Cosmic Microwave Background from Effective Field Theory ^†

Abstract

In this work, we study the key role of generic Effective Field Theory (EFT) framework to quantify the correlation functions in a quasi de Sitter background for an arbitrary initial choice of the quantum vacuum state. We perform the computation in unitary gauge, in which we apply the Stückelberg trick in lowest dimensional EFT operators which are broken under time diffeomorphism. In particular, using this non-linear realization of broken time diffeomorphism and truncating the action by considering the contribution from two derivative terms in the metric, we compute the two-point and three-point correlations from scalar perturbations and two-point correlation from tensor perturbations to quantify the quantum fluctuations observed in the Cosmic Microwave Background (CMB) map. We also use equilateral limit and squeezed limit configurations for the scalar three-point correlations in Fourier space. To give future predictions from EFT setup and to check the consistency of our derived results for correlations, we use the results obtained from all classes of the canonical single-field and general single-field $P(X,\varphi )$ model. This analysis helps us to fix the coefficients of the relevant operators in EFT in terms of the slow-roll parameters and effective sound speed. Finally, using CMB observations from Planck we constrain all these coefficients of EFT operators for the single-field slow-roll inflationary paradigm.

1. Introduction

The basic idea of Effective Field Theory (EFT) is very useful in many branches in theoretical physics including particle physics [1,2], condensed matter physics [3], gravity [4,5], cosmology [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and hydrodynamics [27,28]. In a more technical ground, EFT framework is an approximated model-independent version of the underlying physical theory which is valid up to a specified cut-off scale at high energies, commonly known as UV cut-off scale (${\Lambda }_{\mathrm{UV}}$), which is in usual practice fixed at the Planck scale $M_p$. EFT prescription deal with all possible relevant and irrelevant operators allowed by the underlying symmetry in the effective action and all the higher-dimensional non-renormalizable operators are accordingly suppressed by the UV cut-off scale (${\Lambda }_{\mathrm{UV}}\sim M_p$). There are two possible approaches within the framework of quantum field theory (QFT) using which one can explain the origin of EFT, which are appended below:

• Top-down approach:
  In this case, the usual idea is to start with a UV complete fundamental QFT framework which contain all possible degrees of freedom. Furthermore, using this setup one can finally derive the EFT of relevant degrees of freedom at low energy scale ${\Lambda }_s<{\Lambda }_{\mathrm{UV}}\sim M_p$ by doing path integration over all irrelevant field contents [12,14]. To demonstrate this idea in a more technical ground let us consider a visible sector light scalar field $\varphi$ which has a very small mass $m_{\varphi }<{\Lambda }_{\mathrm{UV}}\sim M_p$ and heavy scalar fields ${\Psi }_i\forall i=1,2,\cdots ,N$ with mass $M_{{\Psi }_i}>{\Lambda }_{\mathrm{UV}}\sim M_p$, in the hidden sector of the theory. the representative action of the theory is described by the following action [12,14]:

  $$\begin{array}{c}\hfill S[\varphi ,{\Psi }_i,g_{\mu \nu }]=\int d^{4}x\sqrt{-g}\left[\frac{M_p^2}{2}R+{\mathcal{L}}_{\mathrm{vis}}\left[\varphi \right]+\sum _{i=1}^N{\mathcal{L}}_{\mathrm{hid}}^{\left(i\right)}\left[{\Psi }_i\right]+\sum _{j=1}^N{\mathcal{L}}_{\mathrm{int}}^{\left(j\right)}[\varphi ,{\Psi }_j]\right],\end{array}$$

  (1)

  where $g_{\mu \nu }$ is the classical background metric, ${\mathcal{L}}_{\mathrm{vis}}\left[\varphi \right]$ is the Lagrangian density of the visible sector light field, ${\mathcal{L}}_{\mathrm{hid}}^{\left(i\right)}\left[{\Psi }_i\right]\forall i=1,2,\cdots ,N$ is the Lagrangian density of the hidden-sector heavy field and ${\mathcal{L}}_{\mathrm{int}}^{\left(j\right)}[\varphi ,{\Psi }_j]\forall j=1,2,\cdots ,N$ is the Lagrangian density of the interaction between hidden-sector and visible-sector field. Furthermore, using Equation (1) one can construct an EFT by performing path integration over the contributions from all hidden-sector heavy fields and all possible high-frequency contributions as given by:

  $$\begin{array}{c}\hfill S_{EFT}\left[\varphi ,g_{\mu \nu }\right]=-iln\left[\prod _{j=1}^N\int \left[\mathcal{D}{\Psi }_j\right]e^{iS[\varphi ,{\Psi }_j,g_{\mu \nu }]}\right]=-i\sum _{j=1}^{N}ln\left[\int \left[\mathcal{D}{\Psi }_j\right]e^{S[\varphi ,{\Psi }_j,g_{\mu \nu }]}\right].\end{array}$$

  (2)

  Finally, one can express the EFT action in terms of the systematic series expansion of visible sector light degrees of freedom and classical gravitational background as [12,14]:

  $$\begin{array}{c}\hfill S_{EFT}\left[\varphi ,g_{\mu \nu }\right]=\int d^{4}x\sqrt{-g}\left[\frac{M_p^2}{2}R+{\mathcal{L}}_{\mathrm{vis}}\left[\varphi \right]+\sum _{\gamma }\sum _{j=1}^N{\mathcal{C}}_{\gamma }^{\left(j\right)}\left(g_c\right)\frac{{\tilde{\mathcal{O}}}_{\gamma }^{\left(j\right)}\left[\varphi \right]}{M_{{\Psi }_j}^{{\Delta }_{\gamma }-4}}\right],\end{array}$$

  (3)

  where ${\mathcal{C}}_{\gamma }^{\left(j\right)}\left(g_c\right)\forall \gamma ,\forall j=1,2,\cdots ,N$ represent dimensionless coupling constants which depend on the parameter $g_c$ of the UV complete QFT. Also ${\tilde{\mathcal{O}}}_{\gamma }^{\left(j\right)}\left[\varphi \right]\forall \gamma ,\forall j=1,2,\cdots ,N$ represent ${\Delta }_{\gamma }$ mass dimensional local EFT operators suppressed by the scale $M_{{\Psi }_j}^{{\Delta }_{\gamma }-4}$. In this connection one of the best possible example of UV complete field theoretic setup is string theory from which one can derive an EFT setup at the string scale ${\Lambda }_s$ which is identified with $M_{{\Psi }_j}$ in Equation (3).
• Bottom-up approach:
  In this case, the usual idea is to start with a low-energy model-independent effective action allowed by the symmetry requirements. Using such a setup, the prime job is to find out the appropriate UV complete field theoretic setup allowed by the underlying symmetries [12,14]. This identification allows us to determine the coefficients of the EFT operators in terms of the model parameters of UV complete field theories. In this paper we follow this approach to write down the most generic EFT framework using which we describe the theory of quantum fluctuations observed in CMB around a quasi de Sitter inflationary background solution of Einstein’s equations.

In this paper, our prime objective is to compute the expressions for the cosmological two- and three-point correlation functions in unitary gauge using the well-known Stückelberg trick [29,30] along with the arbitrary choice of initial quantum vacuum state. The working principle of the Stückelberg trick in quasi de Sitter background is to break the time diffeomorphism symmetry to generate all the required quantum fluctuations observed in CMB. This is exactly same as applicable in the context of $SU\left(N\right)$ non-abelian gauge theory to describe the spontaneous symmetry breaking. In the present context the scalar modes which are appearing from the quantum fluctuation exactly mimic the role of Goldstone mode as appearing in $SU\left(N\right)$ non-abelian gauge theory. After breaking the time diffeomorphism in the unitary gauge, scalar Goldstone-like degrees of freedom are eaten by the metric. In unitary gauge, to write a most generic EFT in terms of operators which breaks time diffeomorphism symmetry, the following contributions play significant roles in quasi de Sitter background:

• Polynomial powers of the time fluctuation of the component in the metric, $g^{00}$ such as, $\delta g^{00}=g^{00}+1$,
• Polynomial powers of the time fluctuation in the extrinsic curvature at constant time surfaces, $K_{\mu \nu }$ such as, $\delta K_{\mu \nu }=\left(K_{\mu \nu }-a^{2}H{h}_{\mu \nu }\right)$, where a is the scale factor in quasi de Sitter background.

Construction of EFT action using the Stückelberg trick also allows us to characterize all the possible contributions to the model-independent simple versions of field theoretic framework based on the models of the inflationary paradigm described by single field, where the observables are constrained by the CMB observation appearing from Planck data. It is important to note that this idea of constructing EFT action using the Stückelberg trick can also be generalized to the EFT framework guided by multiple number of scalar fields as well.

The main highlighting points of this paper are appended below pointwise:

• We have presented all the results by restricting up to all possible contributions coming from the two derivative terms in the metric which finally give rise to a consistently truncated EFT action1. Consequently, we get consistent predictions for single-field slow-roll [31,32,33,34,35,36,37,38,39,40,41,42] and Generalized Single-Field $P(X,\varphi )$ models of inflation [43,44,45,46,47,48,49,50,51,52,53,54,55,56]. In earlier works various efforts are made to derive cosmological three-point correlation functions by writing a consistent EFT action in the similar theoretical framework. However, the earlier results are not consistent with the single-field slow-roll inflation with effective sound speed $c_S=1$ as it predicts vanishing three-point correlation function for scalar fluctuations. See ref. [6] for more details. The main reason for this inconsistency was ignoring specific contributions from the fluctuation in the EFT action, which give rise to improper truncation.
• We have computed the analytical expression for the two-point and three-point correlation function for the scalar fluctuation in quasi-de Sitter inflationary background in the presence of generalized initial quantum state. Also, for the first time we have presented the result for two-point correlation function for the tensor fluctuation in this context. To simplify our results we have also presented the results for Bunch–Davies vacuum and $\alpha ,\beta$ vacuums2.
• We have presented the exact analytical expressions for all the coefficients of EFT operators for single-field slow-roll and Generalized Single-Field $P(X,\varphi )$ models of inflation in terms of the time-dependent slow-roll parameters as well the parameters which characterize the generalized initial quantum state. To give numerical estimates we have further presented the results for Bunch–Davies vacuum and $\alpha ,\beta$ vacuums.

This paper is organized as follows. In Section 2, we discuss the overview of the EFT framework under consideration, which includes the construction of the EFT action under broken time diffeomorphism in quasi de Sitter background. In Section 3, we derive the expression for the two-point correlation function from EFT using scalar and tensor mode fluctuation. Furthermore, in Section 4, we derive the expression for the scalar three-point function from EFT using scalar mode fluctuation in equilateral and squeezed limit configurations. After that, in Section 5, we derive the exact analytical expressions for coefficients of EFT operators for both single-field slow-roll inflation and generalized single-field $P(X,\varphi )$ models of inflation. Finally, we conclude in Section 6 with some future prospects of the present work.

2. Overview of EFT

2.1. Construction of the Generic EFT Action

In this section, our motivation is to construct the most generic EFT action in the background of quasi de Sitter space. Before going into the further technical details it is important to note that the method of implementing cosmological perturbation using a scalar field is different compared to the generic EFT framework. However, the underlying connection can be explained by interpreting the scalar (inflaton) field as a scalar under all space-time diffeomorphisms in General Relativity:

$$\mathbf{Space}-\mathbf{time}\mathbf{diffeomorphism}:x^{\mu }⟹x^{\mu }+{\xi }^{\mu }(t,\mathbf{x})\forall \mu =0,1,2,3.$$

(4)

Consequently, in the cosmological perturbation the scalar field $\delta \varphi$ transforms like a scalar under the operation of spatial diffeomorphisms; on the other hand, it transforms in non-linear fashion with respect to time diffeomorphisms. The space and time diffeomorphic transformation rules are appended bellow:

$$\begin{array}{cc}\hfill \mathbf{Spatial}\mathbf{diffeomorphism}:t& ⟹t,x^i⟹x^i+{\xi }^i(t,\mathbf{x})\forall i=1,2,3⟶\delta \varphi ⟹\delta \varphi ,\hfill \\ \hfill \mathbf{Time}\mathbf{diffeomorphism}:t& ⟹t+{\xi }^0(t,\mathbf{x}),x^i⟹x^i\forall i=1,2,3⟶\delta \varphi ⟹\delta \varphi +{\dot{\varphi }}_0\left(t\right){\xi }^0(t,\mathbf{x}).\hfill \end{array}$$

(5)

Here ${\xi }^0(t,\mathbf{x})$ and ${\xi }^i(t,\mathbf{x})\forall i=1,2,3$ are the diffeomorphism parameter. In this context one can choose a specific gauge in which we set the background scalar degrees of freedom as, $\varphi (t,\mathbf{x})={\varphi }_0\left(t\right),$ which is consistent with the requirement that the perturbation in the scalar field vanishes:

$$\mathbf{Unitary}\mathbf{gauge}\mathbf{fixing}\Rightarrow \delta \varphi (t,\mathbf{x})=0,$$

(6)

In cosmological perturbation theory this is known as unitary gauge in which all degrees of freedom are preserved in the metric of quasi de Sitter space. This phenomenon is analogous to the spontaneous symmetry breaking as appearing in the context of $SU\left(N\right)$ gauge theory where the Goldstone mode transform in a non-linear fashion and destroyed by the $SU\left(N\right)$ gauge boson in unitary gauge to give a massive spin 1 degrees of freedom after symmetry breaking. In an alternative way one can present the framework of EFT by describing cosmological perturbation theory during inflation where time diffeomorphisms are realized in non-linear fashion.

Now to construct a most general structure of the EFT action suitable for the inflationary paradigm we need to follow the step appended below:

• One must write down the EFT operators that are functions of the metric $g_{\mu \nu }$. Here one of the possibilities is Riemann tensor.
• Also the EFT operators are invariant under the linearly realized time-dependent spatial diffeomorphic transformation:

  $$\mathbf{Spatial}\mathbf{diffeomorphism}:t⟹t,x^i⟹x^i+{\xi }^i(t,\mathbf{x})\forall i=1,2,3.$$

  (7)

  For an example, one can consider an EFT operator constructed by $g^{00}$ or its polynomials without derivatives which transform like a scalar under Equation (7).
• Due to the reduced symmetry of the physical system many more extra contributions are allowed in the EFT action.
• In the EFT action one can also allow geometrical quantities in a preferred space-time slice. For example, one can consider the extrinsic curvature $K_{\mu \nu }$ of surfaces at constant time, which transform like a tensor under Equation (7).

Consequently, the most general EFT action can be written in terms of all possible allowed operators by the space-time diffeomorphism as [6,20]:

$$\begin{array}{cc}\hfill S=\int & d^{4}x\sqrt{-g}\left[\frac{M_p^2}{2}R+M_p^2\dot{H}{g}^{00}-M_p^2\left(3H^2+\dot{H}\right)+{\displaystyle \sum _{n=2}^{\infty }}\frac{M_n^4\left(t\right)}{n!}{\left(\delta g^{00}\right)}^n\right.\hfill \\ & \left.-{\displaystyle \sum _{q=0}^{\infty }}\frac{{\overline{M}}_1^{3-q}\left(t\right)}{(q+2)!}\delta g^{00}{\left(\delta K_{\mu }^{\mu }\right)}^{q+1}-{\displaystyle \sum _{m=0}^{\infty }}\frac{{\overline{M}}_2^{2-m}\left(t\right)}{(m+2)!}{\left(\delta K_{\mu }^{\mu }\right)}^{m+2}-{\displaystyle \sum _{m=0}^{\infty }}\frac{{\overline{M}}_3^{2-m}\left(t\right)}{(m+2)!}{\left[\delta K\right]}^{m+2}+\cdots \right].\hfill \end{array}$$

(8)

where the dots stand for higher-order fluctuations in the EFT action which contains operators with more derivatives in space-time metric. Here we use the following sets of definitions for extrinsic curvature $K_{\mu \nu }$, unit normal $n_{\mu }$ and induced metric $h_{\mu \nu }$:

$$\begin{array}{ccc}\hfill K_{\mu \nu }& =& h_{\mu }^{\sigma }{\nabla }_{\sigma }{n}_{\nu }=\frac{{\delta }_{\mu }^0{\partial }_{\nu }{g}^{00}+{\delta }_{\nu }^0{\partial }_{\mu }{g}^{00}}{2{(-g^{00})}^{3/2}}+\frac{{\delta }_{\mu }^0{\delta }_{\nu }^0{g}^{0\sigma }{\partial }_{\sigma }{g}^{00}}{2{(-g^{00})}^{5/2}}-\frac{g^{0\rho }\left({\partial }_{\mu }{g}_{\rho \nu }+{\partial }_{\nu }{g}_{\rho \mu }-{\partial }_{\rho }{g}_{\mu \nu }\right)}{2{(-g^{00})}^{1/2}},\hfill \\ \hfill h_{\mu \nu }& =& g_{\mu \nu }+n_{\mu }{n}_{\nu },n_{\mu }=\frac{{\partial }_{\mu }t}{\sqrt{-g^{\mu \nu }{\partial }_{\mu }t{\partial }_{\nu }t}}=\frac{{\delta }_{\mu }^0}{\sqrt{-g^{00}}}.\hfill \end{array}$$

(9)

Here $\delta K_{\mu \nu }$ represents the variation of the extrinsic curvature of constant time surfaces with respect to the unperturbed background FLRW metric in quasi de Sitter space-time:

$$\begin{array}{ccc}\hfill \delta g^{00}& =& g^{00}+1,\delta K_{\mu \nu }=K_{\mu \nu }-a^{2}H{h}_{\mu \nu }.\hfill \end{array}$$

(10)

Additionally, we have used a shorthand notation $\left[\delta K\right]$ to define the following tensor contraction rule useful to quantify the EFT action [20]:

$$\begin{array}{ccc}\hfill {\left[\delta K\right]}^{m+2}& =& \delta K_{{\mu }_2}^{{\mu }_1}\delta K_{{\mu }_3}^{{\mu }_2}\delta K_{{\mu }_4}^{{\mu }_3}\cdots \delta K_{{\mu }_{m+2}}^{{\mu }_{m+1}}\delta K_{{\mu }_1}^{{\mu }_{m+2}}.\hfill \end{array}$$

(11)

Before going into the further details let us first point out the few important characteristics of the EFT action which are appended bellow:

• In the EFT action the operators $M_p^2\dot{H}{g}^{00}$ and $M_p^2\left(3H^2+\dot{H}\right)$ are completely specified by the Hubble parameter $H\left(t\right)$ which is the solution of Friedman’s equations in unperturbed background.
• The rest of the contributions in EFT action captures the effect of quantum fluctuations, which are characterized by the perturbation around the background FLRW solution of all UV complete theories of inflation.
• The coefficients of the operators appearing in the EFT action are in general time-dependent.

Now as we are interested to compute the two- and three-point correlation function, we have restricted to the following truncated EFT action [6,20]:

$$\begin{array}{cc}\hfill S=\int d^{4}x\sqrt{-g}& {\displaystyle \left[\frac{M_p^2}{2}R+M_p^2\dot{H}{g}^{00}-M_p^2\left(3H^2+\dot{H}\right)+\frac{M_2^4\left(t\right)}{2!}{\left(g^{00}+1\right)}^2+\frac{M_3^4\left(t\right)}{3!}{\left(g^{00}+1\right)}^3\right.}\hfill \\ & \left.{\displaystyle -\frac{{\overline{M}}_1^3\left(t\right)}{2}\left(g^{00}+1\right)\delta K_{\mu }^{\mu }-\frac{{\overline{M}}_2^2\left(t\right)}{2}{\left(\delta K_{\mu }^{\mu }\right)}^2-\frac{{\overline{M}}_3^2\left(t\right)}{2}\delta K_{\nu }^{\mu }\delta K_{\mu }^{\nu }}\right].\hfill \end{array}$$

(12)

where we have considered the terms in two derivatives in the metric3.

2.2. EFT as a Theory of Goldstone Boson

2.2.1. Stückelberg Trick I: An Example from $SU\left(N\right)$ Gauge Theory with Massive-Gauge Boson in Flat Background

In the unitary gauge, the EFT action consists of graviton mode, two helicities, and scalar mode, respectively. In this context first we apply a broken time diffeomorphic transformation on the Goldstone boson. As a result, $SU\left(N\right)$ gauge symmetry [6,58] is non-linearly realized in the framework of EFT. This mechanism is commonly known as the Stückelberg trick. Let us mention two crucial roles of the Stückelberg trick in gauge theory:

• Using this trick in $SU\left(N\right)$ gauge theory [6,58] one can study the physical implications from longitudinal components of a massive-gauge boson degrees of freedom.
• It is expected that in the weak coupling limit the contributions from the mixing terms are very small and consequently Goldstone modes decouple from the theory.

To give a specific example of the Stückelberg trick we consider $SU\left(N\right)$ gauge theory characterized by a non-abelian gauge field $A_{\mu }^a$ in the background of Minkowski flat space-time. In unitary gauge this theory is described by the following action:

$$\begin{array}{c}\hfill S=\int d^{4}x\left[-\frac{1}{4}\mathrm{Tr}\left(F_{\mu \nu }{F}^{\mu \nu }\right)-\frac{m^2}{2}\mathrm{Tr}\left(A_{\mu }{A}^{\mu }\right)\right],\end{array}$$

(13)

where $A_{\mu }=A_{\mu }^a{T}_a$ and $F_{\mu \nu }^a={\partial }_{[\mu }{A}_{\nu ]}^a$. Here the label $a=1,2,\cdots ,N$ for $SU\left(N\right)$ gauge theory. Also $T_a$ are the generators of the non-abelian gauge group which satisfy the following properties:

$$\begin{array}{ccc}\hfill \left[T^a,T^b\right]& =& i{f}^{abc}{T}_c,\mathrm{Tr}\left(T^a\right)=0,\mathrm{Tr}\left(T^a{T}^b\right)=\frac{{\delta }^{ab}}{2}.\hfill \end{array}$$

(14)

Here $f^{abc}\forall a,b,c=1,2,\cdots ,N$ are the structure constants of the non-abelian $SU\left(N\right)$ gauge theory.

It is important to mention that in this context the $SU\left(N\right)$ gauge transformation on the non-abelian gauge field can be written as:

$$\begin{array}{c}\hfill A_{\mu }⟹{\tilde{A}}_{\mu }=\frac{i}{g}U{D}_{\mu }{U}^{†},\mathrm{with}{D}_{\mu }={\partial }_{\mu }-ig{A}_{\mu }\end{array}$$

(15)

where $D_{\mu }$ is the covariant derivative. Here g is the gauge coupling parameter for $SU\left(N\right)$ non-abelian gauge theory. Under this gauge transformation each of the terms in the action stated in Equation (13) transform as:

$$\begin{array}{ccc}\hfill \mathrm{Tr}\left(F_{\mu \nu }{F}^{\mu \nu }\right)& ⟹& \mathrm{Tr}\left({\tilde{F}}_{\mu \nu }{\tilde{F}}^{\mu \nu }\right)=\mathrm{Tr}\left(F_{\mu \nu }{F}^{\mu \nu }\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \frac{m^2}{2}\mathrm{Tr}\left(A_{\mu }{A}^{\mu }\right)& ⟹& \frac{m^2}{2}\mathrm{Tr}\left({\tilde{A}}_{\mu }{\tilde{A}}^{\mu }\right)=\frac{m^2}{2g}\mathrm{Tr}\left[\left(D_{\mu }{U}^{†}\right)\left(D^{\mu }U\right)\right],\hfill \end{array}$$

(17)

where U is the unitary operator in $SU\left(N\right)$ non-abelian gauge theory.

Consequently, after doing $SU\left(N\right)$ gauge transformation action can be expressed as:

$$\begin{array}{cc}\hfill S⟹\tilde{S}& =S+\underset{\mathbf{Additional}\mathbf{part}\mathbf{which}\mathbf{breaks}\mathbf{SU}\left(\mathbf{N}\right)\mathbf{gauge}\mathbf{symmetry}}{\underbrace{\int d^{4}x\left[\frac{m^2}{2}\mathrm{Tr}\left(A_{\mu }{A}^{\mu }\right)-\frac{m^2}{2g}\mathrm{Tr}\left[\left(D_{\mu }{U}^{†}\right)\left(D^{\mu }U\right)\right]\right]}}.\hfill \end{array}$$

(18)

where $\underbrace{}$ term signifies the gauge symmetry breaking contribution in the unitary gauge.

