Priomordial non-Gaussian features from DBI Galileon inflation

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.

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 [1,2]. In this context the study of primordial non-Gaussian feature acts as a powerful computational tool to discriminate among inflationary models.
The DBI inflationary model [20], [21] is one of the most interesting possibilities to realize large non-Gaussianity [22] of the cosmic microwave background (CMB) temperature fluctuations. But this class of models has a generic problem which states that the non-Gaussian parameters derived from bispectrum and trispectrum analysis is highly sensitive to the tensor to scalar ratio [23][24][25][26] or in other simpler language one can say that it is very difficult to get the large tensor modes and large non-Gaussianity at the same time. This results in a tension between these two parameters from observational ground. To be specific, the parameter space of non-Gaussianity and tensor to scalar ratio as obtained from DBI models contradicts the results of Planck [2]. Very recently, a natural extension to the wide class of DBI models has been brought forward by tagging Galileon with the good old DBI model, resulting in so-called "DBI Galileon" [27][28][29] which, amidst its successes, results in unwanted degrees of freedom like ghosts, Laplacian and Tachyonic instabilities. In a recent paper [30] the present authors have demonstrated how these unwanted debris can be removed in spite of keeping all the good features of [27][28][29] intact by proposing DBI Galileon in D3 brane in the background of N =1,D=4 SUGRA derived from D4 brane in N =2,D=5 bulk SUGRA background. Starting from this model, In the present paper, our prime objective is to carry forward this rich structure of DBI Galileon as proposed in [30] in order to produce sufficient non-Gaussinanity and resolve the problem of DBI inflation by studying bispectrum and trispectrum calculated from three and four point correlation function originated through cosmological perturbation, working upon third and fourth order action in a most generalized situation which includes quasi-exponential type of behavior. Subsequently, we demonstrate that it is indeed possible to have a parameter space for both non-Gaussianity and tensor-to-scalar ratio (r) consistent with Planck data.
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 f N 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 g N L and τ N 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 shown that the violation of standard Suyama-Yamaguchi relation [33], [34] leading to the solution of the generic problem of non-Gaussian parameters in the context of DBI inflation.

II. THE BACKGROUND MODEL
For systematic development of the formalism, let us briefly review from our previous paper [30] 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 S (5) T otal = S (2.2) where T (4) is the D4 brane tension, α ′ is the Regge Slope, exp(−Φ) is the closed string dilaton and C 0 is the Axion.
Here γ (5) , B (5) and F (5) represent the determinant of the 5D induced metric (γ AB ) and the gauge fields (B AB , F AB ) respectively. Additionally here ν 0 and ν 4 represent the constants characterizing the interaction strength between D4-D4 brane. In the present context 5-dimensional coordinates X A = (x α , y), where y parameterizes the extra dimension compactified on the closed interval [−πR, +πR].
It is useful to introduce the 5D metric in conformal form and ds 2 4 = g αβ dx α dx β is FLRW counterpart. The parameter β determines the slope of the warp factor and R represents the compactification radius. Applying dimensional reduction technique via S 1 /Z 2 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 [30]: , .
where α (4) ,l 1 ,l 3 ,l 4 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 := − 1 2 g µν ∂ 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 [30]: where D 0 = 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 [30]: whereg 1 represents the effective 4D Gauss-Bonnet coupling dependent function on FLRW background and Λ (4) is the 4D cosmological constant.

