3 . ′ ′′ λ λ Mabc ′′′ λ = 7 X babc p E. Functions appearing in four scalar correlator 1. Contact interaction The functional dependence of the momentum dependent functions appearing in the context of contact interaction of four scalar correlation can be expressed as: 4  4 4 4 4 X X X 3 3 X 3 ki kj km + iG14 ki kj , G4 = iG1 ki2 + G1 G1 = − νs , G2 = iK̄G1 , G3 = G14 ki2 kj , 2 i>j>m=1 i>j=1 i=1 i6=j=1 4 X p 3 ki2 kj2 + G14 G5 = G1 1 4 G7 = G1 i>j=1 4 Y i>j>m=1 G9 = 4 Y i=1 4 X ki2 kj km + G1 i=1 i>j>m=1 p 2 ki2 kj2 km + G1 4 Y 4 Y i>j=1 i,j,m=1 3 4 ki2 kj2 km kn , G8 = iG1 i<j,m<n,i6=m,j6=n=1 4 Y ki2 , Z̄1 = 1, Z̄2 = iK̄, Z̄3 = 4 X p ki2 kj2 km + i ki , G6 = i G1 ki kj , Z̄4 = 4 Y i>j>m=1 4 Y ki2 kj km kn , i6=j>m>n=1 2 ki2 kj2 km kn , i>j>m6=n=1 4 Y ki kj km , Z̄5 = 4 Y ki i=1 (6.18) and h p p Ī(i, j; m, n) = K̄ 4 G1 Γ(10 − 6νs ) + K̄ 3 G1 (ki + kj + km + kn )Γ(11 − 6νs ) n p p p o 1 2 2 kn − iG14 km kn (km + kn ) − km kn G1 − ki kj G1 − (ki + kj )(km + kn ) G1 Γ(12 − 6νs ) − K̄ 2 km i n p o p p + K̄ ki kj (km + kn ) G1 − (ki + kj )km kn G1 Γ(13 − 6νs ) + km kn G1 Γ(14 − 6νs ) . (6.19) 29 2. Scalar exchange The functional dependence of the momentum dependent functions appearing in the context of scalar exchange contribution of four scalar correlation can be expressed as:
6 6 6 3 XX (−1)b+p−12νs (i(k4 + k5 + k6 ))b+p−6νs 2 − νs Sb (k1 , k2 , k3 )Sp (k4 , k5 , k6 ) Ξ1 (k1 , k2 , k3 , k4 , k5 , k6 ) := ν s (k5 k6 ) (k4 + k5 + k6 )4 b=0 p=0   3 3 ×Γ + b − 3νs Γ(3 + b + p − 6νs ) 2 F1REG 3 + b + p − 6νs ; + b − 3νs ; 2 2  (k1 + k2 + k3 ) 5 , + b − 3νs ; − 2 (k4 + k5 + k6 ) (6.20)
2 4 4 XX − νs (−1)2(m+n)−9νs (i(k4 + k5 + k6 ))1−m−n+6νs E (k , k , k )E (k , k , k ) Ξ2 (k1 , k2 , k3 , k4 , k5 , k6 ) := m 1 2 3 n 4 5 6 cS (k5 k6 )νs m=0 n=0 (7 + 2m − 6νs )(k4 + k5 + k6 )8   7 9 (k1 + k2 + k3 ) , × Γ(7 + m + n − 6νs ) 2 F1 7 + m + n − 6νs ; + m − 3νs ; + b − 3νs ; − 2 2 (k4 + k5 + k6 ) (6.21) 3 2
4 6 4 cS 32 − νs X X (−1)2(p+q)−12νs +3 (i(k4 + k5 + k6 ))1−p−q+6νs Ξ3 (k1 , k2 , k3 , k4 , k5 , k6 ) := S (k , k , k )E (k , k , k ) p 1 2 3 q 4 5 6 (k5 k6 )νs p=0 q=0 (k4 + k5 + k6 )6    3 3 ×Γ + p − 3νs Γ(5 + p + q − 6νs ) 2 F1REG 5 + p + q − 6νs ; + p − 3νs ; 2 2  (k1 + k2 + k3 ) 5 , + p − 3νs ; − 2 (k4 + k5 + k6 ) (6.22)
3 4 6 cS 32 − νs X X (−1)2(t+r)−12νs +3 (i(k4 + k5 + k6 ))1−t−r+6νs E (k , k , k )S (k , k , k ) t 1 2 3 r 4 5 6 ν (k5 k6 ) s t=0 r=0 (k4 + k5 + k6 )6    3 3 ×Γ + t − 3νs Γ(5 + t + r − 6νs ) 2 F1REG 5 + t + r − 6νs ; + t − 3νs ; 2 2  (k1 + k2 + k3 ) 5 + t − 3νs ; − 2 (k4 + k5 + k6 ) (6.23) 2 F1 [a ; b ; c ; d] REG where we use Regularized Hypergeometric function defined as: 2 F1 [a ; b ; c ; d] = . Additionally Γ [c] here we define two new sets of momentum dependent functions given by: 3 3   3 3 − νs , S1 (ka , kb , kc ) = −i(ka + kb + kc ) − νs , S0 (ka , kb , kc ) = 2 2     2    3 2 3 3 3 3 S2 (ka , kb , kc ) = − (ka2 + kb2 ) + − νs − νs ka kb − kc (ka + kb ) − νs − kc2 − νs , 2 2 2 2     2 2    2 3 3 3 3 2 2 2 S3 (ka , kb , kc ) = ika kb (ka + kb ) (ka + kb ) + − νs + ikc − νs − νs ka kb + ikc (ka + kb ) − νs , 2 2 2 2  2 2        3 3 3 3 S4 (ka , kb , kc ) = ka2 kb2 − νs + ka kb kc (ka + kb ) − νs + − νs kc2 (ka2 + kb2 ) + − νs ka kb , 2  2 2 2    3 3 S5 (ka , kb , kc ) = −ika2 kb2 kc − νs − ika kb kc2 (ka + kb ) − νs , S6 (ka , kb , kc ) = −ka2 kb2 kc2 , 2 2       3 3 3 − νs , E1 (ka , kb , kc ) = −i(ka + kb + kc ) − νs , E2 (ka , kb , kc ) = −ka kb − νs E0 (ka , kb , kc ) = 2 2  2
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