Furthermore, it is important to note that the $SU\left(N\right)$ gauge symmetry can be restored by defining the previously mentioned unitary operator in a following fashion:

$$U=exp\left[i{T}^a{\pi }^a(t,\mathbf{x})\right],$$

(19)

where one can identify the ${\pi }^a\forall a=1,2,\cdots ,N$ s with the Goldstone modes, which transform in a linear fashion under the action of the following gauge transformation:

$$\begin{array}{ccc}\hfill U⟹\tilde{U}& =& exp\left[i{T}^a{\tilde{\pi }}^a(t,\mathbf{x})\right]=\Sigma (t,\mathbf{x})exp\left[i{T}^a{\pi }^a(t,\mathbf{x})\right]=\underset{Localoperator}{\underbrace{\Sigma (t,\mathbf{x})}}U.\hfill \end{array}$$

(20)

For the sake of simplicity one can rescale the Goldstone modes by absorbing the mass of the $SU\left(N\right)$ gauge field m and the $SU\left(N\right)$ gauge coupling parameter g by introducing the following canonical normalization as given by:

$$\begin{array}{c}\hfill \mathrm{Canonical}\mathrm{normalization}:{\pi }_c=\frac{m}{g}\pi .\end{array}$$

(21)

Consequently, the action in terms of canonically normalized field ${\pi }_c$ can be written after $SU\left(N\right)$ gauge transformation as:

$$\begin{array}{cc}S⟹\tilde{S}\hfill & {\displaystyle =S+\int d^{4}x\left[\frac{m^2}{2}\mathrm{Tr}\left(A_{\mu }{A}^{\mu }\right)-\underset{\mathbf{Kinetic}\mathbf{term}\mathbf{of}\mathbf{Goldstone}}{\underbrace{\frac{1}{2}\mathrm{Tr}\left[\left({\partial }_{\mu }{\pi }_c\right)\left({\partial }^{\mu }{\pi }_c\right)\right]}}\right.}\hfill \\ & \left.{\displaystyle \underset{\mathbf{Mixing}\mathbf{terms}\mathbf{after}\mathbf{canonical}\mathbf{normalization}}{\underbrace{-\frac{2g^2}{m}\mathrm{Tr}\left(A_{\mu }{\partial }^{\mu }{\pi }_c\right)+\frac{g^2}{2}\mathrm{Tr}\left(A_{\mu }{A}^{\mu }{\pi }_c^2\right)+ig\mathrm{Tr}\left({\pi }_c{A}_{\mu }{\partial }^{\mu }{\pi }_c\right)}}}\right].\hfill \end{array}$$

(22)

It is important to note the important facts from Equation (22) which are appended below:

• The last two terms in Equation (22) are the mixing terms between the transverse component of the $SU\left(N\right)$ gauge field, the Goldstone boson, and its kinetic term, respectively.
• Here one can neglect all such mixing contributions at the energy scale $E_{mix}>>m$. Consequently, two sectors decouple from each other as they are weakly coupled in the energy scale $E_{mix}>>m$ and Equation (22) takes the following form:

  $$\begin{array}{cc}\hfill S⟹\tilde{S}& {\displaystyle =S+\int d^{4}x\left[\frac{m^2}{2}\mathrm{Tr}\left(A_{\mu }{A}^{\mu }\right)-\frac{1}{2}\mathrm{Tr}\left[\left({\partial }_{\mu }{\pi }_c\right)\left({\partial }^{\mu }{\pi }_c\right)\right]\right].}\hfill \end{array}$$

  (23)

2.2.2. Stückelberg Trick II: Broken Time Diffeomorphism in Quasi-de Sitter Background

Here one needs to perform a time diffeomorphism with a local parameter ${\xi }^0(t,\mathbf{x})$, which is interpreted as a Goldstone field $\pi (t,\mathbf{x})$. These Goldstone modes shifts under the application of time diffeomorphism, as given by:

$$\begin{array}{cc}\hfill \mathbf{Time}\mathbf{diffeomorphism}:t& ⟹t+{\xi }^0(t,\mathbf{x}),x^i⟹x^i\forall i=1,2,3⟶\pi (t,\mathbf{x})\to \pi (t,\mathbf{x})-{\xi }^0(t,\mathbf{x}).\hfill \end{array}$$

(24)

The $\pi$ is the Goldstone mode which describes the scalar perturbations around the background FLRW metric. The effective action in the unitary gauge can be reproduced by gauge-fixing the time diffeomorphism as:

$$\mathbf{Unitary}\mathbf{gauge}\mathbf{fixing}\Rightarrow \pi (t,\mathbf{x})=0\Rightarrow \tilde{\pi }(t,\mathbf{x})=-{\xi }^0(t,\mathbf{x}).$$

(25)

To construct the EFT action, it is important to write down the transformation property of each operators under the application of broken time diffeomorphism, which are given by:

• Rule for metric: Under broken time diffeomorphism contravariant and covariant metric transform as:

  $$\begin{array}{cc}\hfill \mathbf{Contravariant}\mathbf{metric}:g^{00}⟹& {(1+\dot{\pi })}^2{g}^{00}+2(1+\dot{\pi })g^{0i}{\partial }_i\pi +g^{ij}{\partial }_i\pi {\partial }_j\pi ,\hfill \\ \hfill g^{0i}⟹& (1+\dot{\pi })g^{0i}+g^{ij}{\partial }_j\pi ,\hfill \\ \hfill g^{ij}⟹& g^{ij}.\hfill \end{array}$$

  (26)

  $$\begin{array}{cc}\hfill \mathbf{Covariant}\mathbf{metric}:g_{00}& ⟹{(1+\dot{\pi })}^2{g}_{00},\hfill \\ \hfill g_{0i}& ⟹(1+\dot{\pi })g_{0i}+g_{00}\dot{\pi }{\partial }_i\pi ,\hfill \\ \hfill g_{ij}& ⟹g_{ij}+g_{0j}{\partial }_i\pi +g_{i0}{\partial }_j\pi .\hfill \end{array}$$

  (27)
• Rule for Ricci scalar and Ricci tensor: Under broken time diffeomorphism Ricci scalar and the spatial component of the Ricci tensor on 3-hypersurface transform as:

  $$\begin{array}{cc}\hfill \mathbf{Ricci}\mathbf{scalar}:{}^{\left(3\right)}R& {\displaystyle ⟹{}^{\left(3\right)}R+\frac{4}{a^2}H\left({\partial }^2\pi \right),}\hfill \\ \hfill \mathbf{Spatial}\mathbf{Ricci}\mathbf{tensor}:{}^{\left(3\right)}{R}_{ij}& ⟹{}^{\left(3\right)}{R}_{ij}+H({\partial }_i{\partial }_j\pi +{\delta }_{ij}{\partial }^2\pi ).\hfill \end{array}$$

  (28)
• Rule for extrinsic curvature: Under broken time diffeomorphism trace and the spatial, time and mixed component of the extrinsic curvature transform as:

  $$\begin{array}{cc}\hfill \mathbf{Trace}:\delta K& {\displaystyle ⟹\delta K-3\pi \dot{H}-\frac{1}{a^2}\left({\partial }^2\pi \right),}\hfill \\ \hfill \mathbf{Spatial}\mathbf{extrinsic}\mathbf{curvature}:\delta K_{ij}& ⟹\delta K_{ij}-\pi \dot{H}{h}_{ij}-{\partial }_i{\partial }_j\pi \hfill \\ \hfill \mathbf{Temporal}\mathbf{extrinsic}\mathbf{curvature}:\delta K_0^0& ⟹\delta K_0^0,\hfill \\ \hfill \mathbf{Mixed}\mathbf{extrinsic}\mathbf{curvature}:\delta K_i^0& ⟹\delta K_i^0,\hfill \\ \hfill \mathbf{Mixed}\mathbf{extrinsic}\mathbf{curvature}:\delta K_0^i& ⟹\delta K_0^i+2H{g}^{ij}{\partial }_j\pi .\hfill \end{array}$$

  (29)
• Rule for time-dependent EFT coefficients: Under broken time diffeomorphism time-dependent EFT coefficients transform after canonical normalization ${\pi }_c=F^2\left(t\right)\pi$ as:

  $$\begin{array}{cc}\hfill \mathbf{EFT}\mathbf{coefficient}:F\left(t\right)⟹F(t+\pi )& {\displaystyle =\left[\sum _{n=0}^{\infty }\frac{{\pi }^n}{n!}\frac{d^n}{d{t}^n}\right]F\left(t\right)}\hfill \\ & {\displaystyle =\left[\sum _{n=0}^{\infty }\underset{\mathbf{Suppression}}{\underbrace{\frac{{\pi }_c^n}{n!F^{2n}}}}\frac{d^n}{d{t}^n}\right]F\left(t\right)\approx F\left(t\right).}\hfill \end{array}$$

  (30)

  Here $F\left(t\right)$ corresponds to all EFT coefficients mention in the EFT action.
• Rule for Hubble parameter: Under broken time diffeomorphism, time-dependent EFT coefficients transform after using the following canonical normalization:

  $$\mathbf{Canonical}\mathbf{normalization}:{\pi }_c=F^2\left(t\right)\pi ,$$

  (31)

  as given by:

  $$\begin{array}{cc}\hfill \mathbf{Hubble}\mathbf{parameter}:H\left(t\right)⟹H(t+\pi )& {\displaystyle =\left[\sum _{n=0}^{\infty }\frac{{\pi }^n}{n!}\frac{d^n}{d{t}^n}\right]H\left(t\right)}\hfill \\ & {\displaystyle =\left[1-\underset{\mathbf{Correction}\mathbf{terms}}{\underbrace{\pi H\left(t\right)ϵ-\frac{{\pi }^{2}H\left(t\right)}{2}\left(\dot{ϵ}-2{ϵ}^2\right)+\cdots }}\right]H\left(t\right).}\hfill \end{array}$$

  (32)
  Here $ϵ=-\dot{H}/H^2$ is the slow-roll parameter.

Now to construct the EFT action we need to also understand the behavior of all the operators appearing in the weak coupling regime of EFT. In this regime one can neglect the mixing contributions between the gravity and Goldstone modes. To demonstrate this explicitly let us start with the EFT operator:

$${\mathcal{O}}_1\left(t\right)=-\dot{H}{M}_p^2{g}^{00}.$$

(33)

Under broken time diffeomorphism, the operator $\mathcal{O}\left(t\right)$ transform as:

$${\mathcal{O}}_1\left(t\right)⟹\left[1+\frac{\pi }{ϵ}\left(\dot{ϵ}-2H{ϵ}^2\right)+\cdots \right]\left[{(1+\dot{\pi })}^2{\mathcal{O}}_1\left(t\right)-\dot{H}{M}_p^2\left(2(1+\dot{\pi }){\partial }_i\pi g^{0i}+g^{ij}{\partial }_i\pi {\partial }_j\pi \right)\right].$$

(34)

For further simplification the temporal component of the metric $g^{00}$ can be written as, $g^{00}={\overline{g}}^{00}+\delta g^{00}$, where the background metric is given by, ${\overline{g}}^{00}=-1$ and the metric fluctuation is characterized by $\delta g^{00}$ [6,20]. Using this in Equation (34) and considering only the first term in Equation (34) we get a kinetic term, $M_p^2\dot{H}{\dot{\pi }}^2\overline{g^{00}}$ and a mixing contribution, $M_p^2\dot{H}\dot{\pi }\delta g^{00}$ respectively. Furthermore, we use a canonical normalized metric fluctuation from the mixing contribution as given by:

$$\mathbf{Canonical}\mathbf{normalization}:\delta g_c^{00}=M_p\delta g^{00},$$

(35)

in terms of which one can write, $M_p^2\dot{H}\dot{\pi }\delta g^{00}=\sqrt{\dot{H}}{\dot{\pi }}_c\delta g_c^{00}$. Consequently, at above the energy scale $E_{mix}=\sqrt{\dot{H}}$, we can neglect this mixing term in the weak coupling regime.

One can also consider mixing contributions $M_p^2\dot{H}{\dot{\pi }}^2\delta g^{00}$ and $\pi M_p^2\ddot{H}\dot{\pi }{\overline{g}}^{00}$, which can be recast after canonical normalization as, $M_p^2\dot{H}{\dot{\pi }}^2\delta g^{00}={\dot{\pi }}_c^2\delta g_c^{00}/M_p$ and $\pi M_p^2\ddot{H}\dot{\pi }{\overline{g}}^{00}=\ddot{H}{\pi }_c{\dot{\pi }}_c{\overline{g}}^{00}/\dot{H}$ with $\ddot{H}/\dot{H}<<1$. Here all higher-order terms in $\dot{\pi }$ will lead to additional Planck-suppression after canonical normalization. Consequently, we can neglect the contribution from $M_p^2\dot{H}\dot{\pi }\delta g^{00}$ term at the scale $E>E_{mix}$. Finally, in the weak coupling regime one can recast Equation (34) as:

$${\mathcal{O}}_1\left(t\right)⟹{\mathcal{O}}_1\left(t\right)\left[{\dot{\pi }}^2-\frac{1}{a^2}{\left({\partial }_i\pi \right)}^2\right].$$

(36)

2.2.3. The Goldstone Action from EFT

Finally, in the weak coupling limit (or decoupling limit) we get the following simplified EFT action:

$$\begin{array}{ccc}\hfill S_{EFT}& =& S_g+S_{\pi },\hfill \end{array}$$

(37)

where the gravitational part and the Goldstone action is given by:

$$\begin{array}{ccc}\hfill S_g& =& \int d^{4}x\sqrt{-g}\left[\frac{M_p^2}{2}R-M_p^2\left(3H^2+\dot{H}\right)\right],\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill S_{\pi }& =& S_{\pi }^{\left(2\right)}+S_{\pi }^{\left(3\right)}+\cdots ,\hfill \end{array}$$

(39)

where the second and third-order Goldstone action can be written as:

$$\begin{array}{cc}\hfill S_{\pi }^{\left(2\right)}& {\displaystyle =\int d^{4}x{a}^3\left[-M_p^2\dot{H}\left({\dot{\pi }}^2-\frac{1}{a^2}{\left({\partial }_i\pi \right)}^2\right)+2M_2^4{\dot{\pi }}^2\right.}\hfill \\ & \left.{\displaystyle +\frac{1}{2}\left({\overline{M}}_3^2+3{\overline{M}}_2^2\right)H^2(1-ϵ)\frac{{\left({\partial }_i\pi \right)}^2}{a^2}-\left({\overline{M}}_3^2+3{\overline{M}}_2^2\right)H^2\frac{{\left({\partial }_i\pi \right)}^2}{a^2}-{\overline{M}}_1^3\dot{\pi }\frac{1}{a^2}\left({\partial }_i^2\pi \right)}\right].\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill S_{\pi }^{\left(3\right)}& {\displaystyle =\int d^{4}x{a}^3\left[\left(2M_2^4-\frac{4}{3}{M}_3^4\right){\dot{\pi }}^3-2M_2^4\dot{\pi }\frac{1}{a^2}{\left({\partial }_i\pi \right)}^2\right.}\hfill \\ & {\displaystyle -{\overline{M}}_3^2\pi \dot{H}\frac{1}{a^2}{\partial }_i^2\pi -3{\overline{M}}_2^2\dot{H}\pi \frac{1}{a^2}\left({\partial }_i^2\pi \right)+\frac{3}{2}{\overline{M}}_1^3\pi \dot{H}\frac{1}{a^2}{\left({\partial }_i\pi \right)}^2}\hfill \\ & \left.{\displaystyle -\frac{3}{2}{\overline{M}}_1^3\dot{H}\pi {\dot{\pi }}^2-{\overline{M}}_1^3\dot{\pi }\frac{1}{a^2}{\left({\partial }_i\pi \right)}^2}\right].\hfill \end{array}$$

(41)

Here we introduce EFT sound speed $c_S$ as:

$$\begin{array}{c}\hfill c_S\equiv \frac{1}{\sqrt{1-\frac{2M_2^4}{\dot{H}{M}_p^2}}}.\end{array}$$

(42)

Here if we set $M_2=0$ or equivalently if we say that $\frac{M_2^4}{2!}{(g^{00}+1)}^2$ term is absent in the effective Lagrangian then Equation (42) suggests that in that case sound speed $c_S=1$, which is true for single-field canonical slow-roll inflation. Next using Equation (42) and applying integration by parts in the Goldstone part of the Lagrangian we get4:

$$\begin{array}{c}\hfill S_{\pi }^{\left(2\right)}=\int d^{4}x{a}^3\left(-\frac{M_p^2\dot{H}}{c_S^2}\right)\left[{\dot{\pi }}^2-c_S^2\left(1-\frac{{\overline{M}}_1^{3}H}{M_p^2\dot{H}}-\left[{\overline{M}}_3^2+3{\overline{M}}_2^2\right]\frac{H^2(1+ϵ)}{2M_p^2\dot{H}}\right)\frac{1}{a^2}{\left({\partial }_i\pi \right)}^2\right].\end{array}$$

(47)

$$\begin{array}{cc}\hfill S_{\pi }^{\left(3\right)}& {\displaystyle =\int d^{4}x{a}^3\left[\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}{\dot{\pi }}^3\right.}\hfill \\ & {\displaystyle -\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\frac{1}{a^2}\dot{\pi }{\left({\partial }_i\pi \right)}^2}\hfill \\ & \left.{\displaystyle -\frac{9}{2}{\overline{M}}_1^3{H}^2\pi {\dot{\pi }}^2+\frac{3}{2}{\overline{M}}_1^{3}H\frac{1}{a^2}\pi \frac{d}{dt}{\left({\partial }_i\pi \right)}^2}\right].\hfill \end{array}$$

(48)

In the present context metric fluctuation of the spatial components are given by:

$$\begin{array}{c}\hfill g_{ij}=a^2\left(t\right)\left[\left(1+2\zeta (t,\mathbf{x})\right){\delta }_{ij}+{\gamma }_{ij}\right]\forall i=1,2,3,\end{array}$$

(49)

where $a\left(t\right)$ is the scale factor in FLRW quasi de Sitter background space-time. Also $\zeta (t,\mathbf{x})$ is known as curvature perturbation which signifies scalar fluctuation. On the other hand, tensor fluctuations are identified with ${\gamma }_{ij}$, which is spin-2, transverse, and traceless rank 2 tensor. Here under the broken time diffeomorphism the scale factor $a\left(t\right)$ transforms in the following fashion:

$$\begin{array}{c}\hfill a\left(t\right)⟹a(t-\pi (t,\mathbf{x}))=a\left(t\right)-H\pi (t,\mathbf{x})a\left(t\right)+\cdots \approx a\left(t\right)\left(1-H\pi (t,\mathbf{x})\right).\end{array}$$

(50)

Furthermore, using Equations (45) and (46), we get:

$$\begin{array}{c}\hfill a^2\left(t\right){\left(1-H\pi (t,\mathbf{x})\right)}^2\approx a^2\left(t\right)\left(1-2H\pi (t,\mathbf{x})\right)=a^2\left(t\right)\left(1+2\zeta (t,\mathbf{x})\right).\end{array}$$

(51)

This implies that the curvature perturbation $\zeta (t,\mathbf{x})$ can be written in terms of Goldstone modes $\pi (t,\mathbf{x})$ in the following way5:

$$\mathbf{Quantum}\mathbf{fluctuation}\mathbf{in}\mathbf{terms}\mathbf{of}\mathbf{Goldstone}\mathbf{mode}:\zeta (t,\mathbf{x})=-H\pi (t,\mathbf{x}).$$

(53)

Furthermore, using Equation (53), the effective action for the Goldstone part of the Lagrangian can be recast in terms of curvature perturbation $\zeta (t,\mathbf{x})$ as:

$$\begin{array}{c}\hfill S_{\zeta }^{\left(2\right)}\approx \int d^{4}x{a}^3\left(\frac{M_p^2ϵ}{c_S^2}\right)\left[{\dot{\zeta }}^2-c_S^2\left(1-\frac{{\overline{M}}_1^{3}H}{M_p^2\dot{H}}-\left[{\overline{M}}_3^2+3{\overline{M}}_2^2\right]\frac{H^2(1+ϵ)}{2M_p^2\dot{H}}\right)\frac{1}{a^2}{\left({\partial }_i\zeta \right)}^2\right].\end{array}$$

(54)

$$\begin{array}{cc}\hfill S_{\zeta }^{\left(3\right)}& {\displaystyle \approx \int d^{4}x\frac{a^3}{H^3}\left[-\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}{\dot{\zeta }}^3\right.}\hfill \\ & {\displaystyle +\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\frac{1}{a^2}\dot{\zeta }{\left({\partial }_i\zeta \right)}^2}\hfill \\ & \left.{\displaystyle +\frac{9}{2}{\overline{M}}_1^3{H}^2\zeta {\dot{\zeta }}^2-\frac{3}{2}{\overline{M}}_1^{3}H\frac{1}{a^2}\zeta \frac{d}{dt}{\left({\partial }_i\zeta \right)}^2}\right].\hfill \end{array}$$

(55)

For further simplification we introduce a few new parameters which are appended bellow6:

• First we define an effective sound speed ${\tilde{c}}_S$, which can be expressed in terms of the usual EFT sound speed $c_S$ as7:

  $$\begin{array}{c}\hfill {\tilde{c}}_S=c_S\sqrt{1-\frac{{\overline{M}}_1^{3}H}{M_p^2\dot{H}}-\left[{\overline{M}}_3^2+3{\overline{M}}_2^2\right]\frac{H^2(1+ϵ)}{2M_p^2\dot{H}}}.\end{array}$$

  (56)

  Since the following approximations:

  $$\begin{array}{ccc}\hfill \left|\frac{{\overline{M}}_1^{3}H}{M_p^2\dot{H}}\right|& <<& 1,\left|\left[{\overline{M}}_3^2+3{\overline{M}}_2^2\right]\frac{H^2(1+ϵ)}{2M_p^2\dot{H}}\right|<<1,\hfill \end{array}$$

  (57)

  are valid in the present context of discussion, one can recast the effective sound speed in the following simplified form as:

  $$\begin{array}{ccc}\hfill {\tilde{c}}_S& \approx & c_S\left\{1+\frac{1}{2ϵH{M}_p^2}\left[{\overline{M}}_1^3+\left({\overline{M}}_3^2+3{\overline{M}}_2^2\right)\frac{H(1+ϵ)}{2}\right]\right\}.\hfill \end{array}$$

  (58)
• Secondly, we introduce the following connecting relationship between $M_3$ and $M_2$ given by:

  $$\begin{array}{ccc}\hfill M_3^4{c}_S^2& =& -{\tilde{c}}_3{M}_2^4.\hfill \end{array}$$

  (59)

  When $M_2=0$ then from Equation (42) we can see that the sound speed $c_S=1$ and Equation (59) also implies that $M_3=0$ in that case.
• Next we define the following connecting relationship between $M_3$ and ${\overline{M}}_1$ given by:

  $$\begin{array}{ccc}\hfill M_3^4{\tilde{c}}_4& =& -H{\overline{M}}_1^3{\tilde{c}}_3.\hfill \end{array}$$

  (60)

  When $M_2=0$ then from Equation (42) we can see that the sound speed $c_S=1$ (which is actually the result for single-field canonical slow-roll models of inflation) and Equations (59) and (60) also implies the following possibilities:
  • $M_3=0$, ${\overline{M}}_1\ne 0$ and $\frac{{\tilde{c}}_3}{{\tilde{c}}_4}\to 0$. We will look into this possibility in detail during our computation for $c_S=1$ case as this will finally give rise to non-vanishing three-point function (non-Gaussianity).
  • $M_3=0$, ${\overline{M}}_1=0$ and $\frac{{\tilde{c}}_3}{{\tilde{c}}_4}\ne 0$. We do not consider this possibility for $c_S=1$ case because for this case third ($S_{\zeta }^{\left(3\right)}$) action for curvature perturbation vanishes, which will give rise to zero three-point function (non-Gaussianity).
• For further simplification one can also assume that:

  $${\overline{M}}_3^2+3{\overline{M}}_2^2=\frac{{\overline{M}}_1^3}{H{\tilde{c}}_5}$$

  (61)

  so that one can write:

  $$\frac{1}{ϵH{M}_p^2}\left[{\overline{M}}_1^3+\left({\overline{M}}_3^2+3{\overline{M}}_2^2\right)\frac{H(1+ϵ)}{2}\right]=\frac{{\overline{M}}_1^3}{ϵH{M}_p^2}\left[1+\frac{(1+ϵ)}{2{\tilde{c}}_5}\right].$$

  (62)

  For $c_S=1$ this implies the following two possibilities:
  • ${\overline{M}}_1\ne 0$ and ${\tilde{c}}_5=-\frac{1}{2}(1+ϵ)$. We will look into this possibility in detail during our computation for $c_S=1$ case as this will finally give rise to non-vanishing three-point function (non-Gaussianity).
  • ${\overline{M}}_1=0$. We do not consider this possibility for $c_S=1$ case because for this case third ($S_{\zeta }^{\left(3\right)}$) action for curvature perturbation vanishes, which will give rise to zero three-point function (non-Gaussianity).
  Consequently, the effective sound speed can be recast as:

  $$\begin{array}{ccc}\hfill {\tilde{c}}_S& =& c_S\sqrt{1+\frac{\Delta {\overline{M}}_1^3}{2ϵH{M}_p^2}}\approx c_S\left\{1+\frac{\Delta {\overline{M}}_1^3}{4ϵH{M}_p^2}\right\}\hfill \end{array}$$

  (63)

  where $\Delta$ is defined as, $\Delta =2+\frac{1+ϵ}{{\tilde{c}}_5}.$ Here $\Delta =0$ for ${\tilde{c}}_5=-\frac{1}{2}(1+ϵ)$ when $c_S=1$. Consequently, we have ${\tilde{c}}_S=c_S=1$ in that case.
• For further simplification one can also assume that:

  $$\begin{array}{c}\hfill {\overline{M}}_3^2\approx {\overline{M}}_2^2=\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}.\end{array}$$

  (64)

  Here $c_S=1$ this implies the following two possibilities:
  • ${\overline{M}}_3^2\approx {\overline{M}}_2^2\ne 0,{\overline{M}}_1\ne 0$ and ${\tilde{c}}_5=-\frac{1}{2}(1+ϵ)$ as mentioned earlier. We will investigate this possibility in detail during our computation for $c_S=1$ case as this will finally give rise to non-vanishing non-Gaussianity.
  • ${\overline{M}}_3^2\approx {\overline{M}}_2^2=0,{\overline{M}}_1=0$. As mentioned earlier here we do not consider this possibility for $c_S=1$ case because for this case second ($S_{\zeta }^{\left(2\right)}$) and third-order ($S_{\zeta }^{\left(3\right)}$) action for curvature perturbation vanishes, which will give rise to zero non-Gaussianity.
• Next we define the following connecting relationship between $M_4$ and $M_3$ given by:

  $$\begin{array}{ccc}\hfill M_4^4{\tilde{c}}_6=M_3^4{\tilde{c}}_4& =& -H{\overline{M}}_1^3{\tilde{c}}_3.\hfill \end{array}$$

  (65)

  When $M_2=0$ then from Equation (42) we can see that the sound speed $c_S=1$ and Equations (59) and (65) also implies the following possibilities:
  • $M_4\ne 0$, $M_3=0$, ${\overline{M}}_1\ne 0$ and $\frac{{\tilde{c}}_3}{{\tilde{c}}_4}\to 0$. We will look into this possibility in detail during our computation for $c_S=1$ case as this will finally give rise to non-vanishing three-point function (non-Gaussianity).
  • $M_4=0$, $M_3=0$, ${\overline{M}}_1=0$ and $\frac{{\tilde{c}}_3}{{\tilde{c}}_4}\ne 0$. We do not consider this possibility for $c_S=1$ case because for this case third ($S_{\zeta }^{\left(3\right)}$) order action for curvature perturbation vanishes, which will give rise to zero three-point function (non-Gaussianity).