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: where can be calculated from the second order action [30] S (4) HereC i (i = 1, 2, 3, ......, 8) are dimensionless co-efficients defined as: and the co-efficient of δL2 δζ | 1 involves spatial and time derivative in equation(3.1) is defined by the following expression: In this context R → 0 as k → 0 at large scale. Additionally It is important to mention here that for scalar and tensor modes ghosts and Laplacian instabilities can be avoided iff c 2 s > 0, Y s > 0. Throughout the paper we use the required parameters from [30] to compute the bispectrum and trispectrum.
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: where the total Hamiltonian in the interaction picture can be expressed in terms of the third order Lagrangian density as (H int (η)) ζζζ = with m = (S[scalar], T [tensor]). Moreover following the momentum dependent ansatz given in [15], [32] the bispectrum B ζζζ ( k 1 , k 2 , k 3 ) is defined as: where the symbol ; 1 is used for three scalar correlation. Here A ζζζ ( k 1 , k 2 , k 3 ) is the shape function for bispectrum and P 2 ζ 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 The Power spectra for scalar (P ζ (k)) and tensor modes (P T (k)) at the horizon crossing can be written as: (3.11) Here for tensor modes we use (P T (k)) ij;kl = |u h (η, k)| 2 N ij;kl , P T (k) = (P T (k)) ij;ij with the following helicity/spin dependent normalization factor: N ij;kl = λ e λ ij ( k)e †(λ) kl ( k). In this context f N L represents the non linear parameter carrying the signature of primordial non-Gaussianities of the curvature perturbation in bispectrum. The explicit form of f N L characterizing the bispectrum can be expressed as: where the functional form of the momentum dependent functions I i (x)∀i are explicitly mentioned in the Appendix B.1. From the coefficients of I i (ν) with i = 1, 3, 5, 7, 8 it seems that the non-Gaussian parameter f N 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 I i (n ζ − 1) with i = 1, 3, 6 and I 4 (ν) 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) = 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 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 = k 1 + k 2 + k 3 , x = (n ζ − 1,ν) and with four new constants ρ 3 , ρ 4 , T 3 , T 4 . In the present context s S V =ċ s Hcs is an extra slow-roll parameter appearing due to the sound speed, c s = 1 as defined in [30]. For the numerical estimation we have further used the equilateral configuration (k 1 = k 2 = k 3 = k and K = 3k) in which the non-linear parameter f N L can be simplified to the following form as: (3.14) Now using the tensor-to-scalar ratio at the pivot scale k * : the sound speed c s can be eliminated from the equation(3.14) also.
Here s T V =ċ T HcT appearing due to the sound speed, c T = 1. See [30] for the details. The numerical value of f equil N L;1 in the equilateral limit is obtained from our set up as 4 < f equil N L;1 < 7 within the window for tensor-to-scalar ratio 0.213 < r < 0.250 [30]. This is extremely interesting result as it is different from other class of DBI models. The most impressing fact is that the upper bound of f equil N L;1 in the quasi-exponential limit are in good agreement with Planck [2] 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: where the dimensionful coefficients F i (i = 1, 2.....7) are defined as: where we use σ =φXG 5X . 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: where the total Hamiltonian in the interaction picture can be expressed in terms of the third order Lagrangian where the symbol ; 2 stands for one scalar two tensor correlation. Here (A ζhh ) ij;kl ( k 1 , k 2 , k 3 ) is the shape function for bispectrum and the polarization indices . We adopt the following normalization depending on the polarization in which we are interested: (3.20) Consequently f N L;2 u ij;kl represents the non-linear parameter which carries the signature of primordial non-Gaussianities of the one scalar two tensor interaction. The explicit form of f N L;2 u ij;kl characterizing the bispectrum can be calculated as: The overall normalization factor for three types of polarization can be expressed as: 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: in terms of which the one scalar two tensor correlation is defined as: where the cross reduced bispectrum is defined as: Applying the basis transformation the explicit form of f N L;2 (λ2;λ3) characterizing the crossed bispectrum can be u;λ2;λ3 equil (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: where the dimensionful coefficients Y i (i = 1, 2.....6) are defined as: 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: where the total Hamiltonian can be expressed in terms of the third order Lagrangian density as Here the cross bispectrum {B ζζh } kl is defined as: where (A ζζh ) kl is the two scalar one tensor correlation shape function and the symbol ; 3 represents two scalar one tensor correlation. Consequently the non-linear parameter f N L;3 u kl can be expressed as: where the functional dependence of the co-efficients ∇ v ij ∀v are explicitly mentioned in the Appendix B.3. In For quasi-exponential limit the overall normalization factor for three types of polarization can be expressed as: As mentioned in the previous sub-section, performing basis transformation cross bispectrum for two scalars and one tensor can be expressed as: where we have used the following parameterization: The polarized non-Gaussian parameter for two scalar and one tensor mode f N L;3 u;λ can be rewritten as: where all the co-efficients ∇ v λ ′ ∀v after basis transformation are explicitly written in the Appendix B.3. In the equilateral limit the expression for the non-Gaussian parameter (f N L ) reduces to the following form:

D. Three tensor correlation
The interactions involving three tensors are given by the following third order action: 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: where the total Hamiltonian is expressed in terms of the third order Lagrangian density as In this context the bispectrum for three tensor correlation can be expressed as: where the symbol ; 4 represents three tensor correlation. Also, the non-Gaussian parameter is given by: For quasi-exponential limit the overall normalization factor for three types of polarization can be expressed as: :u=1(E-mode) 64 :u=2(E B mode) 1024 :u=3(B-mode).

(3.42)
After performing basis transformation the relevant three point correlation function for three tensor interaction can be expressed in terms of bispectrum as: where the the non-linear parameter is given by: In the equilateral limit we have: 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: where S CI , S SE and S GE 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: where the co-efficientsŪ i (i = 1, 2, 3) for effective DBI Galileon are defined as: whereK(φ, X) andG(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: where in the interaction picture the Hamiltonian can be written as: Here following the ansatz used in [7] the trispectrum T CI ζ ( k 1 , k 2 , k 3 , k 4 ) for contact interaction is defined as: where and τ CI N L and g CI N L are the two non linear parameters which carry the signatures of primordial non-Gaussianities of the curvature perturbation in trispectrum analysis. By knowing τ CI N L the other parameter g CI N L can be calculated by making use of the following relation [38]: whereK = k 1 + k 2 + k 3 + k 4 . So, there is only one independent piece of information, namely τ CI N 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 = j) by Θ ij then subject to the constraint Cos(Θ 1 )+Cos(Θ 2 )+Cos(Θ 3 ) = −1comes from the conservation of momentum. Additionally we have used (4.10) The explicit form of τ CI N L characterizing the trispectrum obtained from contact interaction can be expressed for our model as:  17 (4.11) where the functional dependence of the momentum dependent functions G i ∀i, Z q ∀q andĪ(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 that the non-Gaussian parameter τ CI N 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 behavior of the τ CI N L changes. Further, using the equilateral configuration (k 1 = k 2 = k 3 = k 4 = k andK = 4k) and incorporating the contribution from the maximum shape of the trispectrum (Cos(Θ 1 ) = Cos(Θ 2 ) = Cos(Θ 3 ) = − 1 3 and k ij (f or i < j) = 2k √ 3 ) the non linear parameter can be expressed as: . (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: where the co-efficients (A, B) are defined as: (4.14) Using in-in procedure the four point correlation function for quasi-exponential limit can be expressed in the following form: (4.15) where in the interaction picture the Hamiltonian can be written in terms of the third order Lagrangian density Hence following the ansatz used in [7] the trispectrum T SE ζ ( k 1 , k 2 , k 3 , k 4 ) is defined as: 16) where τ SE N L and g SE N L are the two non linear parameters which carry the signatures of primordial non-Gaussianities of the curvature perturbation obtained from scalar exchange contribution in trispectrum analysis. By knowing τ SE N L the other parameter g SE N L can be calculated by making use of the following relation [38]: whereK = k 1 + k 2 + k 3 + k 4 . The explicit form of τ SE N L characterizing the scalar exchange trispectrum can be expressed for our model as: where the momentum dependent functions Ξ i ∀i are mentioned in the Appendix. Further, using the equilateral configuration the non-Gaussian parameter from scalar exchange contribution can be expressed as:

C. 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: 20) where Y 1 = Y S c 2 S . Using in-in procedure the four point correlation function both for quasi-exponential limit can be expressed in the following form: (4.21) where in the interaction picture the Hamiltonian can be written in terms of the third order Lagrangian density as: Here following the ansatz used in [7] the trispectrum T GE ζ ( k 1 , k 2 , k 3 , k 4 ) obtained from the graviton exchange contribution is defined as: 22) where τ GE N L and g GE N L are the two non linear parameters which carry the signatures of primordial non-Gaussianities of the curvature perturbation in trispectrum analysis. By knowing τ GE N L the other parameter g GE N L can be calculated by making use of the following relation [38]: whereK = k 1 + k 2 + k 3 + k 4 . The explicit form of τ GE N L characterizing the trispectrum obtained from the graviton exchange contribution can be expressed for our model as:
24) The momentum dependent functions ϑ abcd (η ⋆ ) are given in the Appendix. Here to write equation(4.24) we have used the fact that the exchange momentum dependent polarization tensor ǫ λ ij ( k ab ) 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: 25) where we define lim η⋆→0 ϑ abcd (η ⋆ ) :=θ abcd . There are divergent contributions in the limit η * → 0 appear with a logarithmic dependence on the momenta, but the additive cumulative contribution of I abcd and I cdab give rise to a finite contribution at late times.
To represent Eq (4.25) in a simpler form, let us start with the polarization sum s ǫ s ij ( k 12 )ǫ s lm ( k 34 )k i 1 k j 2 k l 3 k m 4 in terms of the relative angles between the k a and k 12 . The polarization tensors ǫ s ij can be rewritten as where e and ē are orthogonal unit vectors perpendicular to exchange momentum vector k 12 . It is convenient to write the momentum vector k a in a spherical polar coordinate system having { e, ē, k 12 ≡ k 12 /k 12 } as basis. In this coordinate system one can express the momentum vector as: k a = k a (sin θ a cos φ a , sin θ a sin φ a , cos θ a ) , where cos θ a ≡ k a · k 12 and cosφ a ≡ k a · e. This implies with an identical relation holding for ǫ + ij k i 3 k j 4 and ǫ × ij k i 3 k j 4 which will contribute to the polarization sum also. Since the projections of the momentum vectors k 1 and k 2 ( similarly for k 3 and k 4 ) on the plane orthogonal to exchange momentum vector k 12 ( k 34 ) have the same amplitude but opposite directions. Consequently we have two additional sets of constraint relationships given by: k 2 sin θ 2 = k 1 sin θ 1 and φ 2 = φ 1 + π, k 4 sin θ 4 = k 3 sin θ 3 and φ 4 = φ 3 + π. (4.28) Using these relations we get: where we define a new angular coordinate Υ ab,cd ≡ φ a − φ c with a = 1, (b, c) = 2, 3, 4, d = 3, 4 and b > a, d > c, a = b = c = d, which physically represents the angle between the projections of the two momentum vectors k a and k c on the plane orthogonal to k 12 . Alternatively this can be interpreted as the angle between the two planes formed by the pair of momentum vectors { k 1 , k 2 } and { k 3 , k 4 }. Thus, the expression for the non-Gaussian parameter calculated from the graviton exchange contribution from the trispectrum can be simplified to the following expression:

13
(k 1 k 3 k 2 k 4 ) 2νs cos 2Υ 13,24 · θ 1324 +θ 2413 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 exchange interaction the numerical value of τ equil N L in the equilateral limit is obtained from our set up as 48 < τ equil N 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. Consequently this model is consistent with Planck [2].

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: 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: Here using k 12 → 0, θ 1 , θ 3 → π the polarization sum appearing in Eq (5.2) can be simplified to the following expression as: 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 (k 12 << k 1 ≈ k 2 , k 3 ≈ k 4 ) reduces to the following expression: 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: where n is a label for individual states or particle number within the momentum eigenspace. 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: As the second term in Eq (5.6) is always positive definite we conclude that: . Further using this result in quasi-exponential limit we get: lim q→0 k1 Hence using Eq (5.7) finally we get: resulting in a generic outcome of DBI Galileon inflation, viz, whereτ N L andf N L are used to represent soft limits of the three and four point correlation functions. This relation directly confirms the partial violation of standard Suyama-Yamaguchi relation [33], [34], [35]τ N L = 36 25 f N L 2 thereby solving the generic problem that it is difficult to get the large tensor-to-scalar ratio and large non-Gaussianity at the same time in the context of DBI inflation. The other novel aspects of the violation of well known consistency relations in the context of single field inflation has been studied in [39,40].  ] u;(λ 1 λ 2 λ 3 ) ) parameters related to the primordial bispectrum for A=1 (three scalar), 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 λ 1 , λ 2 and λ 3 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.

VI. SUMMARY AND OUTLOOK
In this article we have explored the 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 f N L and τ N L for equilateral type of non-Gaussian configurations in the nontrivial polarization modes. Hence we have demonstrated the partial violation of the Suyama-Yamaguchi relation leading to a better observational bound for the above mentioned non-linear parameters as estimated from WMAP9 [1] dataset. Nevertheless, the results confront well with the Planck [2] results. The most significant outcome of our model is that it solves the generic problems of DBI inflation by introducing Galileon in the present setup. The detectable features of primordial non-Gaussianity lead to the conclusion that this type of models can be directly confronted with Planck data.
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 [41], [42] 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.
Additionally in the squeezed limit these functions are reduced to the following expressions:

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:  u ij;kl = k 1m k 1m ′ X 123 +X 132 + k 2m k 2m ′ X 231 +X 213 + k 3m k 3m ′ X 312 +X 321 N u ij,kl N u mn,m ′ n .