Furthermore, using all such new defined parameters the EFT action for the Goldstone boson can be recast as8:

$$\begin{array}{c}\hfill \mathbf{For}{\mathbf{c}}_{\mathbf{S}}=\mathbf{1}:\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{\zeta }^{\left(2\right)}& {\displaystyle \approx \int d^{4}x{a}^3{M}_p^2ϵ\left[{\dot{\zeta }}^2-\frac{1}{a^2}{\left({\partial }_i\zeta \right)}^2\right].}\hfill \end{array}\end{array}$$

(66)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{\zeta }^{\left(3\right)}& {\displaystyle \approx \int d^{4}x\frac{a^3}{H^3}\left[-\left\{\frac{3}{2}{\overline{M}}_1^{3}H\right\}{\dot{\zeta }}^3+\left\{\frac{3}{2}{\overline{M}}_1^{3}H\right\}\frac{1}{a^2}\dot{\zeta }{\left({\partial }_i\zeta \right)}^2\right.}\hfill \\ & \left.{\displaystyle +\frac{9}{2}{\overline{M}}_1^3{H}^2\zeta {\dot{\zeta }}^2-\frac{3}{2}{\overline{M}}_1^{3}H\frac{1}{a^2}\zeta \frac{d}{dt}{\left({\partial }_i\zeta \right)}^2}\right].\hfill \end{array}\end{array}$$

(67)

$$\begin{array}{c}\hfill \mathbf{For}{\mathbf{c}}_{\mathbf{S}}<\mathbf{1}:\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{\zeta }^{\left(2\right)}& {\displaystyle \approx \int d^{4}x{a}^3\left(\frac{M_p^2ϵ}{c_S^2}\right)\left[{\dot{\zeta }}^2-{\tilde{c}}_S^2\frac{1}{a^2}{\left({\partial }_i\zeta \right)}^2\right].}\hfill \end{array}\end{array}$$

(68)

$$\begin{array}{cc}\hfill S_{\zeta }^{\left(3\right)}& {\displaystyle \approx \int d^{4}x{a}^3\frac{ϵM_p^2}{H}\left(1-\frac{1}{c_S^2}\right)\left[\left\{1+\frac{3{\tilde{c}}_4}{4c_S^2}+\frac{2{\tilde{c}}_3}{3c_S^2}\right\}{\dot{\zeta }}^3-\left\{1+\frac{3{\tilde{c}}_4}{4c_S^2}\right\}\frac{1}{a^2}\dot{\zeta }{\left({\partial }_i\zeta \right)}^2\right.}\hfill \\ & \left.{\displaystyle -\frac{9H{\tilde{c}}_4}{4c_S^2}\zeta {\dot{\zeta }}^2+\frac{3{\tilde{c}}_4}{4c_S^2}\frac{1}{a^2}\zeta \frac{d}{dt}{\left({\partial }_i\zeta \right)}^2}\right].\hfill \end{array}$$

(69)

3. Two-Point Correlation Function from EFT

3.1. For Scalar Modes

3.1.1. Mode Equation and Solution for Scalar Perturbation

Here we compute the two-point correlation from scalar perturbation. For this purpose we consider the second-order perturbed action as given by9:

$$\begin{array}{ccc}\hfill S_{\zeta }^{\left(2\right)}& \approx & \int d^{4}x{a}^3\left(\frac{M_p^2ϵ}{c_S^2}\right)\left[{\dot{\zeta }}^2-c_S^2\left(1-\frac{{\overline{M}}_1^{3}H}{M_p^2\dot{H}}-\left[{\overline{M}}_3^2+3{\overline{M}}_2^2\right]\frac{H^2(1+ϵ)}{2M_p^2\dot{H}}\right)\frac{1}{a^2}{\left({\partial }_i\zeta \right)}^2\right],\hfill \end{array}$$

(70)

which can be recast for $c_S=1$ and $c_S<1$ case as:

$$\mathbf{For}{\mathbf{c}}_{\mathbf{S}}=\mathbf{1}:S_{\zeta }^{\left(2\right)}\approx \int d^{4}x{a}^3{M}_p^2ϵ\left[{\dot{\zeta }}^2-\frac{1}{a^2}{\left({\partial }_i\zeta \right)}^2\right],$$

(71)

$$\mathbf{For}{\mathbf{c}}_{\mathbf{S}}<\mathbf{1}:S_{\zeta }^{\left(2\right)}\approx \int d^{4}x{a}^3\left(\frac{M_p^2ϵ}{c_S^2}\right)\left[{\dot{\zeta }}^2-{\tilde{c}}_S^2\frac{1}{a^2}{\left({\partial }_i\zeta \right)}^2\right],$$

(72)

where the effective sound speed ${\tilde{c}}_S$ is defined earlier.

Next we define Mukhanov–Sasaki variable $v(\eta ,\mathbf{x})$ which is defined as:

$$\mathbf{Mukhanov}–\mathbf{Sasaki}\mathbf{variable}:v(\eta ,\mathbf{x})=z\zeta (\eta ,\mathbf{x})M_p=-zH\pi (\eta ,\mathbf{x})M_p.$$

(73)

In general, the parameter z is defined for the present EFT setup as, $z=\frac{a\sqrt{2ϵ}}{{\tilde{c}}_S}.$ Now in terms of $v(\eta ,\mathbf{x})$ the second-order action for the curvature perturbation can be recast as:

$$\begin{array}{c}\hfill S_{\zeta }^{\left(2\right)}\approx \int d^{3}xd\eta \left[v^{{}^{\prime }2}-{\tilde{c}}_S^2{\left({\partial }_{i}v\right)}^2\frac{1}{a^2}{\left({\partial }_i\zeta \right)}^2-m_{eff}^2\left(\eta \right)v^2\right],\end{array}$$

(74)

where the effective mass parameter $m_{eff}\left(\eta \right)$ is defined as, $m_{eff}^2\left(\eta \right)=-\frac{1}{z}\frac{d^{2}z}{d{\eta }^2}.$ Here $\eta$ is the conformal time which can be expressed in terms of physical time t as, $\eta =\int \frac{dt}{a\left(t\right)}.$ The conformal time described here is negative and lying within $-\infty <\eta <0$. During inflation, the scale factor and the parameter z can be expressed in terms of the conformal time $\eta$ as:

$$\begin{array}{c}{\displaystyle a\left(\eta \right)=\left\{\begin{array}{cc}{\displaystyle -\frac{1}{H\eta }}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle -\frac{1}{H\eta }\left(1+ϵ\right)}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.}\hfill \end{array}$$

(75)

and

$$\begin{array}{c}{\displaystyle z=\frac{a\sqrt{2ϵ}}{c_S}=\left\{\begin{array}{cc}{\displaystyle -\frac{1}{H\eta }\frac{\sqrt{2ϵ}}{c_S}}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle -\frac{1}{H\eta }\frac{\sqrt{2ϵ}}{c_S}\left(1+ϵ\right)}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.}\hfill \end{array}$$

(76)

Additionally, it is important to note that for de Sitter and quasi de Sitter case the relation between conformal time $\eta$ and physical time t can be expressed as, $t=-\frac{1}{H}ln(-H\eta ).$ Within this setup inflation ends when the conformal time $\eta \sim 0$.

Now further doing the Fourier transform:

$$v(\eta ,x)=\int \frac{d^{3}k}{{\left(2\pi \right)}^3}{v}_{\mathbf{k}}\left(\eta \right)e^{i\mathbf{k}.\mathbf{x}}$$

(77)

one can write down the equation of motion for scalar fluctuation as:

$$\mathbf{Mukhanov}–\mathbf{Sasaki}\mathbf{Eqn}\mathbf{for}\mathbf{scalar}\mathbf{mode}:v_{\mathbf{k}}^{\prime \prime }+\left({\tilde{c}}_S^2{k}^2+m_{eff}^2\left(\eta \right)\right)v_{\mathbf{k}}=0.$$

(78)

Here it is important to note that for de Sitter and quasi de Sitter case the effective mass parameter can be expressed as:

$$\begin{array}{c}{\displaystyle m_{eff}^2\left(\eta \right)=\left\{\begin{array}{cc}{\displaystyle -\frac{2}{{\eta }^2}}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle -\frac{\left({\nu }^2-\frac{1}{4}\right)}{{\eta }^2}}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.}\hfill \end{array}$$

(79)

Here in the de Sitter and quasi de Sitter case the parameter $\nu$ can be written as:

$$\begin{array}{c}{\displaystyle \nu =\left\{\begin{array}{cc}{\displaystyle \frac{3}{2}}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle \frac{3}{2}+3ϵ-\eta +\frac{s}{2}}\hfill & \mathbf{for}\mathbf{qdS},\hfill \end{array}\right.}\hfill \end{array}$$

(80)

where $ϵ$, $\eta$ and s are the slow-roll parameter defined as:

$$\begin{array}{ccc}\hfill ϵ& =& -\frac{\dot{H}}{H^2},\eta =2ϵ-\frac{\dot{ϵ}}{2Hϵ},s=\frac{\dot{c_S}}{H{c}_S}.\hfill \end{array}$$

(81)

In the slow-roll regime of inflation $ϵ<<1$ and $\left|\eta \right|<<1$ and at the end of inflation, the slow-roll condition breaks when any of the criteria satisfy (1) $ϵ=1$ or $\left|\eta \right|=1$, (2) $ϵ=1=\left|\eta \right|$.

The general solution for $v_{\mathbf{k}}\left(\eta \right)$ thus can be written as:

$$\begin{array}{c}{\displaystyle v_{\mathbf{k}}\left(\eta \right)=\left\{\begin{array}{cc}{\displaystyle \sqrt{-\eta }\left[C_1{H}_{\frac{3}{2}}^{\left(1\right)}\left(-k{\tilde{c}}_S\eta \right)+C_2{H}_{\frac{3}{2}}^{\left(2\right)}\left(-k{\tilde{c}}_S\eta \right)\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle \sqrt{-\eta }\left[C_1{H}_{\nu }^{\left(1\right)}\left(-k{\tilde{c}}_S\eta \right)+C_2{H}_{\nu }^{\left(2\right)}\left(-k{\tilde{c}}_S\eta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.}\hfill \end{array}$$

(82)

Here $C_1$ and $C_2$ are the arbitrary integration constants and the numerical values depend on the choice of the initial vacuum. In the present context we consider the following choice of the vacuum for the computation:

• Bunch–Davies vacuum: In this case, we choose, $C_1=1,C_2=0.$
• $\mathit{\alpha },\mathit{\beta }$ vacuum: In this case, we choose $C_1=cosh\alpha ,C_2=e^{i\beta }sinh\alpha .$ Here $\beta$ is a phase factor.

For the most general solution as stated in Equation (82) one can consider the limiting physical situations, as given by, I. Superhorizon regime: $k{\tilde{c}}_S\eta <<-1$, II. Horizon crossing: $k{\tilde{c}}_S\eta =-1$, III. Subhorizon regime: $k{\tilde{c}}_S\eta >>-1$.

Finally, considering the behavior of the mode function in the subhorizon regime and superhorizon regime one can write the expression in de Sitter and quasi de Sitter case as:

$$\begin{array}{ccc}\hfill {\displaystyle v_{\mathbf{k}}\left(\eta \right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{1}{i\eta }\frac{1}{\sqrt{2}{\left(k{\tilde{c}}_S\right)}^{\frac{3}{2}}}\left[C_1{e}^{-ik{\tilde{c}}_S\eta }\left(1+ik{\tilde{c}}_S\eta \right)e^{-i\pi }-C_2{e}^{ik{c}_S\eta }\left(1-ik{\tilde{c}}_S\eta \right)e^{i\pi }\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\nu -\frac{3}{2}}\frac{1}{i\eta }\frac{1}{\sqrt{2}{\left(k{\tilde{c}}_S\right)}^{\frac{3}{2}}}{(-k{\tilde{c}}_S\eta )}^{\frac{3}{2}-\nu }\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|\left[C_1{e}^{-ik{\tilde{c}}_S\eta }\left(1+ik{\tilde{c}}_S\eta \right)e^{-\frac{i\pi }{2}\left(\nu +\frac{1}{2}\right)}\right.}\hfill \\ \left.{\displaystyle -C_2{e}^{ik{c}_S\eta }\left(1-ik{\tilde{c}}_S\eta \right)e^{\frac{i\pi }{2}\left(\nu +\frac{1}{2}\right)}}\right]\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

Furthermore, using Equation (83) one can write down the expression for the curvature perturbation $\zeta (\eta ,\mathbf{k})=\frac{v_{\mathbf{k}}\left(\eta \right)}{z{M}_p}$ as:

$$\begin{array}{ccc}\hfill {\displaystyle \zeta (\eta ,\mathbf{k})}& =& \left\{\begin{array}{cc}{\displaystyle \frac{iH{\tilde{c}}_S}{2M_p\sqrt{ϵ}{\left(k{\tilde{c}}_S\right)}^{\frac{3}{2}}}\left[C_1{e}^{-ik{\tilde{c}}_S\eta }\left(1+ik{\tilde{c}}_S\eta \right)e^{-i\pi }-C_2{e}^{ik{c}_S\eta }\left(1-ik{\tilde{c}}_S\eta \right)e^{i\pi }\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\nu -\frac{3}{2}}\frac{iH{\tilde{c}}_S}{2M_p\sqrt{ϵ}(1+ϵ){\left({\tilde{c}}_{S}k\right)}^{\frac{3}{2}}}{(-k{\tilde{c}}_S\eta )}^{\frac{3}{2}-\nu }\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|\left[C_1{e}^{-ik{\tilde{c}}_S\eta }\left(1+ik{\tilde{c}}_S\eta \right)e^{-\frac{i\pi }{2}\left(\nu +\frac{1}{2}\right)}\right.}\hfill \\ \left.{\displaystyle -C_2{e}^{ik{c}_S\eta }\left(1-ik{\tilde{c}}_S\eta \right)e^{\frac{i\pi }{2}\left(\nu +\frac{1}{2}\right)}}\right]\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

One can further compute the two-point function for scalar fluctuation as:

$$\begin{array}{c}\hfill \langle \zeta (\eta ,\mathbf{k})\zeta (\eta ,\mathbf{q})\rangle ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}(\mathbf{k}+\mathbf{q})P_{\zeta }(k,\eta ),\end{array}$$

(83)

where $P_{\zeta }(k,\eta )$ is the power spectrum at time $\eta$ for scalar fluctuations and in the present context it is defined as:

$$\begin{array}{ccc}\hfill {\displaystyle P_{\zeta }(k,\eta )}& =& \frac{|v_{\mathbf{k}}{\left(\eta \right)|}^2}{z^2{M}_p^2}=\left\{\begin{array}{cc}{\displaystyle \frac{H^2}{4M_p^2ϵ{\tilde{c}}_S}\frac{1}{k^3}{\left|C_1{e}^{-ik{\tilde{c}}_S\eta }\left(1+ik{\tilde{c}}_S\eta \right)e^{-i\pi }-C_2{e}^{ik{c}_S\eta }\left(1-ik{\tilde{c}}_S\eta \right)e^{i\pi }\right|}^2}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{H^2}{4M_p^2ϵ{(1+ϵ)}^2{\tilde{c}}_S}\frac{1}{k^3}{(-k{\tilde{c}}_S\eta )}^{3-2\nu }{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle {\left|C_1{e}^{-ik{\tilde{c}}_S\eta }\left(1+ik{\tilde{c}}_S\eta \right)e^{-\frac{i\pi }{2}\left(\nu +\frac{1}{2}\right)}-C_2{e}^{ik{c}_S\eta }\left(1-ik{\tilde{c}}_S\eta \right)e^{\frac{i\pi }{2}\left(\nu +\frac{1}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

(84)

3.1.2. Primordial Power Spectrum for Scalar Perturbation

Finally, at the horizon crossing one can furthermore write the two-point correlation function as10:

$$\begin{array}{c}\hfill \langle \zeta \left(\mathbf{k}\right)\zeta \left(\mathbf{q}\right)\rangle ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}(\mathbf{k}+\mathbf{q})P_{\zeta }\left(k\right),\end{array}$$

(85)

where $P_{\zeta }\left(k\right)$ is the power spectrum at time $\eta$ for scalar fluctuations and it is defined as:

$$\begin{array}{ccc}\hfill P_{\zeta }\left(k\right)& =& {\left[\frac{|v_{\mathbf{k}}{\left(\eta \right)|}^2}{z^2{M}_p^2}\right]}_{|k{\tilde{c}}_S\eta |=1}=P_{\zeta }\left(k_{*}\right)\frac{1}{k^3}=\left\{\begin{array}{cc}{\displaystyle \frac{H^2}{4M_p^2ϵ{\tilde{c}}_S}\frac{1}{k^3}\left[|C_1{|}^2+{\left|C_2\right|}^2-\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{H^2}{4M_p^2ϵ{(1+ϵ)}^2{\tilde{c}}_S}\frac{1}{k^3}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left[|C_1{|}^2+{\left|C_2\right|}^2\right.}\hfill \\ \left.{\displaystyle -\left(C_1^{*}{C}_2{e}^{i\pi \left(\nu +\frac{1}{2}\right)}+C_1{C}_2^{*}{e}^{-i\pi \left(\nu +\frac{1}{2}\right)}\right)}\right]\hfill & \mathbf{for}\mathbf{qdS},\hfill \end{array}\right.\hfill \end{array}$$

(86)

where $P_{\zeta }\left(k_{*}\right)$ is power spectrum for scalar fluctuation at the pivot scale $k=k_{*}$. For simplicity one can keep $k^3/2{\pi }^2$ dependence outside and further define amplitude of the power spectrum ${\Delta }_{\zeta }\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

$$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_{\zeta }\left(k_{*}\right)}& =\frac{k^3}{2{\pi }^2}{P}_{\zeta }\left(k\right)=\frac{1}{2{\pi }^2}{P}_{\zeta }\left(k_{*}\right)=& \left\{\begin{array}{cc}{\displaystyle \frac{H^2}{8{\pi }^2{M}_p^2ϵ{\tilde{c}}_S}\left[|C_1{|}^2+{\left|C_2\right|}^2-\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{H^2}{8{\pi }^2{M}_p^2ϵ{(1+ϵ)}^2{\tilde{c}}_S}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left[|C_1{|}^2+{\left|C_2\right|}^2\right.}\hfill & \\ \left.{\displaystyle -\left(C_1^{*}{C}_2{e}^{i\pi \left(\nu +\frac{1}{2}\right)}+C_1{C}_2^{*}{e}^{-i\pi \left(\nu +\frac{1}{2}\right)}\right)}\right]\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

(87)

For Bunch–Davies and $\mathit{\alpha },\mathit{\beta }$ vacuum power spectrum can be written as:

• For Bunch–Davies vacuum:
  In this case, by setting $C_1=1$ and $C_2=0$ we get the following expression for the power spectrum:

  $$\begin{array}{ccc}\hfill {\displaystyle P_{\zeta }\left(k\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{H^2}{4M_p^2ϵ{\tilde{c}}_S}\frac{1}{k^3}}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{H^2}{4M_p^2ϵ{(1+ϵ)}^2{\tilde{c}}_S}\frac{1}{k^3}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (88)

  Also, the power spectrum ${\Delta }_{\zeta }\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

  $$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_{\zeta }\left(k_{*}\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{H^2}{8{\pi }^2{M}_p^2ϵ{\tilde{c}}_S}}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{H^2}{8{\pi }^2{M}_p^2ϵ{(1+ϵ)}^2{\tilde{c}}_S}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (89)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  In this case, by setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get the following expression for the power spectrum:

  $$\begin{array}{ccc}\hfill {\displaystyle P_{\zeta }\left(k\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{H^2}{4M_p^2ϵ{\tilde{c}}_S}\frac{1}{k^3}\left[cosh2\alpha -sinh2\alpha cos\beta \right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{H^2}{4M_p^2ϵ{(1+ϵ)}^2{\tilde{c}}_S}\frac{1}{k^3}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle \left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\nu +\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (90)
  Also, the power spectrum ${\Delta }_{\zeta }\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

  $$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_{\zeta }\left(k_{*}\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{H^2}{8{\pi }^2{M}_p^2ϵ{\tilde{c}}_S}\left[cosh2\alpha -sinh2\alpha cos\beta \right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{H^2}{8{\pi }^2{M}_p^2ϵ{(1+ϵ)}^2{\tilde{c}}_S}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle \left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\nu +\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (91)

Finally, at the horizon crossing we get the following expression for the spectral tilt for scalar fluctuation at the pivot scale $k=k_{*}$ as:

$$\begin{array}{c}\hfill {\displaystyle n_{\zeta }\left(k_{*}\right)-1={\left[\frac{dln{\Delta }_{\zeta }\left(k\right)}{dlnk}\right]}_{|k{\tilde{c}}_S\eta |=1}=2\eta -4ϵ-\tilde{s},}\end{array}$$

(92)

where $\tilde{s}$ is defined as, $\tilde{s}=\frac{{\dot{\tilde{c}}}_S}{H{\tilde{c}}_S}.$

3.2. For Tensor Modes

3.2.1. Mode Equation and Solution for Tensor Perturbation

Here we compute the two-point correlation from tensor perturbation. For this purpose we consider the second-order perturbed action as given by11:

$$S_{\gamma }^{\left(2\right)}\approx \int d^{4}x{a}^3\frac{M_p^2}{8}\left[\left(1-\frac{{\overline{M}}_3^2}{M_p^2}\right){\dot{\gamma }}_{ij}{\dot{\gamma }}_{ij}-\frac{1}{a^2}{\left({\partial }_m{\gamma }_{ij}\right)}^2\right]=\int d^{3}xd\eta a^2\frac{M_p^2}{8}\left[\left(1-\frac{{\overline{M}}_3^2}{M_p^2}\right){\gamma }_{ij}^{{}^{\prime }2}-{\left({\partial }_m{\gamma }_{ij}\right)}^2\right].$$

(93)

In Fourier space one can write ${\gamma }_{ij}(\eta ,\mathbf{x})$ as:

$${\gamma }_{ij}(\eta ,\mathbf{x})=\sum _{\lambda =\times ,+}\int \frac{d^{3}k}{{\left(2\pi \right)}^{\frac{3}{2}}}{ϵ}_{ij}^{\lambda }\left(k\right){\gamma }_{\lambda }(\eta ,\mathbf{k})e^{i\mathbf{k}.\mathbf{x}},$$

(94)

where the rank-2 polarization tensor ${ϵ}_{ij}^{\lambda }$ satisfies the properties, ${ϵ}_{ii}^{\lambda }=k^i{ϵ}_{ij}^{\lambda }=0,$ ${\sum }_{i,j}{ϵ}_{ij}^{\lambda }{ϵ}_{ij}^{{\lambda }^{{}^{\prime }}}=2{\delta }_{\lambda {\lambda }^{{}^{\prime }}}.$ Similar to scalar fluctuation here we also define a new variable $u_{\lambda }(\eta ,\mathbf{k})$ in Fourier space as:

$$\begin{array}{c}{\displaystyle u_{\lambda }(\eta ,\mathbf{k})=\frac{a}{\sqrt{2}}{M}_p{\gamma }_{\lambda }(\eta ,\mathbf{k})=\left\{\begin{array}{cc}{\displaystyle -\frac{1}{\sqrt{2}H\eta }{M}_p{\gamma }_{\lambda }(\eta ,\mathbf{k})}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle -\frac{1}{\sqrt{2}H\eta }\left(1+ϵ\right)M_p{\gamma }_{\lambda }(\eta ,\mathbf{k})}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.}\hfill \end{array}$$

(95)

Using $u_{\lambda }(\eta ,\mathbf{k})$ one can further write Equation (93) as:

$$\begin{array}{c}\hfill S_{\gamma }^{\left(2\right)}\approx \int d^{3}xd\eta a^2\frac{M_p^2}{4}\left[\left(1-\frac{{\overline{M}}_3^2}{M_p^2}\right)u_{\lambda }^{{}^{\prime }2}(\eta ,\mathbf{k})-\left(k^2-\frac{a^{{}^{\prime \prime }}}{a}\right){\left(u_{\lambda }(\eta ,\mathbf{k})\right)}^2\right].\end{array}$$

(96)

From this action one can find out the mode equation for tensor fluctuation as:

$$\begin{array}{c}\hfill \mathbf{Mukhanov}–\mathbf{Sasaki}\mathbf{Eqn}\mathbf{for}\mathbf{tensor}\mathbf{mode}:u_{\lambda }^{{}^{\prime \prime }}(\eta ,\mathbf{k})+\frac{\left(k^2-\frac{a^{{}^{\prime \prime }}}{a}\right)}{\left(1-\frac{{\overline{M}}_3^2}{M_p^2}\right)}{u}_{\lambda }(\eta ,\mathbf{k})=0.\end{array}$$

(97)

Furthermore, we introduce a new parameter $c_T$ defined as:

$$c_T=\frac{1}{\sqrt{1-\frac{{\overline{M}}_3^2}{M_p^2}}}.$$

(98)

The general solution for the mode equation for graviton fluctuation can finally written as:

$$\begin{array}{c}{\displaystyle u_{\lambda }(\eta ,\mathbf{k})=\left\{\begin{array}{cc}{\displaystyle \sqrt{-\eta }\left[D_1{H}_{\frac{1}{2}\sqrt{1+8c_T^2}}^{\left(1\right)}\left(-k{c}_T\eta \right)+D_2{H}_{\frac{1}{2}\sqrt{1+8c_T^2}}^{\left(2\right)}\left(-k{c}_T\eta \right)\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle \sqrt{-\eta }\left[D_1{H}_{\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}}^{\left(1\right)}\left(-k{c}_T\eta \right)+D_2{H}_{\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}}^{\left(2\right)}\left(-k{c}_T\eta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.}\hfill \end{array}$$

(99)

Here $D_1$ and $D_2$ are the arbitrary integration constants and the numerical values depend on the choice of the initial vacuum. In the present context we consider the following choice of the vacuum for the computation:

• Bunch–Davies vacuum: In this case, we choose, $D_1=1,D_2=0$.
• $\mathit{\alpha },\mathit{\beta }$ vacuum: In this case, we choose $D_1=cosh\alpha ,D_2=e^{i\beta }sinh\alpha$. Here $\beta$ is a phase factor.

For the most general solution as stated in Equation (99) one can consider the limiting physical situations, as given by, I. Superhorizon regime: $|k{c}_T\eta |<<1$, II. Horizon crossing: $|k{c}_T\eta |=1$, III. Subhorizon regime: $|k{c}_T\eta |>>1$.

Finally, considering the behavior of the mode function in the subhorizon regime and superhorizon regime we get:

$$\begin{array}{c}\hfill {\displaystyle u_{\lambda }(\eta ,\mathbf{k})=\left\{\begin{array}{c}{\displaystyle 2^{\frac{1}{2}\sqrt{1+8c_T^2}-\frac{3}{2}}\frac{1}{i\eta }\frac{1}{\sqrt{2}{\left(k{c}_T\right)}^{\frac{3}{2}}}{(-k{c}_T\eta )}^{\frac{3}{2}-\frac{1}{2}\sqrt{1+8c_T^2}}\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}\hfill \\ {\displaystyle \left[D_1{e}^{-ik{c}_T\eta }\left(1+ik{c}_T\eta \right)e^{-\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}-D_2{e}^{ik{c}_T\eta }\left(1-ik{c}_T\eta \right)e^{\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-\frac{3}{2}}\frac{1}{i\eta }\frac{1}{\sqrt{2}{\left(k{c}_T\right)}^{\frac{3}{2}}}{(-k{c}_T\eta )}^{\frac{3}{2}-\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}}\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}\hfill \\ {\displaystyle \left[D_1{e}^{-ik{c}_T\eta }\left(1+ik{c}_T\eta \right)e^{-\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}-D_2{e}^{ik{c}_T\eta }\left(1-ik{c}_T\eta \right)e^{\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.}\end{array}$$

(100)

Furthermore, using Equation (83) one can write down the expression for the curvature perturbation $\zeta (\eta ,\mathbf{k})$ as:

$$\begin{array}{ccc}\hfill {\displaystyle h_{\lambda }(\eta ,\mathbf{k})}& =& \frac{u_{\lambda }(\eta ,\mathbf{k})}{a{M}_p}=\left\{\begin{array}{c}{\displaystyle 2^{\frac{1}{2}\sqrt{1+8c_T^2}-\frac{3}{2}}\frac{iH}{M_p}\frac{1}{{\left(k{c}_T\right)}^{\frac{3}{2}}}{(-k{c}_T\eta )}^{\frac{3}{2}-\frac{1}{2}\sqrt{1+8c_T^2}}\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}\hfill \\ {\displaystyle \left[D_1{e}^{-ik{c}_T\eta }\left(1+ik{c}_T\eta \right)e^{-\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}-D_2{e}^{ik{c}_T\eta }\left(1-ik{c}_T\eta \right)e^{\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-\frac{3}{2}}\frac{iH}{M_p\left(1+ϵ\right)}\frac{1}{{\left(k{c}_T\right)}^{\frac{3}{2}}}{(-k{c}_T\eta )}^{\frac{3}{2}-\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}}}\hfill \\ {\displaystyle \left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|\left[D_1{e}^{-ik{c}_T\eta }\left(1+ik{c}_T\eta \right)e^{-\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}\right.}\hfill \\ \left.{\displaystyle -D_2{e}^{ik{c}_T\eta }\left(1-ik{c}_T\eta \right)e^{\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}}\right]\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

(101)

3.2.2. Primordial Power Spectrum for Tensor Perturbation

One can further compute the two-point function for tensor fluctuation as:

$$\begin{array}{c}\hfill \langle h(\eta ,\mathbf{k})h(\eta ,\mathbf{q})\rangle =\sum _{\lambda ,{\lambda }^{{}^{\prime }}}\langle h_{\lambda }(\eta ,\mathbf{k})h_{{\lambda }^{{}^{\prime }}}(\eta ,\mathbf{q})\rangle ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}(\mathbf{k}+\mathbf{q})P_h(k,\eta ),\end{array}$$

(102)

where $P_h(k,\eta )$ is the power spectrum at time $\eta$ for tensor fluctuations and in the present context it is defined as:

$$\begin{array}{ccc}\hfill {\displaystyle P_h(k,\eta )}& =& \frac{4|h_{\lambda }{(\eta ,\mathbf{k})|}^2}{a^2{M}_p^2}=\left\{\begin{array}{c}{\displaystyle 2^{\sqrt{1+8c_T^2}-3}\frac{4H^2}{M_p^2}\frac{1}{{\left(k{c}_T\right)}^3}{(-k{c}_T\eta )}^{3-\sqrt{1+8c_T^2}}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle {\left|D_1{e}^{-ik{c}_T\eta }\left(1+ik{c}_T\eta \right)e^{-\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}-D_2{e}^{ik{c}_T\eta }\left(1-ik{c}_T\eta \right)e^{\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-3}\frac{4H^2}{M_p^2{\left(1+ϵ\right)}^2}\frac{1}{{\left(k{c}_T\right)}^3}{(-k{c}_T\eta )}^{3-\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}}}\hfill \\ {\displaystyle {\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left|D_1{e}^{-ik{c}_T\eta }\left(1+ik{c}_T\eta \right)e^{-\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}\right.}\hfill \\ {\left.{\displaystyle -D_2{e}^{ik{c}_T\eta }\left(1-ik{c}_T\eta \right)e^{\frac{i\pi }{2}\left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}}\right|}^2\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

(103)

Finally, at the horizon crossing we get the following two-point correlation function for tensor perturbation as:

$$\begin{array}{c}\hfill \langle h\left(\mathbf{k}\right)h\left(\mathbf{q}\right)\rangle ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}(\mathbf{k}+\mathbf{q})P_h\left(k\right),\end{array}$$

(104)

where $P_h\left(k\right)$ is known as the power spectrum at the horizon crossing for tensor fluctuations and in the present context it is defined as:

$$\begin{array}{ccc}\hfill {\displaystyle P_h\left(k\right)}& =& P_h\left(k_{*}\right)\frac{1}{k^3}=\left\{\begin{array}{c}{\displaystyle 2^{\sqrt{1+8c_T^2}-3}\frac{4H^2}{M_p^2{c}_T^3}\frac{1}{k^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left[|D_1{|}^2+{\left|D_2\right|}^2\right.}\hfill \\ \left.{\displaystyle -\left(D_1^{*}{D}_2{e}^{i\pi \left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}+D_1{D}_2^{*}{e}^{-i\pi \left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}\right)}\right]\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-3}\frac{4H^2}{M_p^2{\left(1+ϵ\right)}^2{c}_T^3}\frac{1}{k^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left[|D_1{|}^2+{\left|D_2\right|}^2\right.}\hfill \\ \left.{\displaystyle -\left(D_1^{*}{D}_2{e}^{i\pi \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}+D_1{D}_2^{*}{e}^{-i\pi \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}\right)}\right]\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

(105)

where $P_h\left(k_{*}\right)$ is power spectrum for tensor fluctuation at the pivot scale $k=k_{*}$. For simplicity one can keep $k^3/2{\pi }^2$ dependence outside and further define amplitude of the power spectrum ${\Delta }_h\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

$$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_h\left(k_{*}\right)}& =& \frac{k^3}{2{\pi }^2}{P}_h\left(k\right)=\frac{1}{2{\pi }^2}{P}_h\left(k_{*}\right)\hfill \\ & =& \left\{\begin{array}{c}{\displaystyle 2^{\sqrt{1+8c_T^2}-3}\frac{2H^2}{{\pi }^2{M}_p^2{c}_T^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left[|D_1{|}^2+{\left|D_2\right|}^2\right.}\hfill \\ \left.{\displaystyle -\left(C_1^{*}{C}_2{e}^{i\pi \left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}+D_1{D}_2^{*}{e}^{-i\pi \left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)}\right)}\right]\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-3}\frac{2H^2}{{\pi }^2{M}_p^2{\left(1+ϵ\right)}^2{c}_T^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left[|D_1{|}^2+{\left|D_2\right|}^2\right.}\hfill \\ \left.{\displaystyle -\left(D_1^{*}{D}_2{e}^{i\pi \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}+D_1{D}_2^{*}{e}^{-i\pi \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)}\right)}\right]\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

(106)

For Bunch–Davies and $\mathit{\alpha },\mathit{\beta }$ vacuums we get:

• For Bunch–Davies vacuum:
  In this case, by setting $D_1=1$ and $D_2=0$ we get the following expression for the power spectrum:

  $$\begin{array}{ccc}\hfill {\displaystyle P_h\left(k\right)}& =& \left\{\begin{array}{cc}{\displaystyle 2^{\sqrt{1+8c_T^2}-3}\frac{4H^2}{M_p^2{c}_T^3}\frac{1}{k^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-3}\frac{4H^2}{M_p^2{\left(1+ϵ\right)}^2{c}_T^3}\frac{1}{k^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (107)
  Also, the power spectrum ${\Delta }_{\zeta }\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

  $$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_h\left(k_{*}\right)}& =& \left\{\begin{array}{cc}{\displaystyle 2^{\sqrt{1+8c_T^2}-3}\frac{2H^2}{{\pi }^2{M}_p^2{c}_T^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-3}\frac{2H^2}{{\pi }^2{M}_p^2{\left(1+ϵ\right)}^2{c}_T^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (108)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  In this case, by setting $D_1=cosh\alpha$ and $D_2=e^{i\beta }sinh\alpha$ we get the following expression for the power spectrum:

  $$\begin{array}{ccc}\hfill {\displaystyle P_h\left(k\right)}& =& \left\{\begin{array}{c}{\displaystyle 2^{\sqrt{1+8c_T^2}-3}\frac{4H^2}{M_p^2{c}_T^3}\frac{1}{k^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle \left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-3}\frac{4H^2}{M_p^2{\left(1+ϵ\right)}^2{c}_T^3}\frac{1}{k^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle \left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (109)
  Also, the power spectrum ${\Delta }_{\zeta }\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

  $$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_h\left(k_{*}\right)}& =& \left\{\begin{array}{c}{\displaystyle 2^{\sqrt{1+8c_T^2}-3}\frac{2H^2}{{\pi }^2{M}_p^2{c}_T^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+8c_T^2}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle \left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\frac{1}{2}\sqrt{1+8c_T^2}+\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}-3}\frac{2H^2}{{\pi }^2{M}_p^2{\left(1+ϵ\right)}^2{c}_T^3}{\left|\frac{\Gamma \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}\right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle \left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\frac{1}{2}\sqrt{1+4c_T^2\left({\nu }^2-\frac{1}{4}\right)}+\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (110)

Now let us consider a special case for tensor fluctuation where $c_T=1$ and it implies the following two possibilities:

• ${\overline{M}}_3=0$. But for this case as we have assumed earlier ${\overline{M}}_3^2\approx {\overline{M}}_3^2={\overline{M}}_1^3/4H{\tilde{c}}_5$, then ${\overline{M}}_1=0$ which is not our matter of interest in this work as this leads to zero three-point function for scalar fluctuation. But if we assume that ${\overline{M}}_3^2\ne {\overline{M}}_3^2$ but ${\overline{M}}_3^2={\overline{M}}_1^3/4H{\tilde{c}}_5$ then by setting ${\overline{M}}_3=0$ one can get ${\overline{M}}_1\ne 0$, which is necessarily required for non-vanishing three-point function for scalar fluctuation.
• ${\overline{M}}_3<<M_p$. In this case if we assume ${\overline{M}}_3^2\approx {\overline{M}}_3^2={\overline{M}}_1^3/4H{\tilde{c}}_5$, then ${\overline{M}}_1^3/4H{\tilde{c}}_5{M}_p^2<<1$ and ${\overline{M}}_2<<M_p$. This is perfectly ok of generating non-vanishing three-point function for scalar fluctuation.

If we set $c_T=1$ then for Bunch–Davies and $\alpha ,\beta$ vacuum power spectrum can be recast into the following simplified form:

• For Bunch–Davies vacuum:
  In this case, by setting $D_1=1$ and $D_2=0$ we get the following expression for the power spectrum:

  $$\begin{array}{ccc}\hfill {\displaystyle P_h\left(k\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{4H^2}{M_p^2}\frac{1}{k^3}}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{4H^2}{M_p^2{\left(1+ϵ\right)}^2}\frac{1}{k^3}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (111)
  Also, the power spectrum ${\Delta }_{\zeta }\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

  $$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_h\left(k_{*}\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{2H^2}{{\pi }^2{M}_p^2}}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{\nu -3}\frac{2H^2}{{\pi }^2{M}_p^2{\left(1+ϵ\right)}^2}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (112)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  In this case, by setting $D_1=cosh\alpha$ and $D_2=e^{i\beta }sinh\alpha$ we get the following expression for the power spectrum:

  $$\begin{array}{ccc}\hfill {\displaystyle P_h\left(k\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{4H^2}{M_p^2}\frac{1}{k^3}\left[cosh2\alpha -sinh2\alpha cos\beta \right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{4H^2}{M_p^2{\left(1+ϵ\right)}^2}\frac{1}{k^3}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2\left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\nu +\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (113)
  Also, the power spectrum ${\Delta }_{\zeta }\left(k_{*}\right)$ at the pivot scale $k=k_{*}$ as:

  $$\begin{array}{ccc}\hfill {\displaystyle {\Delta }_h\left(k_{*}\right)}& =& \left\{\begin{array}{cc}{\displaystyle \frac{2H^2}{{\pi }^2{M}_p^2}\left[cosh2\alpha -sinh2\alpha cos\beta \right]}\hfill & \mathbf{for}\mathbf{dS}\hfill \\ {\displaystyle 2^{2\nu -3}\frac{2H^2}{{\pi }^2{M}_p^2{\left(1+ϵ\right)}^2}{\left|\frac{\Gamma \left(\nu \right)}{\Gamma \left(\frac{3}{2}\right)}\right|}^2}\hfill \\ {\displaystyle \left[cosh2\alpha -sinh2\alpha cos\left(\pi \left(\nu +\frac{1}{2}\right)+\beta \right)\right]}\hfill & \mathbf{for}\mathbf{qdS}.\hfill \end{array}\right.\hfill \end{array}$$

  (114)

4. Scalar Three-Point Correlation Function from EFT

4.1. Basic Setup

Here we compute the three-point correlation function for perturbations from scalar modes. For this purpose we consider the third-order perturbed action for the scalar modes as given by12:

$$\begin{array}{cc}\hfill S_{\zeta }^{\left(3\right)}& {\displaystyle \approx \int d^{4}x\frac{a^3}{H^3}\left[-\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}{\dot{\zeta }}^3\right.}\hfill \\ & {\displaystyle +\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\frac{1}{a^2}\dot{\zeta }{\left({\partial }_i\zeta \right)}^2}\hfill \\ & \left.{\displaystyle \underset{\mathbf{New}\mathbf{contributions}\mathbf{in}\mathbf{EFT}\mathbf{action}}{\underbrace{+\frac{9}{2}{\overline{M}}_1^3{H}^2\zeta {\dot{\zeta }}^2-\frac{3}{2}{\overline{M}}_1^{3}H\frac{1}{a^2}\zeta \frac{d}{dt}{\left({\partial }_i\zeta \right)}^2}}}\right],\hfill \end{array}$$

(115)

which can be recast for $c_S=1$ and $c_S<1$ case as:

$$\begin{array}{cc}\mathbf{For}{\mathbf{c}}_{\mathbf{S}}=\mathbf{1}:\hfill & \\ \hfill S_{\zeta }^{\left(3\right)}& {\displaystyle \approx \int d^{4}x\frac{a^3}{H^3}\left[-\left\{\frac{3}{2}{\overline{M}}_1^{3}H\right\}{\dot{\zeta }}^3+\left\{\frac{3}{2}{\overline{M}}_1^{3}H\right\}\frac{1}{a^2}\dot{\zeta }{\left({\partial }_i\zeta \right)}^2\right.}\hfill \\ & \left.{\displaystyle +\frac{9}{2}{\overline{M}}_1^3{H}^2\zeta {\dot{\zeta }}^2-\frac{3}{2}{\overline{M}}_1^{3}H\frac{1}{a^2}\zeta \frac{d}{dt}{\left({\partial }_i\zeta \right)}^2}\right],\hfill \end{array}$$

(116)

$$\begin{array}{cc}\hfill \mathbf{For}{\mathbf{c}}_{\mathbf{S}}<\mathbf{1}:& \\ \hfill S_{\zeta }^{\left(3\right)}& {\displaystyle \approx \int d^{4}x{a}^3\frac{ϵM_p^2}{H}\left(1-\frac{1}{c_S^2}\right)\left[\left\{1+\frac{3{\tilde{c}}_4}{4c_S^2}+\frac{2{\tilde{c}}_3}{3c_S^2}\right\}{\dot{\zeta }}^3-\left\{1+\frac{3{\tilde{c}}_4}{4c_S^2}\right\}\frac{1}{a^2}\dot{\zeta }{\left({\partial }_i\zeta \right)}^2\right.}\hfill \\ & \left.{\displaystyle -\frac{9H{\tilde{c}}_4}{4c_S^2}\zeta {\dot{\zeta }}^2+\frac{3{\tilde{c}}_4}{4c_S^2}\frac{1}{a^2}\zeta \frac{d}{dt}{\left({\partial }_i\zeta \right)}^2}\right],\hfill \end{array}$$

(117)

To extract further information from third-order action, first one needs to start with the Fourier transform of the curvature perturbation $\zeta (\eta ,\mathbf{x})$ defined as:

$$\begin{array}{ccc}\hfill \zeta (\eta ,\mathbf{x})& =& \int \frac{d^{3}k}{{\left(2\pi \right)}^3}{\zeta }_{\mathbf{k}}\left(\eta \right)exp(i\mathbf{k}.\mathbf{x}),\hfill \end{array}$$

(118)

where ${\zeta }_{\mathbf{k}}\left(\eta \right)$ is the time-dependent part of the curvature fluctuation after Fourier transform and can be expressed in terms of the normalized time-dependent scalar mode function $v_{\mathbf{k}}\left(\eta \right)$ as:

$$\widehat{\zeta }(\eta ,\mathbf{k})=\frac{v_{\mathbf{k}}\left(\eta \right)}{z{M}_p}=\frac{\zeta \left(\eta ,\mathbf{k}\right)a\left(\mathbf{k}\right)+{\zeta }^{*}\left(\eta ,-\mathbf{k}\right)a^{†}\left(-\mathbf{k}\right)}{z{M}_p}$$

(119)

where z is explicitly defined earlier and $a\left(\mathbf{k}\right),a^{†}\left(\mathbf{k}\right)$ are the creation and annihilation operator satisfies the following commutation relations:

$$\begin{array}{ccc}\hfill \left[a\left(\mathbf{k}\right),a^{†}(-{\mathbf{k}}^{{}^{\prime }})\right]& =& {\left(2\pi \right)}^3{\delta }^3(\mathbf{k}+{\mathbf{k}}^{{}^{\prime }}),\left[a\left(\mathbf{k}\right),a\left({\mathbf{k}}^{{}^{\prime }}\right)\right]=0,\left[a^{†}\left(\mathbf{k}\right),a^{†}\left({\mathbf{k}}^{{}^{\prime }}\right)\right]=0.\hfill \end{array}$$

(120)

4.2. Computation of Scalar Three-Point Function in Interaction Picture

Presently our prime objective is to compute the three-point function of the curvature fluctuation in momentum space from $S_{\zeta }^2$ with respect to the arbitrary choice of vacuum, which leads to important result in the context of primordial cosmology. For more details see Appendix B where we have explicitly discussed about the background physics of choosing various types of quantum vacuum. Furthermore, using the In-In formalism discussed in Appendix A in the interaction picture the three-point function of the curvature fluctuation in momentum space can be expressed as:

$$\langle \zeta \left({\mathbf{k}}_{\mathbf{1}}\right)\zeta \left({\mathbf{k}}_{\mathbf{2}}\right)\zeta \left({\mathbf{k}}_{\mathbf{3}}\right)\rangle =-i\underset{{\eta }_i=-\infty }{\overset{{\eta }_f=0}{\int }}d\eta a\left(\eta \right)\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),H_{int}\left(\eta \right)\right]|0\rangle ,$$

(121)

where $a\left(\eta \right)$ is the scale factor defined in the earlier section in terms of Hubble parameter H and conformal time scale $\eta$. Here $|0\rangle$ represents any arbitrary vacuum state and for discussion we will only derive the results for Bunch–Davies vacuum and $\alpha ,\beta$ vacuum. In the interaction picture the Hamiltonian can written as13:

$$\begin{array}{cc}\hfill H_{int}\left(\eta \right)& {\displaystyle =-\frac{1}{H^3}\left[-\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}{\dot{\zeta }}^3\right.}\hfill \\ & {\displaystyle +\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\frac{1}{a^2}\dot{\zeta }{\left({\partial }_i\zeta \right)}^2}\hfill \\ & \left.{\displaystyle +\frac{9}{2}{\overline{M}}_1^3{H}^2\zeta {\dot{\zeta }}^2-\frac{3}{2}{\overline{M}}_1^{3}H\frac{1}{a^2}\zeta \frac{d}{dt}{\left({\partial }_i\zeta \right)}^2}\right].\hfill \end{array}$$

(122)

which gives the primary information to compute the explicit expression for the three-point function in the present context. After substituting the Hamiltonian interaction, we finally get the following expression for the three-point function for the scalar fluctuation:

$$\begin{array}{ccc}\hfill \langle \zeta \left({\mathbf{k}}_{\mathbf{1}}\right)\zeta \left({\mathbf{k}}_{\mathbf{2}}\right)\zeta \left({\mathbf{k}}_{\mathbf{3}}\right)\rangle & =& -i\underset{{\eta }_i=-\infty }{\overset{{\eta }_f=0}{\int }}d\eta a\left(\eta \right)\int \frac{d^{3}x}{H^3}\int \int \int \frac{d^3{k}_4}{{\left(2\pi \right)}^3}\frac{d^3{k}_5}{{\left(2\pi \right)}^3}\frac{d^3{k}_6}{{\left(2\pi \right)}^3}{e}^{i({\mathbf{k}}_{\mathbf{4}}+{\mathbf{k}}_{\mathbf{5}}+{\mathbf{k}}_{\mathbf{6}}).\mathbf{x}}\hfill \\ & & \left\{{\alpha }_1\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),{\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{4}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{5}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \right.\hfill \\ & & -{\alpha }_2({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),{\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{4}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{5}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \hfill \\ & & +{\alpha }_{3}a\left(\eta \right)\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{4}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{5}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \hfill \\ & & -{\alpha }_4({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{4}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{5}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \hfill \\ & & \left.-{\alpha }_5({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{4}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{5}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \right\},\hfill \end{array}$$

(123)

where the coefficients ${\alpha }_j\forall j=1,2,3,4,5$ are defined as14:

$$\begin{array}{ccc}\hfill {\alpha }_1& =& \left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\},\hfill \end{array}$$

(124)

$$\begin{array}{ccc}\hfill {\alpha }_2& =& -\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\},\hfill \end{array}$$

(125)

$$\begin{array}{ccc}\hfill {\alpha }_3& =& -\frac{9}{2}{\overline{M}}_1^3{H}^2,\hfill \end{array}$$

(126)

$$\begin{array}{ccc}\hfill {\alpha }_4& =& \frac{3}{2}{\overline{M}}_1^{3}H,\hfill \end{array}$$

(127)

$$\begin{array}{ccc}\hfill {\alpha }_5& =& \frac{3}{2}{\overline{M}}_1^{3}H.\hfill \end{array}$$

(128)

Now let us evaluate the coefficients of ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ in the present context using Wick’s theorem:

• Coefficient of ${\alpha }_1$:

  $$\begin{array}{c}\hfill \langle 0\left|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),{\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{4}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{5}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]\right|0\rangle \\ \hfill =\langle 0\left|a\left({\mathbf{k}}_{\mathbf{1}}\right)a\left({\mathbf{k}}_{\mathbf{2}}\right)a\left({\mathbf{k}}_{\mathbf{3}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{4}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{5}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{6}}\right)\right|0\rangle \\ \hfill \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{6}})\\ \hfill +\langle 0\left|a\left({\mathbf{k}}_{\mathbf{4}}\right)a\left({\mathbf{k}}_{\mathbf{5}}\right)a\left({\mathbf{k}}_{\mathbf{6}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{1}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{2}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{3}}\right)\right|0\rangle \\ \hfill {\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{6}}).\end{array}$$

  (129)
• Coefficient of ${\alpha }_2$:

  $$\begin{array}{c}\hfill ({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),{\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{4}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{5}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \\ \hfill =({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|a\left({\mathbf{k}}_{\mathbf{1}}\right)a\left({\mathbf{k}}_{\mathbf{2}}\right)a\left({\mathbf{k}}_{\mathbf{3}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{4}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{5}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{6}}\right)|0\rangle \\ \hfill \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{6}})\\ \hfill +({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|a\left({\mathbf{k}}_{\mathbf{4}}\right)a\left({\mathbf{k}}_{\mathbf{5}}\right)a\left({\mathbf{k}}_{\mathbf{6}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{1}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{2}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{3}}\right)|0\rangle \\ \hfill {\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}({\eta }_f,{\mathbf{k}}_{\mathbf{4}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{5}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{6}}).\end{array}$$

  (130)
• Coefficient of ${\alpha }_3$:

  $$\begin{array}{c}\hfill \langle 0\left|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{4}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{5}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]\right|0\rangle \\ \hfill =\langle 0\left|a\left({\mathbf{k}}_{\mathbf{1}}\right)a\left({\mathbf{k}}_{\mathbf{2}}\right)a\left({\mathbf{k}}_{\mathbf{3}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{4}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{5}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{6}}\right)\right|0\rangle \\ \hfill \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{6}})\\ \hfill +\langle 0\left|a\left({\mathbf{k}}_{\mathbf{4}}\right)a\left({\mathbf{k}}_{\mathbf{5}}\right)a\left({\mathbf{k}}_{\mathbf{6}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{1}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{2}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{3}}\right)\right|0\rangle \\ \hfill {\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{{}^{\prime }}({\eta }_f,{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{{}^{\prime }}({\eta }_f,{\mathbf{k}}_{\mathbf{6}}).\end{array}$$

  (131)
• Coefficient of ${\alpha }_4$:

  $$\begin{array}{c}\hfill ({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{4}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{5}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \\ \hfill =({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|a\left({\mathbf{k}}_{\mathbf{1}}\right)a\left({\mathbf{k}}_{\mathbf{2}}\right)a\left({\mathbf{k}}_{\mathbf{3}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{4}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{5}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{6}}\right)|0\rangle \\ \hfill \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{6}})\\ \hfill +({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|a\left({\mathbf{k}}_{\mathbf{4}}\right)a\left({\mathbf{k}}_{\mathbf{5}}\right)a\left({\mathbf{k}}_{\mathbf{6}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{1}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{2}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{3}}\right)|0\rangle \\ \hfill {\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{{}^{\prime }}({\eta }_f,{\mathbf{k}}_{\mathbf{5}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{6}}).\end{array}$$

  (132)
• Coefficient of ${\alpha }_5$:

  $$\begin{array}{c}\hfill ({\mathbf{k}}_{\mathbf{5}}.{\mathbf{k}}_{\mathbf{6}})\langle 0|\left[\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\widehat{\zeta }({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{4}})\widehat{\zeta }(\eta ,{\mathbf{k}}_{\mathbf{5}}){\widehat{\zeta }}^{{}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{6}})\right]|0\rangle \\ \hfill =\langle 0\left|a\left({\mathbf{k}}_{\mathbf{1}}\right)a\left({\mathbf{k}}_{\mathbf{2}}\right)a\left({\mathbf{k}}_{\mathbf{3}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{4}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{5}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{6}}\right)\right|0\rangle \\ \hfill \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{4}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{{}^{\prime }*}({\eta }_f,-{\mathbf{k}}_{\mathbf{6}})\\ \hfill +\langle 0\left|a\left({\mathbf{k}}_{\mathbf{4}}\right)a\left({\mathbf{k}}_{\mathbf{5}}\right)a\left({\mathbf{k}}_{\mathbf{6}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{1}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{2}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{3}}\right)\right|0\rangle \\ \hfill {\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{4}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{5}}){\overline{v}}^{{}^{\prime }}({\eta }_f,{\mathbf{k}}_{\mathbf{6}}).\end{array}$$

  (133)

where we define $\overline{v}$ as:

$$\overline{v}(\eta ,\mathbf{k})=\frac{v(\eta ,\mathbf{k})}{z{M}_p}.$$

(134)

Furthermore, we also use the following result in simplify the coefficients of ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$:

$$\begin{array}{cc}& \langle 0\left|a\left({\mathbf{k}}_{\mathbf{1}}\right)a\left({\mathbf{k}}_{\mathbf{2}}\right)a\left({\mathbf{k}}_{\mathbf{3}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{4}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{5}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{6}}\right)\right|0\rangle \hfill \\ =& \langle 0\left|a\left({\mathbf{k}}_{\mathbf{4}}\right)a\left({\mathbf{k}}_{\mathbf{5}}\right)a\left({\mathbf{k}}_{\mathbf{6}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{1}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{2}}\right)a^{†}\left(-{\mathbf{k}}_{\mathbf{3}}\right)\right|0\rangle \hfill \\ =& {\left(2\pi \right)}^9\left\{{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{4}}+{\mathbf{k}}_{\mathbf{1}})\left[{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{5}}+{\mathbf{k}}_{\mathbf{2}}){\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{6}}+{\mathbf{k}}_{\mathbf{3}})+{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{5}}+{\mathbf{k}}_{\mathbf{3}}){\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{6}}+{\mathbf{k}}_{\mathbf{2}})\right]\right.\hfill \\ & +{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{4}}+{\mathbf{k}}_{\mathbf{2}})\left[{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{5}}+{\mathbf{k}}_{\mathbf{1}}){\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{6}}+{\mathbf{k}}_{\mathbf{3}})+{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{5}}+{\mathbf{k}}_{\mathbf{3}}){\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{6}}+{\mathbf{k}}_{\mathbf{1}})\right]\hfill \\ & \left.+{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{4}}+{\mathbf{k}}_{\mathbf{3}})\left[{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{5}}+{\mathbf{k}}_{\mathbf{1}}){\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{6}}+{\mathbf{k}}_{\mathbf{3}})+{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{5}}+{\mathbf{k}}_{\mathbf{2}}){\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{6}}+{\mathbf{k}}_{\mathbf{1}})\right]\right\}.\hfill \end{array}$$

(135)

Finally, one can write the following expression for the three-point function for the scalar fluctuation15:

$$\begin{array}{c}\hfill \langle \zeta \left({\mathbf{k}}_{\mathbf{1}}\right)\zeta \left({\mathbf{k}}_{\mathbf{2}}\right)\zeta \left({\mathbf{k}}_{\mathbf{3}}\right)\rangle ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}+{\mathbf{k}}_{\mathbf{3}})B_{EFT}(k_1,k_2,k_3).\end{array}$$

(136)

where $B_{EFT}(k_1,k_2,k_3)$ is the bispectrum for scalar fluctuation. In the present computation, one can further write down the expression for the bispectrum as:

$$\begin{array}{c}\hfill B_{EFT}(k_1,k_2,k_3)=\sum _{j=1}^5{\alpha }_j{\mathsf{\Theta }}_j(k_1,k_2,k_3),\end{array}$$

(137)

where ${\mathsf{\Theta }}_j(k_1,k_2,k_3)\forall j=1,2,3,4,5$ is defined in the next subsections. Here it is important to note that we have derived the expression for the three-point function and the associated bispectrum for effective sound speed $c_S=1$ and $c_S<1$ with a choice of general quantum vacuum state.

4.2.1. Function ${\mathsf{\Theta }}_1(k_1,k_2,k_3)$

Here we can write the function ${\mathsf{\Theta }}_1(k_1,k_2,k_3)$ as:

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_1(k_1,k_2,k_3)& =& 6i{\int }_{{\eta }_i=-\infty }^{{\eta }_f=0}d\eta \frac{a\left(\eta \right)}{H^3}\left[\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{3}})\right.\hfill \\ & & \left.+{\overline{v}}^{*}({\eta }_f,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\right].\hfill \end{array}$$

(138)

Furthermore, using the integrals from the Appendix C we finally get the following simplified expression for the three-point function for the scalar fluctuations16:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_1(k_1,k_2,k_3)& {\displaystyle =\frac{3H^2}{16{ϵ}^3{M}_p^6}\frac{1}{k_1{k}_2{k}_3}\left[\left\{\frac{1}{K^3}\left[{\left(C_1-C_2\right)}^3\left(C_1^{*3}+C_2^{*3}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3\left(C_1^3+C_2^3\right)\right]\right.\right.}\hfill \\ & \left.\left.{\displaystyle +\left[{\left(C_1-C_2\right)}^3{C}_1^{*}{C}_2^{*}\left(C_1^{*}-C_2^{*}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3{C}_1{C}_2\left(C_1-C_2\right)\right]\sum _{i=1}^3\frac{1}{{(2k_i-K)}^3}}\right\}\right],\hfill \end{array}$$

(140)

Finally, for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following contribution in the three-point function for scalar fluctuations:

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get:

  $$\begin{array}{c}\hfill {\mathsf{\Theta }}_1(k_1,k_2,k_3)=\frac{6H^2}{16{ϵ}^3{M}_p^6}\frac{1}{k_1{k}_2{k}_3}\frac{1}{K^3},\end{array}$$

  (141)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

  $$\begin{array}{cc}\hfill {\mathsf{\Theta }}_1(k_1,k_2,k_3)& {\displaystyle =\frac{3H^2}{16{ϵ}^3{M}_p^6}\frac{1}{k_1{k}_2{k}_3}\left\{\frac{1}{K^3}\left[{\left(cosh\alpha -e^{i\beta }sinh\alpha \right)}^3\left({cosh}^3\alpha +e^{-3i\beta }{sinh}^3\alpha \right)\right.\right.}\hfill \\ & \left.{\displaystyle +{\left(cosh\alpha -e^{-i\beta }sinh\alpha \right)}^3\left({cosh}^3\alpha +e^{3i\beta }{sinh}^3\alpha \right)}\right]\hfill \\ & {\displaystyle +\frac{1}{2}\left[{\left(cosh\alpha -e^{i\beta }sinh\alpha \right)}^3{e}^{-i\beta }sinh2\alpha \left(cosh\alpha -e^{-i\beta }sinh\alpha \right)\right.}\hfill \\ & \left.{\displaystyle \left.{\displaystyle +{\left(cosh\alpha -e^{-i\beta }sinh\alpha \right)}^3{e}^{i\beta }sinh2\alpha \left(cosh\alpha -e^{i\beta }sinh\alpha \right)}\right]\sum _{i=1}^3\frac{1}{{(2k_i-K)}^3}}\right\}.\hfill \end{array}$$

  (142)

4.2.2. Function ${\mathsf{\Theta }}_2(k_1,k_2,k_3)$

Here we can write the function ${\mathsf{\Theta }}_2(k_1,k_2,k_3)$ as:

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_2(k_1,k_2,k_3)& =& i{\int }_{{\eta }_i=-\infty }^{{\eta }_f=0}d\eta \frac{a\left(\eta \right)}{H^3}\left\{2({\mathbf{k}}_{\mathbf{2}}.{\mathbf{k}}_{\mathbf{3}})\left[\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}})\right.\right.\hfill \\ & & \left.+{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\right]\hfill \\ & & +2({\mathbf{k}}_{\mathbf{3}}.{\mathbf{k}}_{\mathbf{1}})\left[\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}})\right.\hfill \\ & & \left.+{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\right]\hfill \\ & & +2({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}})\left[\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}})\right.\hfill \\ & & \left.\left.+{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\right]\right\}.\hfill \end{array}$$

(143)

Using the results derived in Appendix C we finally get the following simplified expression for the three-point function for the scalar fluctuations17:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_2(k_1,k_2,k_3)& {\displaystyle =\frac{H^2}{32{ϵ}^3{M}_p^6{\tilde{c}}_S^2}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[k_1^2({\mathbf{k}}_{\mathbf{2}}.{\mathbf{k}}_{\mathbf{3}})G_1(k_1,k_2,k_3)\right.}\hfill \\ & \left.{\displaystyle +k_2^2({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{3}})G_2(k_1,k_2,k_3)+k_3^2({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}})G_3(k_1,k_2,k_3)}\right],\hfill \end{array}$$

(145)

where the momentum dependent functions $G_1(k_1,k_2,k_3)$, $G_2(k_1,k_2,k_3)$ and $G_3(k_1,k_2,k_3)$ are defined as:

$$\begin{array}{ccc}\hfill G_1(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_1-K)}^3}\left[K^2+2k_2{k}_3+K(K-5k_1)-2(K-k_1)k_1+4k_1^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_2{k}_3+K(k_3-5k_2)+6k_2^2\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_2{k}_3+K(k_2-5k_3)+6k_3^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(146)

$$\begin{array}{ccc}\hfill G_2(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_2-K)}^3}\left[K^2+2k_1{k}_3+K(K-5k_2)-2(K-k_2)k_2+4k_2^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_3+K(k_3-5k_1)+6k_1^2\right]\hfill \\ & & \left.+\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_1{k}_3+K(k_1-5k_3)+6k_3^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(147)

$$\begin{array}{ccc}\hfill G_3(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_3-K)}^3}\left[K^2+2k_1{k}_2+K(K-5k_3)-2(K-k_3)k_3+4k_3^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_1{k}_2+K(k_1-5k_2)+6k_2^2\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_2+K(k_2-5k_1)+6k_1^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(148)

Here ${\sum }_{i=1}^3{\mathbf{k}}_{\mathbf{i}}={\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}+{\mathbf{k}}_{\mathbf{3}}=0.$ Consequently, one can write:

$$\begin{array}{ccc}\hfill {\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}}& =& \frac{1}{2}\left(k_3^2-k_2^2-k_1^2\right),{\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{3}}=\frac{1}{2}\left(k_2^2-k_1^2-k_3^2\right),{\mathbf{k}}_{\mathbf{2}}.{\mathbf{k}}_{\mathbf{3}}=\frac{1}{2}\left(k_1^2-k_2^2-k_3^2\right),\hfill \end{array}$$

(149)

and using these results one can further recast the three-point function for the scalar fluctuation as:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_2(k_1,k_2,k_3)& {\displaystyle =\frac{H^2}{64{ϵ}^3{M}_p^6{\tilde{c}}_S^2}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[k_1^2\left(k_1^2-k_2^2-k_3^2\right)G_1(k_1,k_2,k_3)\right.}\hfill \\ & {\displaystyle \left.+k_2^2\left(k_2^2-k_1^2-k_3^2\right)G_2(k_1,k_2,k_3)+k_3^2\left(k_3^2-k_2^2-k_1^2\right)G_3(k_1,k_2,k_3)\right].}\hfill \end{array}$$

(150)

Finally, for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following contribution in the three-point function for scalar fluctuations:

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get:

  $$\begin{array}{ccc}\hfill G_1(k_1,k_2,k_3)& =& \frac{2}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right].\hfill \end{array}$$

  (151)

  $$\begin{array}{ccc}\hfill G_2(k_1,k_2,k_3)& =& \frac{2}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right].\hfill \end{array}$$

  (152)

  $$\begin{array}{ccc}\hfill G_3(k_1,k_2,k_3)& =& \frac{2}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right].\hfill \end{array}$$

  (153)
  Consequently, we get:

  $$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_2(k_1,k_2,k_3)& =& \frac{H^2}{32{ϵ}^3{M}_p^6{\tilde{c}}_S^2}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\frac{1}{K^3}\left[k_1^2\left(k_1^2-k_2^2-k_3^2\right)\left[K^2+2k_2{k}_3+K(K-k_1)\right]\right.\hfill \\ & & +k_2^2\left(k_2^2-k_1^2-k_3^2\right)\left[K^2+2k_1{k}_3+K(K-k_2)\right]\hfill \\ & & \left.+k_3^2\left(k_3^2-k_2^2-k_1^2\right)\left[K^2+2k_1{k}_2+K(K-k_3)\right]\right].\hfill \end{array}$$

  (154)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

  $$\begin{array}{ccc}\hfill G_1(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_1-K)}^3}\left[K^2+2k_2{k}_3+K(K-5k_1)-2(K-k_1)k_1+4k_1^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_2{k}_3+K(k_3-5k_2)+6k_2^2\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_2{k}_3+K(k_2-5k_3)+6k_3^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (155)

  $$\begin{array}{ccc}\hfill G_2(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_2-K)}^3}\left[K^2+2k_1{k}_3+K(K-5k_2)-2(K-k_2)k_2+4k_2^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_3+K(k_3-5k_1)+6k_1^2\right]\hfill \\ & & \left.+\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_1{k}_3+K(k_1-5k_3)+6k_3^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (156)

  $$\begin{array}{ccc}\hfill G_3(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_3-K)}^3}\left[K^2+2k_1{k}_2+K(K-5k_3)-2(K-k_3)k_3+4k_3^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_1{k}_2+K(k_1-5k_2)+6k_2^2\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_2+K(k_2-5k_1)+6k_1^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (157)

  where $J_1(\alpha ,\beta )$ and $J_2(\alpha ,\beta )$ are defined as:

  $$\begin{array}{ccc}\hfill J_1(\alpha ,\beta )& =& \left[{\left(cosh\alpha -e^{i\beta }sinh\alpha \right)}^3\left({cosh}^3\alpha +e^{-3i\beta }{sinh}^3\alpha \right)\right.\hfill \\ & & \left.+{\left(cosh\alpha -e^{-i\beta }sinh\alpha \right)}^3\left({cosh}^3\alpha +e^{3i\beta }{sinh}^3\alpha \right)\right],\hfill \end{array}$$

  (158)

  $$\begin{array}{ccc}\hfill J_2(\alpha ,\beta )& =& \frac{1}{2}\left[{\left(cosh\alpha -e^{i\beta }sinh\alpha \right)}^3{e}^{-i\beta }sinh2\alpha \left(cosh\alpha -e^{-i\beta }sinh\alpha \right)\right.\hfill \\ & & \left.+{\left(cosh\alpha -e^{-i\beta }sinh\alpha \right)}^3{e}^{i\beta }sinh2\alpha \left(cosh\alpha -e^{i\beta }sinh\alpha \right)\right].\hfill \end{array}$$

  (159)

  Consequently the three-point function for the scalar fluctuation can also be written after substituting all the momentum dependent functions $G_1(k_1,k_2,k_3)$, $G_2(k_1,k_2,k_3)$ and $G_3(k_1,k_2,k_3)$ for $\alpha ,\beta$ vacuum.

4.2.3. Function ${\mathsf{\Theta }}_3(k_1,k_2,k_3)$

Here we can write the function ${\mathsf{\Theta }}_3(k_1,k_2,k_3)$ as:

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_3(k_1,k_2,k_3)& =& 2i{\int }_{{\eta }_i=-\infty }^{{\eta }_f=0}d\eta \frac{a^2\left(\eta \right)}{H^3}\left\{\left[\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{3}})\right.\right.\hfill \\ & & \left.+{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\right]\hfill \\ & & +\left[\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{3}})\right.\hfill \\ & & \left.+{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\right]\hfill \\ & & +\left[\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{2}})\right.\hfill \\ & & \left.\left.+{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\right]\right\}.\hfill \end{array}$$

(160)

Using the results obtained in the Appendix C we finally get the following simplified expression for the three-point function for the scalar fluctuations18:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_3(k_1,k_2,k_3)& {\displaystyle =\frac{H}{32{ϵ}^3{M}_p^6}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[{\left(k_2{k}_3\right)}^2{M}_1(k_1,k_2,k_3)\right.}\hfill \\ & \left.{\displaystyle +{\left(k_1{k}_3\right)}^2{M}_2(k_1,k_2,k_3)+{\left(k_1{k}_2\right)}^2{M}_3(k_1,k_2,k_3)}\right],\hfill \end{array}$$

(162)

where the momentum dependent functions $M_1(k_1,k_2,k_3)$, $M_2(k_1,k_2,k_3)$ and $M_3(k_1,k_2,k_3)$ are defined as:

$$\begin{array}{ccc}\hfill M_1(k_1,k_2,k_3)& =& \frac{1}{K^2}(K+k_1)\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{(K-3k_1)}{{(2k_1-K)}^2}+\frac{(K+k_1-2k_2)}{{(2k_2-K)}^2}+\frac{(K+k_1-2k_3)}{{(2k_3-K)}^2}\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(163)

$$\begin{array}{ccc}\hfill M_2(k_1,k_2,k_3)& =& \frac{1}{K^2}(K+k_2)\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{(K-3k_2)}{{(2k_2-K)}^2}+\frac{(K+k_2-2k_1)}{{(2k_1-K)}^2}+\frac{(K+k_2-2k_3)}{{(2k_3-K)}^2}\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(164)

$$\begin{array}{ccc}\hfill M_3(k_1,k_2,k_3)& =& \frac{1}{K^2}(K+k_3)\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{(K-3k_3)}{{(2k_3-K)}^2}+\frac{(K+k_3-2k_2)}{{(2k_2-K)}^2}+\frac{(K+k_3-2k_1)}{{(2k_1-K)}^2}\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(165)

Finally, for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following contribution in the three-point function for scalar fluctuations:

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get:

  $$\begin{array}{ccc}\hfill M_1(k_1,k_2,k_3)& =& \frac{2}{K^2}(K+k_1).\hfill \end{array}$$

  (166)

  $$\begin{array}{ccc}\hfill M_2(k_1,k_2,k_3)& =& \frac{2}{K^2}(K+k_2).\hfill \end{array}$$

  (167)

  $$\begin{array}{ccc}\hfill M_3(k_1,k_2,k_3)& =& \frac{2}{K^2}(K+k_3).\hfill \end{array}$$

  (168)

  Consequently, we get the following contribution:

  $$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_3(k_1,k_2,k_3)& =& \frac{H}{16{ϵ}^3{M}_p^6}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\frac{1}{K^2}\left[{\left(k_2{k}_3\right)}^2(K+k_1)]+{\left(k_1{k}_3\right)}^2(K+k_2)+{\left(k_1{k}_2\right)}^2(K+k_3)\right],\hfill \end{array}$$

  (169)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

  $$\begin{array}{ccc}\hfill M_1(k_1,k_2,k_3)& =& \frac{(K+k_1)J_1(\alpha ,\beta )}{K^2}+\left\{\frac{(K-3k_1)}{{(2k_1-K)}^2}+\frac{(K+k_1-2k_2)}{{(2k_2-K)}^2}+\frac{(K+k_1-2k_3)}{{(2k_3-K)}^2}\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (170)

  $$\begin{array}{ccc}\hfill M_2(k_1,k_2,k_3)& =& \frac{(K+k_2)J_1(\alpha ,\beta )}{K^2}+\left\{\frac{(K-3k_2)}{{(2k_2-K)}^2}+\frac{(K+k_2-2k_1)}{{(2k_1-K)}^2}+\frac{(K+k_2-2k_3)}{{(2k_3-K)}^2}\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (171)

  $$\begin{array}{ccc}\hfill M_3(k_1,k_2,k_3)& =& \frac{(K+k_3)J_1(\alpha ,\beta )}{K^2}+\left\{\frac{(K-3k_3)}{{(2k_3-K)}^2}+\frac{(K+k_3-2k_2)}{{(2k_2-K)}^2}+\frac{(K+k_3-2k_1)}{{(2k_1-K)}^2}\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (172)

  where $J_1(\alpha ,\beta )$ and $J_2(\alpha ,\beta )$ are defined earlier. Consequently the three-point function for the scalar fluctuation can also be written after substituting all the momentum dependent functions $M_1(k_1,k_2,k_3)$, $M_2(k_1,k_2,k_3)$ and $M_3(k_1,k_2,k_3)$ for $\alpha ,\beta$ vacuum.

4.2.4. Function ${\mathsf{\Theta }}_4(k_1,k_2,k_3)$

Here we can write the function ${\mathsf{\Theta }}_4(k_1,k_2,k_3)$ as:

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_4(k_1,k_2,k_3)& =& i{\int }_{{\eta }_i=-\infty }^{{\eta }_f=0}d\eta \frac{a\left(\eta \right)}{H^3}\left\{({\mathbf{k}}_{\mathbf{2}}.{\mathbf{k}}_{\mathbf{3}})X_1(k_1,k_2,k_3)+({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{3}})X_2(k_1,k_2,k_3)+({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}})X_3(k_1,k_2,k_3)\right\},\hfill \end{array}$$

(173)

where the momentum dependent functions $X_1(k_1,k_2,k_3)$, $X_2(k_1,k_2,k_3)$ and $X_3(k_1,k_2,k_3)$ can be expressed in terms of the various combinations of the scalar mode functions as:

$$\begin{array}{ccc}\hfill X_1(k_1,k_2,k_3)& =& \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}}),\hfill \end{array}$$

(174)

$$\begin{array}{ccc}\hfill X_2(k_1,k_2,k_3)& =& \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}}),\hfill \end{array}$$

(175)

$$\begin{array}{ccc}\hfill X_3(k_1,k_2,k_3)& =& \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\hfill \\ & & +({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}}),\hfill \end{array}$$

(176)

Using the results obtained in the Appendix C we finally get the following simplified expression for the three-point function for the scalar fluctuations19:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_4(k_1,k_2,k_3)& {\displaystyle =-\frac{H^2}{64{\tilde{c}}_S^2{ϵ}^3{M}_p^6}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[k_2^2({\mathbf{k}}_{\mathbf{2}}.{\mathbf{k}}_{\mathbf{3}}){\mathcal{F}}_1(k_1,k_2,k_3)+k_3^2({\mathbf{k}}_{\mathbf{2}}.{\mathbf{k}}_{\mathbf{3}}){\mathcal{F}}_2(k_1,k_2,k_3)\right.}\hfill \\ & {\displaystyle +k_1^2({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{3}}){\mathcal{F}}_3(k_1,k_2,k_3)+k_3^2({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{3}}){\mathcal{F}}_4(k_1,k_2,k_3)}\hfill \\ & \left.{\displaystyle +k_1^2({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}}){\mathcal{F}}_5(k_1,k_2,k_3)+k_2^2({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}}){\mathcal{F}}_6(k_1,k_2,k_3)}\right],\hfill \end{array}$$

(178)

where the momentum dependent functions ${\mathcal{F}}_i(k_1,k_2,k_3)\forall i=1,2,\cdots ,6$ are defined as:

$$\begin{array}{ccc}\hfill {\mathcal{F}}_1(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_3+K(k_3-5k_1)+6k_1^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[(K-2k_2)(K-2k_2+k_1)+(K+2k_1-2k_2)k_3\right]\hfill \\ & & \left.+\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_1{k}_3+K(k_1-5k_3)+6k_3^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(179)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_2(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_2+K(k_2-5k_1)+6k_1^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_3-K)}^3}\left[(K-2k_3)(K-2k_3+k_1)+(K+2k_1-2k_3)k_2\right]\hfill \\ & & \left.+\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_1{k}_2+K(k_1-5k_2)+6k_2^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(180)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_3(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_2{k}_3+K(k_3-5k_2)+6k_2^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_1-K)}^3}\left[(K-2k_1)(K-2k_1+k_2)+(K+2k_2-2k_1)k_2\right]\hfill \\ & & \left.+\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_2{k}_3+K(k_2-5k_3)+6k_3^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(181)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_4(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_1{k}_2+K(k_1-5k_2)+6k_2^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_3-K)}^3}\left[(K-2k_3)(K-2k_3+k_2)+(K+2k_2-2k_3)k_2\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_2+K(k_2-5k_1)+6k_1^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(182)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_5(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_2{k}_3+K(k_2-5k_3)+6k_3^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_1-K)}^3}\left[(K-2k_1)(K-2k_1+k_2)+(K+2k_2-2k_1)k_3\right]\hfill \\ & & \left.+\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_2{k}_3+K(k_3-5k_2)+6k_2^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(183)

$$\begin{array}{ccc}\hfill {\mathcal{F}}_6(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right]\left[{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})+{(C_1-C_2)}^3(C_1^{*3}+C_2^{*3})\right]\hfill \\ & & +\left\{\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_1{k}_3+K(k_1-5k_3)+6k_3^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[(K-2k_2)(K-2k_2+k_3)+(K+2k_3-2k_2)k_1\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_3+K(k_3-5k_1)+6k_1^2\right]\right\}\hfill \\ & & \left[{(C_1-C_2)}^3{C}_1^{*}{C}_2^{*}(C_1^{*}+C_2^{*})+{(C_1^{*}-C_2^{*})}^3{C}_1{C}_2(C_1+C_2)\right].\hfill \end{array}$$

(184)

Furthermore, after simplification one can recast the three-point function for the scalar fluctuation as:

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_4(k_1,k_2,k_3)& {\displaystyle =-\frac{H^2}{128{\tilde{c}}_S^2{ϵ}^3{M}_p^6}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[{\displaystyle k_2^2\left(k_1^2-k_2^2-k_3^2\right){\mathcal{F}}_1(k_1,k_2,k_3)+k_3^2\left(k_1^2-k_2^2-k_3^2\right){\mathcal{F}}_2(k_1,k_2,k_3)}\right.}\hfill \\ & {\displaystyle +k_1^2\left(k_2^2-k_1^2-k_3^2\right){\mathcal{F}}_3(k_1,k_2,k_3)+k_3^2\left(k_2^2-k_1^2-k_3^2\right){\mathcal{F}}_4(k_1,k_2,k_3)}\hfill \\ & \left.{\displaystyle +k_1^2\left(k_3^2-k_2^2-k_1^2\right){\mathcal{F}}_5(k_1,k_2,k_3)+k_2^2\left(k_3^2-k_2^2-k_1^2\right){\mathcal{F}}_6(k_1,k_2,k_3)}\right],\hfill \end{array}$$

(185)

Finally, for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following contribution in the three-point function for scalar fluctuations:

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get:

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_1(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right].\hfill \end{array}$$

  (186)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_2(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right].\hfill \end{array}$$

  (187)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_3(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right].\hfill \end{array}$$

  (188)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_4(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right].\hfill \end{array}$$

  (189)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_5(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right].\hfill \end{array}$$

  (190)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_6(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right].\hfill \end{array}$$

  (191)
  Consequently, the three-point function for the scalar fluctuation can be expressed as:

  $$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_4(k_1,k_2,k_3)& =& -\frac{H^2}{128{\tilde{c}}_S^2{ϵ}^3{M}_p^6}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[k_2^2\left(k_1^2-k_2^2-k_3^2\right)\left[K^2+2k_1{k}_3+K(K-k_2)\right]\right.\hfill \\ & & +k_3^2\left(k_1^2-k_2^2-k_3^2\right)\left[K^2+2k_1{k}_2+K(K-k_3)\right]\hfill \\ & & +k_1^2\left(k_2^2-k_1^2-k_3^2\right)\left[K^2+2k_2{k}_3+K(K-k_1)\right]\hfill \\ & & +k_3^2\left(k_2^2-k_1^2-k_3^2\right)\left[K^2+2k_1{k}_2+K(K-k_3)\right]\hfill \\ & & +k_1^2\left(k_3^2-k_2^2-k_1^2\right)\left[K^2+2k_2{k}_3+K(K-k_1)\right]\hfill \\ & & \left.+k_2^2\left(k_3^2-k_2^2-k_1^2\right)\left[K^2+2k_1{k}_3+K(K-k_2)\right]\right],\hfill \end{array}$$

  (192)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_1(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_3+K(k_3-5k_1)+6k_1^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[(K-2k_2)(K-2k_2+k_1)+(K+2k_1-2k_2)k_3\right]\hfill \\ & & \left.+\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_1{k}_3+K(k_1-5k_3)+6k_3^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (193)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_2(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_2+K(k_2-5k_1)+6k_1^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_3-K)}^3}\left[(K-2k_3)(K-2k_3+k_1)+(K+2k_1-2k_3)k_2\right]\hfill \\ & & \left.+\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_1{k}_2+K(k_1-5k_2)+6k_2^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (194)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_3(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_2{k}_3+K(k_3-5k_2)+6k_2^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_1-K)}^3}\left[(K-2k_1)(K-2k_1+k_2)+(K+2k_2-2k_1)k_2\right]\hfill \\ & & \left.+\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_2{k}_3+K(k_2-5k_3)+6k_3^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (195)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_4(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_2+K(K-k_3)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_1{k}_2+K(k_1-5k_2)+6k_2^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_3-K)}^3}\left[(K-2k_3)(K-2k_3+k_2)+(K+2k_2-2k_3)k_2\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_2+K(k_2-5k_1)+6k_1^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (196)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_5(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_2{k}_3+K(K-k_1)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_2{k}_3+K(k_2-5k_3)+6k_3^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_1-K)}^3}\left[(K-2k_1)(K-2k_1+k_2)+(K+2k_2-2k_1)k_3\right]\hfill \\ & & \left.+\frac{1}{{(2k_2-K)}^3}\left[K^2-4k_2{k}_3+K(k_3-5k_2)+6k_2^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (197)

  $$\begin{array}{ccc}\hfill {\mathcal{F}}_6(k_1,k_2,k_3)& =& \frac{1}{K^3}\left[K^2+2k_1{k}_3+K(K-k_2)\right]J_1(\alpha ,\beta )\hfill \\ & & +\left\{\frac{1}{{(2k_3-K)}^3}\left[K^2-4k_1{k}_3+K(k_1-5k_3)+6k_3^2\right]\right.\hfill \\ & & +\frac{1}{{(2k_2-K)}^3}\left[(K-2k_2)(K-2k_2+k_3)+(K+2k_3-2k_2)k_1\right]\hfill \\ & & \left.+\frac{1}{{(2k_1-K)}^3}\left[K^2-4k_1{k}_3+K(k_3-5k_1)+6k_1^2\right]\right\}{J}_2(\alpha ,\beta ).\hfill \end{array}$$

  (198)

  where $J_1(\alpha ,\beta )$ and $J_2(\alpha ,\beta )$ are defined earlier. Consequently the three-point function for the scalar fluctuation can also be written after substituting all the momentum dependent functions ${\mathcal{F}}_i(k_1,k_2,k_3)\forall i=1,2,\cdots ,6$ for $\alpha ,\beta$ vacuum.

4.2.5. Function ${\mathsf{\Theta }}_5(k_1,k_2,k_3)$

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_5(k_1,k_2,k_3)& =& i{\int }_{{\eta }_i=-\infty }^{{\eta }_f=0}d\eta \frac{a\left(\eta \right)}{H^3}\left\{({\mathbf{k}}_{\mathbf{2}}.{\mathbf{k}}_{\mathbf{3}})Y_1(k_1,k_2,k_3)+({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{3}})Y_2(k_1,k_2,k_3)\right.\hfill \\ & & \left.+({\mathbf{k}}_{\mathbf{1}}.{\mathbf{k}}_{\mathbf{2}})Y_3(k_1,k_2,k_3)\right\},\hfill \end{array}$$

(199)

where the momentum dependent functions $Y_1(k_1,k_2,k_3)$, $Y_2(k_1,k_2,k_3)$ and $Y_3(k_1,k_2,k_3)$ can be expressed in terms of the various combinations of the scalar mode functions as:

$$\begin{array}{ccc}\hfill Y_1(k_1,k_2,k_3)& =& \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{*}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{*}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{2}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}}),\hfill \end{array}$$

(200)

$$\begin{array}{ccc}\hfill Y_2(k_1,k_2,k_3)& =& \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{*}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\hfill \\ & & +\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{*}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{1}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}}),\hfill \end{array}$$

(201)

$$\begin{array}{ccc}\hfill Y_3(k_1,k_2,k_3)& =& \overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }*}(\eta ,{\mathbf{k}}_{\mathbf{2}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}})\hfill \\ & & +\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\overline{v}({\eta }_f,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{*}(\eta ,{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{*}^{\prime }}(\eta ,{\mathbf{k}}_{\mathbf{1}})\hfill \\ & & +{\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{1}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{*}({\eta }_f,-{\mathbf{k}}_{\mathbf{3}})\overline{v}(\eta ,-{\mathbf{k}}_{\mathbf{3}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{2}}){\overline{v}}^{{}^{\prime }}(\eta ,-{\mathbf{k}}_{\mathbf{1}}),\hfill \end{array}$$

(202)

Here we get the following contribution in the three-point function for scalar fluctuations20:

$$\begin{array}{c}\hfill {\mathsf{\Theta }}_5(k_1,k_2,k_3)={\mathsf{\Theta }}_4(k_1,k_2,k_3),\end{array}$$

(204)

where ${\mathsf{\Theta }}_4(k_1,k_2,k_3)$ is defined earlier. Here the result is exactly same as derived for the coefficient ${\alpha }_4$.

4.3. Limiting Configurations of Scalar Bispectrum

To analyze the features of the bispectrum computed from the present setup here we further consider the following two configurations:

4.3.1. Equilateral Limit Configuration

Equilateral limit configuration is characterized by the condition, $k_1=k_2=k_3=k,$ where $k_i=\left|{\mathbf{k}}_{\mathbf{i}}\right|\forall i=1,2,3$. Consequently, we have, $K=3k.$

For this case, the bispectrum can be written as:

$$B_{EFT}(k,k,k)=\sum _{j=1}^5{\alpha }_j{\mathsf{\Theta }}_j(k,k,k),$$

(205)

where ${\alpha }_j\forall j=1,2,\cdots ,5$ are defined earlier and ${\mathsf{\Theta }}_j(k,k,k)\forall j=1,2,\cdots ,5$ are given by:

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_1(k,k,k)& =& \frac{3H^2}{16{ϵ}^3{M}_p^6}\frac{1}{k^6}\left[\frac{1}{27}{U}_1-3U_2\right],\hfill \end{array}$$

(206)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_2(k,k,k)& =& -\frac{3H^2}{64{ϵ}^3{M}_p^6{\tilde{c}}_S^2}\frac{1}{k^6}\left[\frac{17}{27}{U}_1-3U_2\right],\hfill \end{array}$$

(207)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_3(k,k,k)& =& \frac{3H}{32{ϵ}^3{M}_p^6}\frac{1}{k^6}\left[\frac{10}{9}{U}_1-\frac{22}{49}{U}_2\right],\hfill \end{array}$$

(208)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_4(k,k,k)& =& \frac{3H^2}{64{\tilde{c}}_S^2{ϵ}^3{M}_p^6}\frac{1}{k^6}\left[\frac{17}{27}{U}_1-3U_2\right]={\mathsf{\Theta }}_5(k,k,k),\hfill \end{array}$$

(209)

where $U_1$ and $U_2$ are defined as:

$$\begin{array}{ccc}\hfill U_1& =& \left[{\left(C_1-C_2\right)}^3\left(C_1^{*3}+C_2^{*3}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3\left(C_1^3+C_2^3\right)\right],\hfill \\ \hfill U_2& =& \left[{\left(C_1-C_2\right)}^3{C}_1^{*}{C}_2^{*}\left(C_1^{*}-C_2^{*}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3{C}_1{C}_2\left(C_1-C_2\right)\right].\hfill \end{array}$$

(210)

Furthermore, substituting the explicit expressions for ${\alpha }_j\forall j=1,2,\cdots ,5$ and ${\mathsf{\Theta }}_j(k,k,k)\forall j=1,2,\cdots ,5$ we get the following expression for the bispectrum for scalar fluctuations:

$$\begin{array}{c}\hfill B_{EFT}(k,k,k)=\frac{3H^2}{16{ϵ}^3{\tilde{c}}_S{M}_p^6}\frac{1}{k^6}\sum _{p=1}^2{f}_p{U}_p,\end{array}$$

(211)

where $f_p\forall p=1,2$ are defined as:

$$\begin{array}{ccc}\hfill f_1& =& \frac{{\tilde{c}}_s}{27}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}+\frac{17}{108{\tilde{c}}_s}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\hfill \\ & & -\frac{5}{2}{\overline{M}}_1^{3}H{\tilde{c}}_s+\frac{17}{36{\tilde{c}}_s}{\overline{M}}_1^{3}H,\hfill \end{array}$$

(212)

$$\begin{array}{ccc}\hfill f_2& =& -3{\tilde{c}}_s\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}-\frac{3}{4{\tilde{c}}_s}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\hfill \\ & & +\frac{99}{98}{\overline{M}}_1^{3}H{\tilde{c}}_s-\frac{9}{4{\tilde{c}}_s}{\overline{M}}_1^{3}H.\hfill \end{array}$$

(213)

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get:

  $$\begin{array}{c}\hfill U_1=2,U_2=0.\end{array}$$

  (214)

  Consequently, we get the following expression for the bispectrum for scalar fluctuations:

  $$\begin{array}{ccc}\hfill B_{EFT}(k,k,k)& =& \frac{H^2}{4ϵ{\tilde{c}}_S{M}_p^2}\frac{1}{M_p^4{ϵ}^2}\frac{1}{k^6}\left[\frac{{\tilde{c}}_s}{18}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}\right.\hfill \\ & & \left.+\frac{17}{72{\tilde{c}}_s}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}-\frac{15}{4}{\overline{M}}_1^{3}H{\tilde{c}}_s+\frac{17}{24{\tilde{c}}_s}{\overline{M}}_1^{3}H\right].\hfill \end{array}$$

  (215)

  For ${\tilde{c}}_S=1=c_S$ case we know that $M_2=0$ and $M_3=0$ which we have already shown earlier. As a result the bispectrum for scalar fluctuation can be expressed in the following simplified form:

  $$\begin{array}{ccc}\hfill B_{EFT}(k,k,k)& =& -\frac{H^2}{4ϵM_p^2}\frac{1}{M_p^4{ϵ}^2}\frac{1}{k^6}\frac{125}{48}{\overline{M}}_1^{3}H.\hfill \end{array}$$

  (216)

  For ${\tilde{c}}_S<1$ and $c_S<1$ case one can also recast the bispectrum for scalar fluctuations in the following simplified form:

  $$\begin{array}{ccc}\hfill B_{EFT}(k,k,k)& =& \frac{H^2}{4ϵ{\tilde{c}}_S{M}_p^2}\frac{1}{M_p^4{ϵ}^2}\frac{1}{k^6}{\overline{M}}_1^{3}H\left[\frac{{\tilde{c}}_S}{18}\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\right.\hfill \\ & & \left.+\frac{17}{72{\tilde{c}}_S}\left\{\frac{2c_S^2}{{\tilde{c}}_4}+\frac{3}{2}\right\}-\frac{15}{4}{\tilde{c}}_S+\frac{17}{24{\tilde{c}}_S}\right].\hfill \end{array}$$

  (217)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

  $$\begin{array}{c}\hfill U_1=J_1(\alpha ,\beta ),U_2=J_2(\alpha ,\beta ),\end{array}$$

  (218)

  where $J_1(\alpha ,\beta )$ and $J_2(\alpha ,\beta )$ are defined earlier.
  Consequently, we get the following expression for the bispectrum for scalar fluctuations:

  $$\begin{array}{ccc}\hfill B_{EFT}(k,k,k)& =& \frac{H^2}{4ϵ{\tilde{c}}_S{M}_p^2}\frac{1}{M_p^4{ϵ}^2}\frac{1}{k^6}\left[\left(\frac{{\tilde{c}}_s}{36}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}\right.\right.\hfill \\ & & \left.+\frac{17}{144{\tilde{c}}_s}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}-\frac{15}{8}{\overline{M}}_1^{3}H{\tilde{c}}_s+\frac{17}{48{\tilde{c}}_s}{\overline{M}}_1^{3}H\right)J_1(\alpha ,\beta )\hfill \\ & & +\left(-\frac{9}{4}{\tilde{c}}_s\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}\right.\hfill \\ & & \left.\left.-\frac{9}{16{\tilde{c}}_s}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}+\frac{297}{392}{\overline{M}}_1^{3}H{\tilde{c}}_s-\frac{27}{16{\tilde{c}}_s}{\overline{M}}_1^{3}H\right)J_2(\alpha ,\beta )\right].\hfill \end{array}$$

  (219)

  For ${\tilde{c}}_S=1=c_S$ case we know that $M_2=0$ and $M_3=0$ which we have already shown earlier. As a result the bispectrum for scalar fluctuation can be expressed in the following simplified form:

  $$\begin{array}{ccc}\hfill B(k,k,k)& =& -\frac{H^2}{4ϵM_p^2}\frac{1}{M_p^4{ϵ}^2}\frac{1}{k^6}{\overline{M}}_1^{3}H\left[\frac{125}{96}{J}_1(\alpha ,\beta )+\frac{8073}{1568}{J}_2(\alpha ,\beta )\right].\hfill \end{array}$$

  (220)

  For ${\tilde{c}}_S<1$ and $c_S<1$ case one can also recast the bispectrum for scalar fluctuations in the following simplified form:

  $$\begin{array}{ccc}\hfill B_{EFT}(k,k,k)& =& \frac{H^2}{4ϵ{\tilde{c}}_S{M}_p^2}\frac{1}{M_p^4{ϵ}^2}\frac{1}{k^6}{\overline{M}}_1^{3}H\left[\left(\frac{{\tilde{c}}_S}{36}\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}+\frac{17}{144{\tilde{c}}_S}\left\{\frac{2c_S^2}{{\tilde{c}}_4}+\frac{3}{2}\right\}\right.\right.\hfill \\ & & \left.-\frac{15}{8}{\tilde{c}}_S+\frac{17}{48{\tilde{c}}_S}\right)J_1(\alpha ,\beta )+\left(-\frac{9}{4}{\tilde{c}}_S\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}-\frac{9}{16{\tilde{c}}_S}\left\{\frac{2c_S^2}{{\tilde{c}}_4}+\frac{3}{2}\right\}\right.\hfill \\ & & \left.\left.+\frac{297}{392}{\tilde{c}}_S-\frac{27}{16{\tilde{c}}_S}\right)J_2(\alpha ,\beta )\right].\hfill \end{array}$$

  (221)

4.3.2. Squeezed Limit Configuration

Squeezed limit configuration is characterized by the condition, $k_1\approx k_2(=k_L)>>k_3(=k_S),$ where $k_i=\left|{\mathbf{k}}_{\mathbf{i}}\right|\forall i=1,2,3$. Also $k_L$ and $k_S$ characterize long and short mode momentum, respectively. Consequently, we have, $K=2k_L+k_S.$ For this case, the bispectrum can be written as:

$$B_{EFT}(k_L,k_L,k_S)=\sum _{j=1}^5{\alpha }_j{\mathsf{\Theta }}_j(k_L,k_L,k_S),$$

(222)

where ${\alpha }_j\forall j=1,2,\cdots ,5$ are defined earlier and ${\mathsf{\Theta }}_j(k_L,k_L,k_S)\forall j=1,2,\cdots ,5$ are given by:

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_1(k_L,k_L,k_S)& \approx & \frac{3H^2}{128{ϵ}^3{M}_p^6}\frac{1}{k_L^5{k}_S}\left[U_1-16U_2{\left(\frac{k_L}{k_S}\right)}^3\right],\hfill \end{array}$$

(223)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_2(k_L,k_L,k_S)& =& -\frac{H^2}{64{ϵ}^3{M}_p^6{\tilde{c}}_S^2}\frac{1}{k_L^5{k}_S}\left\{\frac{3}{4}\left[U_1-3U_2\right]+\frac{3}{4}\left[U_1-U_2\left(1+\frac{8}{3}{\left(\frac{k_L}{k_S}\right)}^2\right)\right]\right.\hfill \\ & & \left.+\frac{5}{4}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\left[U_1-U_2\left(1-\frac{8}{5}{\left(\frac{k_L}{k_S}\right)}^3\right)\right]\right\},\hfill \end{array}$$

(224)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_3(k_L,k_L,k_S)& =& \frac{H}{64{ϵ}^3{M}_p^6}\frac{1}{k_L^5{k}_S}\left\{3\left[U_1-U_2\right]+{\left(\frac{k_L}{k_S}\right)}^2\left[U_1-\left(1+8\left(\frac{k_L}{k_S}\right)\right)U_2\right]\right\},\hfill \end{array}$$

(225)

$$\begin{array}{ccc}\hfill {\mathsf{\Theta }}_4(k_L,k_L,k_S)& =& -\frac{H^2}{64{\tilde{c}}_S^2{ϵ}^3{M}_p^6}\left\{\frac{3}{4}\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)\left[U_1-U_2\left(1+\frac{4}{3}{\left(\frac{k_L}{k_S}\right)}^2\right)\right]\right.\hfill \\ & & +\frac{5}{4}\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)\left[U_1-U_2\left(1-\frac{16}{5}{\left(\frac{k_L}{k_S}\right)}^2\right)\right]\hfill \\ & & \left.+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\left[U_1-U_2\left(1+\frac{8}{3}\left(\frac{k_L}{k_S}\right)+\frac{4}{3}{\left(\frac{k_L}{k_S}\right)}^2\right)\right]\right\}\hfill \\ & =& {\mathsf{\Theta }}_5(k_L,k_L,k_S),\hfill \end{array}$$

(226)

where $U_1$ and $U_2$ are defined earlier.

Furthermore, substituting the explicit expressions for ${\alpha }_j\forall j=1,2,\cdots ,5$ and ${\mathsf{\Theta }}_j(k_L,k_L,k_S)\forall j=1,2,\cdots ,5$ we get the following expression for the bispectrum for scalar fluctuations:

$$\begin{array}{ccc}\hfill B_{EFT}(k_L,k_L,k_S)& =& \frac{H^2}{64{ϵ}^3{\tilde{c}}_S{M}_p^6}\sum _{p=1}^2{g}_p(k_L,k_S)U_p,\hfill \end{array}$$

(227)

where $g_p(k_L,k_S)\forall p=1,2$ are defined as:

$$\begin{array}{ccc}\hfill g_1(k_L,k_S)& =& \frac{3{\tilde{c}}_S}{2k_L^5{k}_S}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}\hfill \\ & & +\frac{1}{{\tilde{c}}_S{k}_L^5{k}_S}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\hfill \\ & & -\frac{9}{2}{\overline{M}}_1^{3}H\frac{{\tilde{c}}_S}{k_L^5{k}_S}\left(3+{\left(\frac{k_L}{k_S}\right)}^2\right)+\frac{3}{{\tilde{c}}_S}{\overline{M}}_1^{3}H\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\},\hfill \end{array}$$

(228)

$$\begin{array}{ccc}\hfill g_2(k_L,k_S)& =& \frac{24{\tilde{c}}_S}{k_L^5{k}_S}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}{\left(\frac{k_L}{k_S}\right)}^2\hfill \\ & & +\frac{1}{{\tilde{c}}_S{k}_L^5{k}_S}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\left\{3+2{\left(\frac{k_S}{k_L}\right)}^2\right.\hfill \\ & & \left.+\frac{5}{4}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\left(1-\frac{8}{5}{\left(\frac{k_S}{k_L}\right)}^3\right)\right\}-\frac{9}{2}{\overline{M}}_1^{3}H\frac{{\tilde{c}}_S}{k_L^5{k}_S}\left\{3+{\left(\frac{k_L}{k_S}\right)}^2\left(1+8\left(\frac{k_L}{k_S}\right)\right)\right\}\hfill \\ & & +\frac{3}{{\tilde{c}}_S}{\overline{M}}_1^{3}H\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\}.\hfill \end{array}$$

(229)

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get:

  $$\begin{array}{c}\hfill U_1=2,U_2=0.\end{array}$$

  (230)

  Consequently, we get the following expression for the bispectrum for scalar fluctuations:

  $$\begin{array}{ccc}\hfill B_{EFT}(k_L,k_L,k_S)& =& \frac{H^2}{4ϵ{\tilde{c}}_S{M}_p^2}\frac{1}{8M_p^4{ϵ}^2}\left[\frac{3{\tilde{c}}_S}{2k_L^5{k}_S}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H-\frac{4}{3}{M}_3^4\right\}\right.\hfill \\ & & +\frac{1}{{\tilde{c}}_S{k}_L^5{k}_S}\left\{\left(1-\frac{1}{c_S^2}\right)\dot{H}{M}_p^2+\frac{3}{2}{\overline{M}}_1^{3}H\right\}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\hfill \\ & & -\frac{9}{2}{\overline{M}}_1^{3}H\frac{{\tilde{c}}_S}{k_L^5{k}_S}\left(3+{\left(\frac{k_L}{k_S}\right)}^2\right)\hfill \\ & & \left.+\frac{3}{{\tilde{c}}_S}{\overline{M}}_1^{3}H\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\}\right].\hfill \end{array}$$

  (231)

  For ${\tilde{c}}_S=1=c_S$ case we know that $M_2=0$ and $M_3=0$ which we have already shown earlier. As a result the bispectrum for scalar fluctuation can be expressed in the following simplified form:

  $$\begin{array}{ccc}\hfill B_{EFT}(k_L,k_L,k_S)& =& \frac{H^2}{4ϵM_p^2}\frac{1}{8M_p^4{ϵ}^2}{\overline{M}}_1^{3}H\left[\frac{9}{4k_L^5{k}_S}+\frac{3}{2k_L^5{k}_S}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\right.\hfill \\ & & \left.-\frac{9}{2}\frac{1}{k_L^5{k}_S}\left(3+{\left(\frac{k_L}{k_S}\right)}^2\right)+3\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\}\right].\hfill \end{array}$$

  (232)

  For ${\tilde{c}}_S<1$ and $c_S<1$ case one can also recast the bispectrum for scalar fluctuations in the following simplified form:

  $$\begin{array}{ccc}\hfill B_{EFT}(k_L,k_L,k_S)& =& \frac{H^2}{4ϵ{\tilde{c}}_S{M}_p^2}\frac{1}{8M_p^4{ϵ}^2}{\overline{M}}_1^{3}H\left[\frac{3{\tilde{c}}_S}{2k_L^5{k}_S}\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\right.\hfill \\ & & +\frac{1}{{\tilde{c}}_S{k}_L^5{k}_S}\left\{\frac{2c_S^2}{{\tilde{c}}_4}+\frac{3}{2}\right\}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\hfill \\ & & -\frac{9}{2}\frac{{\tilde{c}}_S}{k_L^5{k}_S}\left(3+{\left(\frac{k_L}{k_S}\right)}^2\right)\hfill \\ & & \left.+\frac{3}{{\tilde{c}}_S}\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\}\right].\hfill \end{array}$$

  (233)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

  $$\begin{array}{c}\hfill U_1=J_1(\alpha ,\beta ),U_2=J_2(\alpha ,\beta ),\end{array}$$

  (234)

  where $J_1(\alpha ,\beta )$ and $J_2(\alpha ,\beta )$ are defined earlier.
  Consequently, we get the following expression for the bispectrum for scalar fluctuations:

  $$\begin{array}{ccc}\hfill B_{EFT}(k_L,k_L,k_S)& =& \frac{H^2}{4ϵ{\tilde{c}}_S{M}_p^2}\frac{1}{16M_p^4{ϵ}^2}\left[g_1(k_L,k_S)J_1(\alpha ,\beta )+g_2(k_L,k_S)J_2(\alpha ,\beta )\right].\hfill \end{array}$$

  (235)

  For ${\tilde{c}}_S=1=c_S$ case we know that $M_2=0$ and $M_3=0$ which we have already shown earlier. As a result the factors $g_1(k_L,k_S)$ and $g_2(k_L,k_S)$ appearing in the expression for bispectrum for scalar fluctuation can be expressed in the following simplified form:

  $$\begin{array}{ccc}\hfill g_1(k_L,k_S)& =& \frac{9}{4k_L^5{k}_S}{\overline{M}}_1^{3}H+\frac{3}{2k_L^5{k}_S}{\overline{M}}_1^{3}H\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\hfill \\ & & -\frac{9}{2}{\overline{M}}_1^{3}H\frac{1}{k_L^5{k}_S}\left(3+{\left(\frac{k_L}{k_S}\right)}^2\right)\hfill \\ & & +3{\overline{M}}_1^{3}H\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\},\hfill \end{array}$$

  (236)

  $$\begin{array}{ccc}\hfill g_2(k_L,k_S)& =& \frac{36}{k_L^5{k}_S}{\overline{M}}_1^{3}H{\left(\frac{k_L}{k_S}\right)}^2+\frac{3}{2k_L^5{k}_S}{\overline{M}}_1^{3}H\left\{3+2{\left(\frac{k_S}{k_L}\right)}^2\right.\hfill \\ & & \left.+\frac{5}{4}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\left(1-\frac{8}{5}{\left(\frac{k_S}{k_L}\right)}^3\right)\right\}\hfill \\ & & -\frac{9}{2}{\overline{M}}_1^{3}H\frac{1}{k_L^5{k}_S}\left\{3+{\left(\frac{k_L}{k_S}\right)}^2\left(1+8\left(\frac{k_L}{k_S}\right)\right)\right\}\hfill \\ & & +3{\overline{M}}_1^{3}H\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)++\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\}.\hfill \end{array}$$

  (237)

  For ${\tilde{c}}_S<1$ and $c_S<1$ case one can also recast the factors $g_1(k_L,k_S)$ and $g_2(k_L,k_S)$ as appearing in the expression for bispectrum for scalar fluctuations in the following simplified form:

  $$\begin{array}{ccc}\hfill g_1(k_L,k_S)& =& \frac{3{\tilde{c}}_S}{2k_L^5{k}_S}\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\hfill \\ & & +\frac{1}{{\tilde{c}}_S{k}_L^5{k}_S}\left\{\frac{2c_S^2}{{\tilde{c}}_4}+\frac{3}{2}\right\}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\hfill \\ & & -\frac{9}{2}{\overline{M}}_1^{3}H\frac{{\tilde{c}}_S}{k_L^5{k}_S}\left(3+{\left(\frac{k_L}{k_S}\right)}^2\right)\hfill \\ & & +\frac{3}{{\tilde{c}}_S}{\overline{M}}_1^{3}H\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)+\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\},\hfill \end{array}$$

  (238)

  $$\begin{array}{ccc}\hfill g_2(k_L,k_S)& =& \frac{24{\tilde{c}}_S}{k_L^5{k}_S}\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}{\left(\frac{k_L}{k_S}\right)}^2\hfill \\ & & +\frac{1}{{\tilde{c}}_S{k}_L^5{k}_S}\left\{\frac{2c_S^2}{{\tilde{c}}_4}+\frac{3}{2}\right\}\left\{3+2{\left(\frac{k_S}{k_L}\right)}^2+\frac{5}{4}\left(2-\frac{5}{4}{\left(\frac{k_S}{k_L}\right)}^2\right)\left(1-\frac{8}{5}{\left(\frac{k_S}{k_L}\right)}^3\right)\right\}\hfill \\ & & -\frac{9}{2}{\overline{M}}_1^{3}H\frac{{\tilde{c}}_S}{k_L^5{k}_S}\left\{3+{\left(\frac{k_L}{k_S}\right)}^2\left(1+8\left(\frac{k_L}{k_S}\right)\right)\right\}\hfill \\ & & +\frac{3}{{\tilde{c}}_S}{\overline{M}}_1^{3}H\left\{2\left(\frac{1}{k_L^5{k}_S}+\frac{k_S}{k_L^7}\right)++\frac{3}{2k_L^3{k}_S^3}\left(2-{\left(\frac{k_S}{k_L}\right)}^2\right)\right\}.\hfill \end{array}$$

  (239)

5. Determination of EFT Coefficients and Future Predictions

In this section, we compute the exact analytical expression for the EFT coefficients for two specific cases—1. Canonical single-field slow-roll inflation and 2. General single-field $P(X,\varphi )$ models of inflation, where $X=-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ is the kinetic term. To determine the EFT coefficients for the canonical single-field slow-roll model or from general single-field $P(X,\varphi )$ model of inflation we will follow the following strategy:

• First, we will start with the general expression for the three-point function and the bispectrum for scalar perturbations with an arbitrary choice of quantum vacuum. Then we take the Bunch–Davies and $\alpha ,\beta$ vacuum to match with the standard results of scalar three-point function.
• Next we take the equilateral limit and squeezed limit configuration of the bispectrum obtained from the single-field slow-roll model and general single-field $P(X,\varphi )$ model.
• Furthermore, we equate the equilateral limit and squeezed limit configuration of the bispectrum computed from the EFT of inflation with the single-field slow-roll or from the general single-field $P(X,\varphi )$ model.
• Finally, for sound speed $c_S=1$ and $c_S<1$ we get the analytical expressions for all the EFT coefficients for canonical single-field slow-roll models or from generalized single-field $P(X,\varphi )$ models of inflation.

5.1. For Canonical Single-Field Slow-Roll Inflation

Here our prime objective is to derive the EFT coefficients by computing the most general expression for the three-point function for scalar fluctuations from the canonical single-field slow-roll model of inflation for arbitrary vacuum. Then we give specific example for Bunch–Davies and $\alpha ,\beta$ vacuum for completeness.

5.1.1. Basic Setup

Let us start with the action for single scalar field (inflaton) which has canonical kinetic term as given by:

$$\mathbf{Canonical}\mathbf{model}:S=\int d^{4}x\sqrt{-g}\left[\frac{M_p^2}{2}R+X-V\left(\varphi \right)\right],$$

(240)

where $V\left(\varphi \right)$ is the potential which satisfies the slow-roll condition for inflation.

It is important to mention here that perturbations to the homogeneous situation discussed above are introduced in the ADM formalism where the metric takes the form [31]:

$$\mathbf{ADM}\mathbf{metric}:d{s}^2=-N^{2}d{t}^2+g_{ij}\left(d{x}^i+N^{i}dt\right)\left(d{x}^j+N^{j}dt\right),$$

(241)

where $g_{ij}$ is the metric on the spatial three-surface characterized by t, lapse N and shift $N_i$. Here we choose synchronous gauge to maintain diffeomorphism invariance of the theory where the gauge-fixing conditions are given by:

$$\begin{array}{c}\hfill \mathbf{Synchronous}\mathbf{gauge}:N=1,N^i=0,\end{array}$$

(242)

and the perturbed metric is given by:

$$\begin{array}{c}\hfill g_{ij}=a^2\left(t\right)\left[(1+2\zeta (t,\mathbf{x})){\delta }_{ij}+{\gamma }_{ij}\right],{\gamma }_{ii}=0,\end{array}$$

(243)

where $\zeta (t,\mathbf{x})$ and ${\gamma }_{ij}$ are defined earlier. Here it is important to note that the structure of $g_{ij}$ is exactly same that we have mentioned in the case of EFT framework discussed in this paper. Please note that in the context of ADM formalism one can treat the scalar field $\varphi$, induced metric $g_{ij}$ as the dynamical variables. On the other hand, N and $N^i$ mimics the role of Lagrange multipliers in ADM formalism. Consequently, one needs to impose the equations of motion of $N,N^i$ as additional constraints in the synchronous gauge where the gauge condition as stated in Equation (242) holds good perfectly. More precisely, in this context the equations of motion of N and $N^i$ correspond to time and spatial reparametrization invariance.

Furthermore, using the ADM metric as stated in Equation (241), the action for the single scalar field Equation (240) can be recast as [31]:

$$\begin{array}{c}\hfill S=\int d^{4}x\sqrt{-g}\left[\frac{M_p^2}{2}N{}^{\left(3\right)}R-NV+\frac{1}{2N}\left(E_{ij}{E}^{ij}-E^2\right)+\frac{1}{2N}{\left(\dot{\varphi }-N^i{\partial }_i\varphi \right)}^2-N{g}^{ij}{\partial }_i\varphi {\partial }_j\varphi \right],\end{array}$$

(244)

where ${}^{\left(3\right)}R$ is the Ricci scalar curvature of the spatial slice. Also, here $E_{ij}$ and E is defined as [31]:

$$\begin{array}{ccc}\hfill E_{ij}:& =& \frac{1}{2}\left({\dot{g}}_{ij}-{\nabla }_i{N}_j-{\nabla }_j{N}_i\right)=N{K}_{ij},\hfill \end{array}$$

(245)

$$\begin{array}{ccc}\hfill E:& =& E_i^i=g_{ij}{E}^{ij}=g_{ij}{g}^{im}{g}^{jn}{E}_{mn}=N{g}_{ij}{g}^{im}{g}^{jn}{K}_{mn}.\hfill \end{array}$$

(246)

Here the covariant derivative ${\nabla }_i$, is taken with respect to the 3-metric $g_{ij}$. Also, in this context the extrinsic curvature $K_{ij}$ is defined as [31]:

$$\begin{array}{ccc}\hfill K_{ij}& =& \frac{1}{N}{E}_{ij}=\frac{1}{2N}\left({\dot{g}}_{ij}-{\nabla }_i{N}_j-{\nabla }_j{N}_i\right).\hfill \end{array}$$

(247)

Additionally, we choose the following two gauges:

$$\begin{array}{c}\hfill \mathbf{Gauge}\mathbf{I}:\delta \varphi (t,\mathbf{x})=0,\zeta (t,\mathbf{x})\ne 0,{\partial }_i{\gamma }_{ij}=0,{\gamma }_{ii}=0.\end{array}$$

(248)

$$\begin{array}{c}\hfill \mathbf{Gauge}\mathbf{II}:\delta \varphi (t,\mathbf{x})\ne 0,\zeta (t,\mathbf{x})=0,{\partial }_i{\gamma }_{ij}=0,{\gamma }_{ii}=0.\end{array}$$

(249)

For our present computations, we will work in Gauge I as this is exactly same as the unitary gauge that we have used in the context of EFT framework. Also, the tensor perturbation ${\gamma }_{ij}$ is exactly same for the unitary gauge that we have used for EFT setup.

5.1.2. Scalar Three-Point Function for Single-Field Slow-Roll inflation

Before computing the three-point function for scalar mode fluctuation here it is important to note that the two-point function for single-field slow-roll inflation is exactly same with the results obtained for EFT of inflation with sound speed $c_S=1$ and ${\tilde{c}}_S=1$, which can be obtained by setting the EFT coefficients, $M_2=0$, $M_3=0$, ${\overline{M}}_1\ne 0$, $M_4\ne 0$, ${\overline{M}}_2\ne 0$, ${\overline{M}}_3\ne 0$, ${\tilde{c}}_5=-\frac{1}{2}(1+ϵ)$21. Using three-point function we can able to fix all of these coefficients.

We here now proceed to calculate the three-point function for the scalar fluctuation $\zeta (t,\mathbf{x})$ in the interacting picture with arbitrary vacuum. Then we cite results for Bunch–Davies and $\alpha ,\beta$ vacuum. For single-field slow-roll inflation, the third-order term in the action Equation (244) is given by [31]:

$$S_{\zeta }^{\left(3\right)}=\int d^{4}x\left[a^3{ϵ}^2\tilde{\zeta }{\dot{\tilde{\zeta }}}^2+a{ϵ}^2\tilde{\zeta }{(\partial \tilde{\zeta })}^2-2a^3ϵ\dot{\tilde{\zeta }}{\partial }_i\tilde{\zeta }{\partial }_i\left(ϵ{\partial }^{-2}\dot{\tilde{\zeta }}\right)\right],$$

(251)

which is derived from Equation (244) and here after neglecting all the contribution from the terms which are sub-leading in the slow-roll parameters. Additionally, here we use the following field redefinition:

$$\zeta =\tilde{\zeta }+\left\{ϵ-\frac{\eta }{2}\right\}{\tilde{\zeta }}^2,$$

(252)

where $ϵ$, $\eta$, $\delta$ and s are slow-roll parameters which are defined in the context of single-field slow-roll inflation as:

$$\begin{array}{ccc}\hfill ϵ& \sim & \frac{1}{2M_p^2}\frac{{\dot{\varphi }}^2}{H^2},\eta \sim ϵ-\delta ,\delta =\frac{\ddot{\varphi }}{H\dot{\varphi }},s=0.\hfill \end{array}$$

(253)

Here one can also express the slow-roll parameters $ϵ$ and $\eta$ in terms of the slowly varying potential $V\left(\varphi \right)$ as, $ϵ\sim {ϵ}_V,$ $\eta \sim {\eta }_V-{ϵ}_V,$ $\delta \sim 2{ϵ}_V-{\eta }_V.$ where the new slow-roll parameter ${ϵ}_V$ and ${\eta }_V$ are defined as, ${ϵ}_V=\frac{M_p^2}{2}{\left(\frac{V^{{}^{\prime }}\left(\varphi \right)}{V\left(\varphi \right)}\right)}^2,{\eta }_V=M_p^2\left(\frac{V^{{}^{\prime \prime }}\left(\varphi \right)}{V\left(\varphi \right)}\right).$ Here ${}^{\prime }$ represents $d/d\varphi$.

Now it is important to note that in the present context of discussion we are interested in the three-point function for the scalar fluctuation field $\zeta$, not for the redefined scalar field fluctuation $\tilde{\zeta }$ and for this reason one can write down the exact connection between the three-point function for the scalar function field $\zeta$ and redefined scalar fluctuation field $\tilde{\zeta }$ in position space as:

$$\begin{array}{ccc}\hfill \langle \zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\rangle & =& \langle \tilde{\zeta }\left({\mathbf{x}}_{\mathbf{1}}\right)\tilde{\zeta }\left({\mathbf{x}}_{\mathbf{2}}\right)\tilde{\zeta }\left({\mathbf{x}}_{\mathbf{3}}\right)\rangle +\left(2ϵ-\eta \right)\left[\langle \zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\rangle \langle \zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\rangle \right.\hfill \\ & & \left.+\langle \zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\rangle \langle \zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\rangle +\langle \zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\rangle \langle \zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\rangle \right].\hfill \end{array}$$

(254)

After taking the Fourier transform of the scalar function field $\zeta$ and redefined scalar fluctuation field $\tilde{\zeta }$ one can express connection between three-point function in momentum space and this is also our main point of interest.

The interaction Hamiltonian for the redefined scalar fluctuation $\tilde{\zeta }$ can be expressed as:

$$H_{int}=\int d^{3}x\left[a{ϵ}^2\tilde{\zeta }{\tilde{\zeta }}^{{}^{\prime }2}+a{ϵ}^2\tilde{\zeta }{(\partial \tilde{\zeta })}^2-2aϵ{\tilde{\zeta }}^{{}^{\prime }}{\partial }_i\tilde{\zeta }{\partial }_i\left(ϵ{\partial }^{-2}{\tilde{\zeta }}^{{}^{\prime }}\right)\right].$$

(255)

Furthermore, following the in-in formalism in interaction picture the expression for the three-point function for the redefined scalar fluctuation $\tilde{\zeta }$ and then transforming the final result in terms of the scalar fluctuation $\zeta$ in momentum one can write the following expression:

$$\begin{array}{cc}\hfill \langle \zeta \left({\mathbf{k}}_{\mathbf{1}}\right)\zeta \left({\mathbf{k}}_{\mathbf{2}}\right)\zeta \left({\mathbf{k}}_{\mathbf{3}}\right)\rangle & {\displaystyle =-i{\int }_{{\eta }_i=-\infty }^{{\eta }_f=0}d\eta a\left(\eta \right)\langle 0|\left[\zeta ({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\zeta ({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\zeta ({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),H_{int}\left(\eta \right)\right]|0\rangle }\hfill \\ & {\displaystyle ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}+{\mathbf{k}}_{\mathbf{3}})B_{SFSR}(k_1,k_2,k_3),}\hfill \end{array}$$

(256)

where $B_{SFSR}(k_1,k_2,k_3)$ represents the bispectrum of scalar fluctuation $\zeta$, which is computed from single-field slow-roll inflation. Here the final expression for the bispectrum of scalar fluctuation for arbitrary vacuum is given by:

$$\begin{array}{cc}\hfill B_{SFSR}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[2(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\sum _{i=1}^3{k}_i^3\right.}\hfill \\ & {\displaystyle +ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2\right.}\hfill \\ & \left.\left.{\displaystyle +8\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\sum _{m=1}^3\frac{1}{K-2k_m}}\right)\right].\hfill \end{array}$$

(257)

For Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get [31]:

  $$\begin{array}{ccc}\hfill B_{SFSR}(k_1,k_2,k_3)& =& \frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[2(2ϵ-\eta )\sum _{i=1}^3{k}_i^3\right.\hfill \\ & & \left.+ϵ\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)\right].\hfill \end{array}$$

  (258)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get [33]:

  $$\begin{array}{ccc}\hfill B_{SFSR}(k_1,k_2,k_3)& =& \frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[2(2ϵ-\eta ){cosh}^{2}2\alpha \sum _{i=1}^3{k}_i^3\right.\hfill \\ & & +ϵ\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)\hfill \\ & & \left.+ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+8\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\sum _{m=1}^3\frac{1}{K-2k_m}\right)\right].\hfill \end{array}$$

  (259)

Furthermore, we consider equilateral limit and squeezed limit in which we finally get:

• Equilateral limit configuration:
  Here the bispectrum for scalar perturbations in the presence of arbitrary quantum vacuum can be expressed as:

  $$\begin{array}{cc}\hfill B_{SFSR}(k,k,k)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2\right.}\hfill \\ & \left.{\displaystyle +27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2}\right].\hfill \end{array}$$

  (260)

  Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$ we get:

    $$\begin{array}{c}\hfill B_{SFSR}(k,k,k)=\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[23ϵ-6\eta \right].\end{array}$$

    (261)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

    $$\begin{array}{c}\hfill B_{SFSR}(k,k,k)=\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[6(2ϵ-\eta ){cosh}^{2}2\alpha +11ϵ+27ϵ{sinh}^{2}2\alpha {cos}^2\beta \right].\end{array}$$

    (262)
• Squeezed limit configuration:
  Here the bispectrum for scalar perturbations in the presence of arbitrary quantum vacuum can be expressed as:

  $$\begin{array}{c}\hfill B_{SFSR}(k_L,k_L,k_S)=\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}\sum _{j=-1}^3{a}_j{\left(\frac{k_S}{k_L}\right)}^j,\end{array}$$

  (263)

  where the expansion coefficients $a_j\forall j=-1,\cdots ,3$ for arbitrary vacuum are defined as:

  $$\begin{array}{ccc}\hfill a_{-1}& =& 16ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill a_0& =& 4(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+4ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+4ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill a_1& =& 34ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,a_2=10ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+10ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill a_3& =& 2(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-5ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2-ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2.\hfill \end{array}$$

  Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$, we get the following expression for the expansion coefficients $a_j\forall j=-1,\cdots ,3$:

    $$\begin{array}{ccc}\hfill a_{-1}& =& 0,a_0=4(3ϵ-\eta ),a_1=0,a_2=10ϵ,a_3=-(ϵ+2\eta ).\hfill \end{array}$$

    (264)

    Consequently, the bispectrum can be recast as:

    $$\begin{array}{c}\hfill B_{SFSR}(k_L,k_L,k_S)=\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}\left[4(3ϵ-\eta )+10ϵ{\left(\frac{k_S}{k_L}\right)}^2-(ϵ+2\eta ){\left(\frac{k_S}{k_L}\right)}^3\right].\end{array}$$

    (265)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$, we get the following expression for the expansion coefficients $a_j\forall j=-1,\cdots ,3$:

    $$\begin{array}{ccc}\hfill a_{-1}& =& 16ϵ{sinh}^{2}2\alpha {cos}^2\beta ,a_0=4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill a_1& =& 34ϵ{sinh}^{2}2\alpha {cos}^2\beta ,a_2=10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill a_3& =& 2(2ϵ-\eta ){cosh}^{2}2\alpha -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta .\hfill \end{array}$$

    Consequently, the bispectrum can be recast as:

    $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{SFSR}(k_L,k_L,k_S)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}\left[16ϵ{sinh}^{2}2\alpha {cos}^2\beta {\left(\frac{k_S}{k_L}\right)}^{-1}\right.}\hfill \\ & {\displaystyle +\left(4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta \right)}\hfill \\ & {\displaystyle {\displaystyle +34ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(\frac{k_S}{k_L}\right)+\left(10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^2}}\hfill \\ & \left.{\displaystyle +\left(2(2ϵ-\eta ){cosh}^{2}2\alpha -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^3}\right].\hfill \end{array}\end{array}$$

    (266)

5.1.3. Expression for EFT Coefficients for Single-Field Slow-Roll Inflation

Here our prime objective is to derive the analytical expressions for EFT coefficients for single-field slow-roll inflation. To serve this purpose we start with a claim that the three-point function and the associated bispectrum for the scalar fluctuations computed from single-field slow-roll inflation is exactly same as that we have computed from EFT setup for consistent UV completion. Here we use the equilateral limit and squeezed limit configurations to extract the analytical expression for the EFT coefficients. In the two limiting cases the results are as follows:

• Equilateral limit configuration:
  For this case with arbitrary vacuum one can write:

  $$\begin{array}{ccc}\hfill B_{EFT}(k,k,k)& =& B_{SFSR}(k,k,k),\hfill \end{array}$$

  (267)

  which implies that:

  $$\begin{array}{cc}\hfill {\overline{M}}_1& ={\left\{\frac{H{M}_p^2ϵ\left[6(\eta -2ϵ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2-27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right]}{\left[\frac{125}{12}{U}_1+\frac{8073}{196}{U}_2\right]}\right\}}^{\frac{1}{3}},\hfill \\ \hfill {\overline{M}}_2& \approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}={\left\{\frac{2M_p^2ϵ\left[6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right]}{\left(1+ϵ\right)\left[\frac{125}{3}{U}_1+\frac{8073}{49}{U}_2\right]}\right\}}^{\frac{1}{2}},\hfill \\ \hfill {\tilde{c}}_5& =-\frac{1}{2}\left(1+ϵ\right),M_2=0,M_3=0,\hfill \\ \hfill M_4& ={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}={\left\{\frac{{\tilde{c}}_3{H}^2{M}_p^2ϵ\left[6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right]}{{\tilde{c}}_6\left[\frac{125}{12}{U}_1+\frac{8073}{196}{U}_2\right]}\right\}}^{\frac{1}{4}}.\hfill \end{array}$$

  (268)

  where for arbitrary vacuum $U_1$ and $U_2$ are defined as:

  $$\begin{array}{ccc}\hfill U_1& =& \left[{\left(C_1-C_2\right)}^3\left(C_1^{*3}+C_2^{*3}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3\left(C_1^3+C_2^3\right)\right],\hfill \end{array}$$

  (269)

  $$\begin{array}{ccc}\hfill U_2& =& \left[{\left(C_1-C_2\right)}^3{C}_1^{*}{C}_2^{*}\left(C_1^{*}-C_2^{*}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3{C}_1{C}_2\left(C_1-C_2\right)\right].\hfill \end{array}$$

  (270)

  To constraint all these coefficients of EFT operators using CMB observations from Planck TT+low P data we use [32]:

  $$\begin{array}{ccc}\hfill ϵ& <& 0.012(95\%\mathrm{CL}),\eta =-0.{0080}_{-0.0146}^{+0.0088}(68\%\mathrm{CL}),c_S=1(95\%\mathrm{CL}),\hfill \\ \hfill H=H_{inf}& \le & 1.09\times {10}^{-4}{M}_p\sqrt{ϵc_S},\hfill \end{array}$$

  (271)

  where $M_p=2.43\times {10}^{18}\mathrm{GeV}$ is the reduced Planck mass. Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$ we get:

    $$\begin{array}{cc}\hfill {\overline{M}}_1& ={\left\{\frac{6}{125}H{M}_p^2ϵ\left[6\eta -23ϵ\right]\right\}}^{\frac{1}{3}},{\overline{M}}_2\approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}={\left\{\frac{3}{125\left(1+ϵ\right)}{M}_p^2ϵ\left[23ϵ-6\eta \right]\right\}}^{\frac{1}{2}},\hfill \\ \hfill {\tilde{c}}_5& =-\frac{1}{2}\left(1+ϵ\right),M_2=0,M_3=0,M_4={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}={\left\{\frac{6{\tilde{c}}_3}{125{\tilde{c}}_6}{H}^2{M}_p^2ϵ\left[23ϵ-6\eta \right]\right\}}^{\frac{1}{4}}.\hfill \end{array}$$

    (272)

    Furthermore, using the constraint stated in Equation (271) we finally get the following constraints on the coefficients of EFT operators:

    $$\begin{array}{c}1.23\times {10}^{-3}{M}_p<|{\overline{M}}_1|<1.41\times {10}^{-3}{M}_p,8.79\times {10}^{-3}{M}_p<|{\overline{M}}_2|\approx |{\overline{M}}_3|<1.08\times {10}^{-2}{M}_p,\hfill \\ M_2=0,M_3=0,3.86\times {10}^{-4}{M}_p<M_4\times {\left(-{\tilde{c}}_6/{\tilde{c}}_3\right)}^{1/4}<4.29\times {10}^{-4}{M}_p.\hfill \end{array}$$

    (273)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

    $$\begin{array}{cc}\hfill {\overline{M}}_1& ={\left\{\frac{H{M}_p^2ϵ\left[6(\eta -2ϵ){cosh}^{2}2\alpha -11ϵ-27ϵ{sinh}^{2}2\alpha {cos}^2\beta \right]}{\left[\frac{125}{12}{J}_1(\alpha ,\beta )+\frac{8073}{196}{J}_2(\alpha ,\beta )\right]}\right\}}^{\frac{1}{3}},\hfill \\ \hfill {\overline{M}}_2& \approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}={\left\{\frac{2M_p^2ϵ\left[6(2ϵ-\eta ){cosh}^{2}2\alpha +11ϵ+27ϵ{sinh}^{2}2\alpha {cos}^2\beta \right]}{\left(1+ϵ\right)\left[\frac{125}{3}{J}_1(\alpha ,\beta )+\frac{8073}{49}{J}_2(\alpha ,\beta )\right]}\right\}}^{\frac{1}{2}},\hfill \\ \hfill {\tilde{c}}_5& {\displaystyle =-\frac{1}{2}\left(1+ϵ\right),M_2=0,M_3=0,}\hfill \\ \hfill M_4& ={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}={\left\{\frac{{\tilde{c}}_3{H}^2{M}_p^2ϵ\left[6(2ϵ-\eta ){cosh}^{2}2\alpha +11ϵ+27ϵ{sinh}^{2}2\alpha {cos}^2\beta \right]}{{\tilde{c}}_6\left[\frac{125}{12}{J}_1(\alpha ,\beta )+\frac{8073}{196}{J}_2(\alpha ,\beta )\right]}\right\}}^{\frac{1}{4}}.\hfill \end{array}$$

    (274)

    Furthermore, using the constraint stated in Equation (271) we finally get the following constraints on the coefficients of EFT operators for a given value of the parameters $\alpha$ and $\beta$ (say for $\alpha =0.1$ and $\beta =0.1$):

    $$\begin{array}{c}9.1\times {10}^{-4}{M}_p<|{\overline{M}}_1|<1.1\times {10}^{-3}{M}_p,1.11\times {10}^{-2}{M}_p<|{\overline{M}}_2|\approx |{\overline{M}}_3|<1.5\times {10}^{-2}{M}_p,\hfill \\ M_2=0,M_3=0,3.06\times {10}^{-4}{M}_p<M_4\times {\left(-{\tilde{c}}_6/{\tilde{c}}_3\right)}^{1/4}<3.54\times {10}^{-4}{M}_p.\hfill \end{array}$$

    (275)
• Squeezed limit configuration:
  For this case with arbitrary vacuum one can write:

  $$\begin{array}{ccc}\hfill B_{EFT}(k_L,k_L,k_S)& =& B_{SFSR}(k_L,k_L,k_S),\hfill \end{array}$$

  (276)

  which implies that:

  $$\begin{array}{cc}\hfill {\overline{M}}_1& ={\left\{2H{M}_p^2ϵ\frac{{\sum }_{j=-1}^3{a}_j{\left(\frac{k_S}{k_L}\right)}^j}{{\sum }_{m=-1}^3{b}_m{\left(\frac{k_S}{k_L}\right)}^m}\right\}}^{\frac{1}{3}},{\overline{M}}_2\approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}={\left\{-\frac{M_p^2ϵ}{\left(1+ϵ\right)}\frac{{\sum }_{j=-1}^3{a}_j{\left(\frac{k_S}{k_L}\right)}^j}{{\sum }_{m=-1}^3{b}_m{\left(\frac{k_S}{k_L}\right)}^m}\right\}}^{\frac{1}{2}},\hfill \\ \hfill {\tilde{c}}_5& =-\frac{1}{2}\left(1+ϵ\right),M_2=0,M_3=0,M_4={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}={\left\{\frac{2H^2{M}_p^2ϵ{\tilde{c}}_3}{{\tilde{c}}_6}\frac{{\sum }_{j=-1}^3{a}_j{\left(\frac{k_S}{k_L}\right)}^j}{{\sum }_{m=-1}^3{b}_m{\left(\frac{k_S}{k_L}\right)}^m}\right\}}^{\frac{1}{4}}.\hfill \end{array}$$

  (277)

  where the expansion coefficients $a_j\forall j=-1,\cdots ,3$ are defined earlier and here the coefficients $b_m\forall m=-1,\cdots ,3$ for arbitrary vacuum are defined as:

  $$\begin{array}{c}\hfill b_{-1}=-36U_2,b_0=\frac{9}{2}\left(U_1+9U_2\right),b_1=0,b_2=\left(\frac{27}{4}{U}_1+\frac{17}{2}{U}_2\right),b_3=0,\end{array}$$

  (278)

  where $U_1$ and $U_2$ are already defined earlier.
  Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$, we get $U_1=2$ and $U_2=0$. Consequently, the expansion coefficients can be recast as:

    $$\begin{array}{c}\hfill a_{-1}=0,a_0=4(3ϵ-\eta ),a_1=0,a_2=10ϵ,a_3=-(ϵ+2\eta ),\end{array}$$

    (279)

    and

    $$\begin{array}{c}\hfill b_{-1}=0,b_0=9,b_1=0,b_2=\frac{27}{2},b_3=0,\end{array}$$

    (280)

    Finally, the EFT coefficients for scalar fluctuation can be written as:

    $$\begin{array}{cc}\hfill {\overline{M}}_1& ={\left\{\frac{2H{M}_p^2ϵ\left[4(3ϵ-\eta )+10ϵ{\left(\frac{k_S}{k_L}\right)}^2-(ϵ+2\eta ){\left(\frac{k_S}{k_L}\right)}^3\right]}{\left[18-\frac{27}{2}{\left(\frac{k_S}{k_L}\right)}^2\right]}\right\}}^{\frac{1}{3}},\hfill \\ \hfill {\overline{M}}_2& \approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}={\left\{\frac{M_p^2ϵ\left[4(\eta -3ϵ)-10ϵ{\left(\frac{k_S}{k_L}\right)}^2+(ϵ+2\eta ){\left(\frac{k_S}{k_L}\right)}^3\right]}{\left(1+ϵ\right)\left[18-\frac{27}{2}{\left(\frac{k_S}{k_L}\right)}^2\right]}\right\}}^{\frac{1}{2}},\hfill \\ \hfill {\tilde{c}}_5& =-\frac{1}{2}\left(1+ϵ\right),M_2=0,M_3=0,\hfill \\ \hfill M_4& ={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}={\left\{\frac{2H^2{M}_p^2ϵ{\tilde{c}}_3}{{\tilde{c}}_6}\frac{\left[4(\eta -3ϵ)-10ϵ{\left(\frac{k_S}{k_L}\right)}^2+(ϵ+2\eta ){\left(\frac{k_S}{k_L}\right)}^3\right]}{\left[18-\frac{27}{2}{\left(\frac{k_S}{k_L}\right)}^2\right]}\right\}}^{\frac{1}{4}}.\hfill \end{array}$$

    (281)

    Furthermore, using the constraint stated in Equation (271) we finally get the following constraints on the coefficients of EFT operators for a given value of the parameter $k_S/k_L$ (say for $k_S/k_L=0.1$):

    $$\begin{array}{c}1.22\times {10}^{-3}{M}_p<|{\overline{M}}_1|<1.56\times {10}^{-3}{M}_p,8.67\times {10}^{-3}{M}_p<|{\overline{M}}_2|\approx |{\overline{M}}_3|<1.25\times {10}^{-2}{M}_p,\hfill \\ M_2=0,M_3=0,3.75\times {10}^{-4}{M}_p<M_4\times {\left(-{\tilde{c}}_6/{\tilde{c}}_3\right)}^{1/4}<4.51\times {10}^{-4}{M}_p.\hfill \end{array}$$

    (282)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$, we get $U_1=J_1(\alpha ,\beta )$ and $U_2=J_2(\alpha ,\beta )$. Consequently, the expansion coefficients can be recast as:

    $$\begin{array}{ccc}\hfill a_{-1}& =& 16ϵ{sinh}^{2}2\alpha {cos}^2\beta ,a_0=4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill a_1& =& 34ϵ{sinh}^{2}2\alpha {cos}^2\beta ,a_2=10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill a_3& =& 2(2ϵ-\eta ){cosh}^{2}2\alpha -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta .\hfill \end{array}$$

    (283)

    and

    $$\begin{array}{ccc}\hfill b_{-1}& =& -36J_2(\alpha ,\beta ),b_0=\frac{9}{2}\left(J_1(\alpha ,\beta )+9J_2(\alpha ,\beta )\right),\hfill \\ \hfill b_2& =& \left(\frac{27}{4}{J}_1(\alpha ,\beta )+\frac{17}{2}{J}_2(\alpha ,\beta )\right),b_3=0=b_1.\hfill \end{array}$$

    (284)

    Finally, the EFT coefficients for scalar fluctuation can be written as:

    $$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\overline{M}}_1& =\left\{2H{M}_p^2ϵ\left[-36J_2(\alpha ,\beta ){\left(\frac{k_S}{k_L}\right)}^{-1}+9\left(J_1(\alpha ,\beta )+\frac{J_2(\alpha ,\beta )}{2}\right)\right.\right.\hfill \\ & {\left.-\frac{3}{4}\left(9J_1(\alpha ,\beta )-7J_2(\alpha ,\beta )\right){\left(\frac{k_S}{k_L}\right)}^2\right]}^{-1}\hfill \\ & \left[16ϵ{sinh}^{2}2\alpha {cos}^2\beta {\left(\frac{k_S}{k_L}\right)}^{-1}+\left(4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta \right)\right.\hfill \\ & +34ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(\frac{k_S}{k_L}\right)+\left(10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^2\hfill \\ & {\left.\left.+\left(2(2ϵ-\eta ){cosh}^{2}2\alpha -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^3\right]\right\}}^{\frac{1}{3}},\hfill \\ \hfill {\overline{M}}_2& \approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}=\left\{\frac{M_p^2ϵ}{(1+ϵ)}\left[36J_2(\alpha ,\beta ){\left(\frac{k_S}{k_L}\right)}^{-1}-9\left(J_1(\alpha ,\beta )+\frac{J_2(\alpha ,\beta )}{2}\right)\right.\right.\hfill \\ & {\left.+\frac{3}{4}\left(9J_1(\alpha ,\beta )-7J_2(\alpha ,\beta )\right){\left(\frac{k_S}{k_L}\right)}^2\right]}^{-1}\hfill \\ & \left[16ϵ{sinh}^{2}2\alpha {cos}^2\beta {\left(\frac{k_S}{k_L}\right)}^{-1}+\left(4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta \right)\right.\hfill \\ & +34ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(\frac{k_S}{k_L}\right)+\left(10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^2\hfill \\ & {\left.\left.+\left(2(2ϵ-\eta ){cosh}^{2}2\alpha -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^3\right]\right\}}^{\frac{1}{2}},\hfill \\ \hfill {\tilde{c}}_5& =-\frac{1}{2}\left(1+ϵ\right),M_2=0,M_3=0,\hfill \\ \hfill M_4& ={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}=\left\{\frac{2H^2{M}_p^2ϵ{\tilde{c}}_3}{{\tilde{c}}_6}\left[36J_2(\alpha ,\beta ){\left(\frac{k_S}{k_L}\right)}^{-1}-9\left(J_1(\alpha ,\beta )+\frac{J_2(\alpha ,\beta )}{2}\right)\right.\right.\hfill \\ & {\left.+\frac{3}{4}\left(9J_1(\alpha ,\beta )-7J_2(\alpha ,\beta )\right){\left(\frac{k_S}{k_L}\right)}^2\right]}^{-1}\hfill \\ & \left[16ϵ{sinh}^{2}2\alpha {cos}^2\beta {\left(\frac{k_S}{k_L}\right)}^{-1}+\left(4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta \right)\right.\hfill \\ & +34ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(\frac{k_S}{k_L}\right)+\left(10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^2\hfill \\ & {\left.\left.+\left(2(2ϵ-\eta ){cosh}^{2}2\alpha -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^3\right]\right\}}^{\frac{1}{4}}.\hfill \end{array}\end{array}$$

    (285)

    Furthermore, using the constraint stated in Equation (271) we finally get the following constraints on the coefficients of EFT operators for a given value of the parameters $\alpha$, $\beta$ and $k_S/k_L$ (say for $\alpha =0.1$, $\beta =0.1$ and $k_S/k_L=0.1$):

    $$\begin{array}{c}6.05\times {10}^{-4}{M}_p<|{\overline{M}}_1|<7.15\times {10}^{-4}{M}_p,3.03\times {10}^{-3}{M}_p<|{\overline{M}}_2|\approx |{\overline{M}}_3|<3.89\times {10}^{-3}{M}_p,\hfill \\ M_2=0,M_3=0,2.22\times {10}^{-3}{M}_p<M_4\times {\left(-{\tilde{c}}_6/{\tilde{c}}_3\right)}^{1/4}<2.51\times {10}^{-3}{M}_p.\hfill \end{array}$$

    (286)

5.2. For General Single-Field $P(X,\varphi )$ Inflation

Here our prime objective is to derive the EFT coefficients by computing the most general expression for the three-point function for scalar fluctuations from the general single-field $P(X,\varphi )$ model of inflation for arbitrary vacuum. Then we give specific example for Bunch–Davies and $\alpha ,\beta$ vacuum for completeness.

5.2.1. Basic Setup

Let us start with the action for single scalar field (inflaton) which is described by the general function $P(X,\varphi )$, contains non-canonical kinetic term in general and it is minimally coupled to the gravity [33,44]:

$$S=\int d^{4}x\sqrt{-g}\left[\frac{M_p^2}{2}R+P(X,\varphi )\right].$$

(287)

In the case of general structure of $P(X,\varphi )$ the pressure p, the energy density $\rho$ and effective speed of sound parameter $c_S$ can be written as [44]:

$$\begin{array}{ccc}\hfill p& =& P(X,\varphi ),\rho =2X{P}_{,X}(X,\varphi )-P(X,\varphi ),c_S=\sqrt{\frac{P_{,X}(X,\varphi )}{P_{,X}(X,\varphi )+2X{P}_{,XX}(X,\varphi )}}.\hfill \end{array}$$

(288)

In the case of general $P(X,\varphi )$ theory the slow-roll parameters can be expressed as [44]:

$$\begin{array}{ccc}\hfill ϵ& =& \frac{X{P}_X(X,\varphi )}{H^2{M}_p^2},\eta =ϵ-\frac{\delta }{P_{,X}(X,\varphi )}\left[P_{,X}(X,\varphi )+X{P}_{,XX}(X,\varphi )\right],\hfill \\ \hfill s& =& \frac{2X\delta }{P_{,X}(X,\varphi )}\frac{\left[X{P}_{,XX}^2(X,\varphi )-P_{,X}(X,\varphi )P_{,XX}(X,\varphi )-X{P}_{,X}(X,\varphi )P_{,XXX}(X,\varphi )\right]}{\left[P_{,X}(X,\varphi )+X{P}_{,XX}(X,\varphi )\right]}.\hfill \end{array}$$

(289)

In the case of single-field slow-roll inflation we have:

$$P(X,\varphi )=X-V\left(\varphi \right),$$

(290)

where $V\left(\varphi \right)$ is the single-field slowly varying potential. For this case if we compute the effective sound speed then it turns out to be $c_S=1$, which is consistent with our result obtained in the previous section. Also, if we compute the expressions for the slow-roll parameters $ϵ,\eta ,\delta$ and s the results also perfectly match the results obtained in Equation (253).

Similarly, in case DBI inflationary model one can identify the function $P(X,\varphi )$ as [45]:

$$P(X,\varphi )=-\frac{1}{f\left(\varphi \right)}\sqrt{1-2Xf\left(\varphi \right)}+\frac{1}{f\left(\varphi \right)}-V\left(\varphi \right),$$

(291)

where the inflaton $\varphi$ is identified to be the position of a D3 barne which is moving in warped throat geometry and $f\left(\varphi \right)$ characterize the warp factor22. For the effective potential $V\left(\varphi \right)$ one can consider following mathematical structures of the potentials in the UV and IR regime [45]:

• UV regime: In this case, the inflaton moves from the UV regime of the warped geometric space to the IR regime under the influence of the effective potential, $V\left(\varphi \right)\simeq \frac{1}{2}{m}^2{\varphi }^2,$ where the inflaton mass satisfies the constraint $m>>M_p\sqrt{\lambda }$. In this specific situation the inflaton starts rolling very far away from the origin of the effective potential and then rolls down in a relativistic fashion to the minimum of potential situated at the origin.
• IR regime: In this case, the inflaton started moving from the IR regime of the warped space geometry to the UV regime under the influence of the effective potential, $V\left(\varphi \right)\simeq V_0-\frac{1}{2}{m}^2{\varphi }^2,$ where the inflaton mass is comparable to the scale of inflation, as given by, $m\approx H$. In this specific situation, the inflaton starts rolling down near the origin of the effective potential and rolls down in a relativistic fashion away from it.

In the case of the DBI model the pressure p and the energy density $\rho$ can be written as [45]:

$$\begin{array}{ccc}\hfill p& =& \frac{1}{f\left(\varphi \right)}(1-c_S)-V\left(\varphi \right),\rho =\frac{1}{f\left(\varphi \right)}\left(\frac{1}{c_S}-1\right)+V\left(\varphi \right),c_S=\sqrt{1-2Xf\left(\varphi \right)}=\sqrt{1-{\dot{\varphi }}^{2}f\left(\varphi \right)},\hfill \end{array}$$

(292)

where $X={\dot{\varphi }}^2/2$. In this context the slow-roll parameter [45]:

$$\begin{array}{ccc}\hfill ϵ& =& \frac{3{\dot{\varphi }}^2}{2\left[c_{S}V\left(\varphi \right)+\frac{1}{f\left(\varphi \right)}(1-c_S)\right]}\approx \frac{3}{2\left[1+c_{S}f\left(\varphi \right)V\left(\varphi \right)\right]}.\hfill \end{array}$$

(293)

is not small and as a result the effective sound speed is very small, $c_S<<1$. Consequently, the inflaton speed during inflation is given by the expression, $\dot{\varphi }=\pm \frac{1}{\sqrt{f\left(\varphi \right)}}.$ Additionally, it is important to note that in the context of DBI inflation the other slow-roll parameters $\eta$ and s can be computed as:

$$\begin{array}{ccc}\hfill \eta & \approx & \frac{\left[3\sqrt{1+c_{S}f\left(\varphi \right)V\left(\varphi \right)}+\frac{\sqrt{3f\left(\varphi \right)c_S}}{2}{M}_p\dot{\varphi }{c}_S\left\{f\left(\varphi \right)V^{{}^{\prime }}\left(\varphi \right)+V\left(\varphi \right)f^{{}^{\prime }}\left(\varphi \right)-\frac{1}{2c_S^2}\left(2\ddot{\varphi }f\left(\varphi \right)+{\dot{\varphi }}^2{f}^{{}^{\prime }}\left(\varphi \right)\right)\right\}\right]}{{\left[1+c_{S}f\left(\varphi \right)V\left(\varphi \right)\right]}^{\frac{3}{2}}},\hfill \\ \hfill s& =& -\frac{\sqrt{3f\left(\varphi \right)c_S}}{2c_S^2}{M}_p\dot{\varphi }\left[2\ddot{\varphi }f\left(\varphi \right)+{\dot{\varphi }}^2{f}^{{}^{\prime }}\left(\varphi \right)\right].\hfill \end{array}$$

(294)

In the slow-roll regime to validate slow-roll approximation along with $c_S<<1$ we need to satisfy the constraint condition for DBI inflation, $2c_{S}f\left(\varphi \right)V\left(\varphi \right)>>1.$

5.2.2. Scalar Three-Point Function for General Single-Field $P(X,\varphi )$ Inflation

Before computing the three-point function for scalar mode fluctuation here it is important to note that the two-point function for general single-field $P(X,\varphi )$ inflation is exactly same with the results obtained for EFT of inflation with sound speed $c_S<<1$ and ${\tilde{c}}_S<<1$, which can be obtained by setting the EFT coefficients, $M_2\ne 0$, $M_3\ne 0$, ${\overline{M}}_1\ne 0$, $M_4\ne 0$, ${\overline{M}}_2\ne 0$, ${\overline{M}}_3\ne 0$, ${\tilde{c}}_5\ne -\frac{1}{2}(1+ϵ)$23. Using three-point function we can able to fix all of these coefficients.

Now here before going into the details of the computation for three-point function, just using the knowledge of the two-point function, we can easily identify the exact analytical expression for the EFT coefficient $M_2$. For this, we need to identify the effective sound speed computed from general single-field $P(X,\varphi )$ inflation with the result obtained for the proposed EFT set up. Consequently, we get:

$$\begin{array}{c}\hfill {\displaystyle M_2={\left(-\frac{X{P}_{,XX}(X,\varphi )}{P_{,X}(X,\varphi )}\dot{H}{M}_p^2\right)}^{\frac{1}{4}}=\left\{\begin{array}{cc}{\displaystyle 0}\hfill & \mathbf{for}\mathbf{single}-\mathbf{field}\mathbf{slow}-\mathbf{roll}\hfill \\ {\displaystyle {\left[\left(\frac{{\dot{\varphi }}^{2}f\left(\varphi \right)}{{\dot{\varphi }}^{2}f\left(\varphi \right)-1}\right)\frac{\dot{H}{M}_p^2}{2}\right]}^{\frac{1}{4}}}\hfill & \mathbf{for}\mathbf{DBI}.\hfill \end{array}\right.}\end{array}$$

(295)

We here now proceed to calculate the three-point function for the scalar fluctuation $\zeta (t,\mathbf{x})$ in the interacting picture with arbitrary vacuum in the case of general single-field $P(X,\varphi )$ inflation. Then we cite results for Bunch–Davies and $\alpha ,\beta$ vacuum and give a specific example for the DBI model of inflation.

Here we introduce two new parameters [44]:

$$\begin{array}{ccc}\hfill {\Sigma }_1(X,\varphi )& =& X{P}_{,X}(X,\varphi )+2X^2{P}_{,XX}(X,\varphi )=\frac{ϵH^2{M}_p^2}{c_S^2},\hfill \end{array}$$

(296)

$$\begin{array}{ccc}\hfill {\Sigma }_2(X,\varphi )& =& X^2{P}_{,XX}(X,\varphi )+\frac{2}{3}{X}^3{P}_{,XXX}(X,\varphi ).\hfill \end{array}$$

(297)

which will appear in the expression for three-point function for the scalar fluctuation. For single-field slow-roll inflation and DBI inflation we get the following expressions for these parameters [44]:

$$\begin{array}{ccc}\hfill {\displaystyle {\Sigma }_1(X,\varphi )}& =& \left\{\begin{array}{cc}{\displaystyle X=ϵH^2{M}_p^2}\hfill & \mathbf{for}\mathbf{single}-\mathbf{field}\mathbf{slow}-\mathbf{roll}\hfill \\ {\displaystyle \frac{X}{{\left(1-2Xf\left(\varphi \right)\right)}^{\frac{3}{2}}}=\frac{ϵH^2{M}_p^2}{c_S^2}}\hfill & \mathbf{for}\mathbf{DBI}.\hfill \end{array}\right.\hfill \end{array}$$

(298)

$$\begin{array}{ccc}\hfill {\displaystyle {\Sigma }_2(X,\varphi )}& =& \left\{\begin{array}{cc}{\displaystyle 0}\hfill & \mathbf{for}\mathbf{single}-\mathbf{field}\mathbf{slow}-\mathbf{roll}\hfill \\ {\displaystyle \frac{X^{2}f\left(\varphi \right)}{{\left(1-2Xf\left(\varphi \right)\right)}^{\frac{5}{2}}}}\hfill & \mathbf{for}\mathbf{DBI}.\hfill \end{array}\right.\hfill \end{array}$$

(299)

For general single-field $P(X,\varphi )$ inflation the third-order term in the action Equation (287) is given by [44]:

$$\begin{array}{cc}\hfill S_{\zeta }^{\left(3\right)}& {\displaystyle =\int d^{4}x\left[-a^3\left\{{\Sigma }_1(X,\varphi )\left(1-\frac{1}{c_S^2}\right)+2{\Sigma }_2(X,\varphi )\right\}\frac{{\dot{\tilde{\zeta }}}^3}{H^3}+\frac{a^3ϵ\left(ϵ-3+3c_S^2\right)}{c_S^4}\tilde{\zeta }{\dot{\tilde{\zeta }}}^2\right.}\hfill \\ & \left.{\displaystyle +\frac{aϵ\left(ϵ-2s+1-c_S^2\right)}{c_S^2}\tilde{\zeta }{(\partial \tilde{\zeta })}^2-2a^3ϵ\dot{\tilde{\zeta }}{\partial }_i\tilde{\zeta }{\partial }_i\left(\frac{ϵ}{c_S^2}{\partial }^{-2}\dot{\tilde{\zeta }}\right)}\right],\hfill \end{array}$$

(300)

which is derived from Equation (244) and here after neglecting all the contribution from the terms which are sub-leading in the slow-roll parameters. Additionally, here we use the following field redefinition:

$$\zeta =\tilde{\zeta }+\frac{1}{c_S^2}\left\{ϵ-\frac{\eta }{2}\right\}{\tilde{\zeta }}^2,$$

(301)

where $ϵ$, $\eta$, $\delta$ and s are already defined earlier for general single-field $P(X,\varphi )$ inflation.

Now it is important to note that in the present context of discussion we are interested in the three-point function for the scalar fluctuation field $\zeta$, not for the redefined scalar field fluctuation $\tilde{\zeta }$ and for this reason one can write down the exact connection between the three-point function for the scalar function field $\zeta$ and redefined scalar fluctuation field $\tilde{\zeta }$ in position space as:

$$\begin{array}{ccc}\hfill \langle \zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\rangle & =& \langle \tilde{\zeta }\left({\mathbf{x}}_{\mathbf{1}}\right)\tilde{\zeta }\left({\mathbf{x}}_{\mathbf{2}}\right)\tilde{\zeta }\left({\mathbf{x}}_{\mathbf{3}}\right)+\frac{\left(2ϵ-\eta \right)}{c_S^2}\left[\langle \zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\rangle \langle \zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\rangle \right.\hfill \\ & & \left.+\langle \zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\rangle \langle \zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\rangle +\langle \zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\zeta \left({\mathbf{x}}_{\mathbf{1}}\right)\rangle \langle \zeta \left({\mathbf{x}}_{\mathbf{3}}\right)\zeta \left({\mathbf{x}}_{\mathbf{2}}\right)\rangle \right].\hfill \end{array}$$

(302)

After taking the Fourier transform of the scalar fluctuation field $\zeta$ and redefined scalar fluctuation field $\tilde{\zeta }$ one can express connection between three-point function in momentum space and this is also our main point of interest.

The interaction Hamiltonian for the redefined scalar fluctuation $\tilde{\zeta }$ can be expressed as:

$$\begin{array}{ccc}\hfill H_{int}& =& \int d^{3}x\left[-\left\{{\Sigma }_1(X,\varphi )\left(1-\frac{1}{c_S^2}\right)+2{\Sigma }_2(X,\varphi )\right\}\frac{{\tilde{\zeta }}^{{}^{\prime }3}}{H^3}+\frac{aϵ\left(ϵ-3+3c_S^2\right)}{c_S^4}\tilde{\zeta }{\tilde{\zeta }}^{{}^{\prime }2}\right.\hfill \\ & & \left.+\frac{aϵ\left(ϵ-2s+1-c_S^2\right)}{c_S^2}\tilde{\zeta }{(\partial \tilde{\zeta })}^2-2a{\tilde{\zeta }}^{{}^{\prime }}{\partial }_i\tilde{\zeta }{\partial }_i\left(\frac{ϵ}{c_S^2}{\partial }^{-2}{\tilde{\zeta }}^{{}^{\prime }}\right)\right].\hfill \end{array}$$

(303)

Furthermore, following the in-in formalism in interaction picture the expression for the three-point function for the redefined scalar fluctuation $\tilde{\zeta }$ and then transforming the final result in terms of the scalar fluctuation $\zeta$ in momentum one can write the following expression:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \langle \zeta \left({\mathbf{k}}_{\mathbf{1}}\right)\zeta \left({\mathbf{k}}_{\mathbf{2}}\right)\zeta \left({\mathbf{k}}_{\mathbf{3}}\right)\rangle & {\displaystyle =-i{\int }_{{\eta }_i=-\infty }^{{\eta }_f=0}d\eta a\left(\eta \right)\langle 0|\left[\zeta ({\eta }_f,{\mathbf{k}}_{\mathbf{1}})\zeta ({\eta }_f,{\mathbf{k}}_{\mathbf{2}})\zeta ({\eta }_f,{\mathbf{k}}_{\mathbf{3}}),H_{int}\left(\eta \right)\right]|0\rangle }\hfill \\ & ={\left(2\pi \right)}^3{\delta }^{\left(3\right)}({\mathbf{k}}_{\mathbf{1}}+{\mathbf{k}}_{\mathbf{2}}+{\mathbf{k}}_{\mathbf{3}})B_{GSF}(k_1,k_2,k_3),\hfill \end{array}\end{array}$$

(304)

where $B_{GSF}(k_1,k_2,k_3)$ represents the bispectrum of scalar fluctuation $\zeta$, which is computed from general single-field $P(X,\varphi )$ inflation. Here the final expression for the bispectrum of scalar fluctuation for arbitrary vacuum is given by:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[\frac{3}{2}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\frac{{\left(k_1{k}_2{k}_3\right)}^2}{K^3}\right.}\hfill \\ & {\displaystyle +\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\left(\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +\frac{s}{c_S^2}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\left(-2\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +2(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\sum _{i=1}^3{k}_i^3}\hfill \\ & {\displaystyle +ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & +ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2\right.\hfill \\ & \left.\left.{\displaystyle +8\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\sum _{m=1}^3\frac{1}{K-2k_m}}\right)+\mathcal{O}\left(ϵ\right)\right],\hfill \end{array}\end{array}$$

(305)

where $\mathcal{O}\left(ϵ\right)$ characterizes the sub-leading corrections in the three-point function for the scalar fluctuation computed from general single-field $P(X,\varphi )$ inflation.

Furthermore, we consider a very specific class of models, where the following constraint condition24$P_{,X\varphi }(X,\varphi )=0$ perfectly holds good. In this case one can write down the following simplified expression for the bispectrum of scalar fluctuation for arbitrary vacuum as:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\frac{{\left(k_1{k}_2{k}_3\right)}^2}{K^3}\right.}\hfill \\ & {\displaystyle +\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\left(\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +2(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\sum _{i=1}^3{k}_i^3}\hfill \\ & {\displaystyle +ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & \left.{\displaystyle +ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+8\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\sum _{m=1}^3\frac{1}{K-2k_m}\right)}\right],\hfill \end{array}\end{array}$$

(306)

where the new parameter ${ϵ}_X$ is defined as25:

$${ϵ}_X=-\frac{\dot{X}{H}_{,X}}{H^2}.$$

(308)

For Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:

• For Bunch–Davies vacuum:
  After setting $C_1=1$ and $C_2=0$ we get [44]:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{{(}_1{k}_2{k}_3)}^3}\left[\frac{3}{2}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)\frac{{\left(k_1{k}_2{k}_3\right)}^2}{K^3}\right.}\hfill \\ & {\displaystyle +\left(\frac{1}{c_S^2}-1\right)\left(\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +\frac{s}{c_S^2}\left(-2\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +2(2ϵ-\eta )\sum _{i=1}^3{k}_i^3}\hfill \\ & {\displaystyle \left.+ϵ\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)\right].}\hfill \end{array}\end{array}$$

  (309)

  Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, one can further write down the following expression for the bispectrum:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)\frac{{\left(k_1{k}_2{k}_3\right)}^2}{K^3}\right.}\hfill \\ & {\displaystyle +\left(\frac{1}{c_S^2}-1\right)\left(\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +2(2ϵ-\eta )\sum _{i=1}^3{k}_i^3}\hfill \\ & {\displaystyle \left.+ϵ\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)\right],}\hfill \end{array}\end{array}$$

  (310)
• For $\mathit{\alpha },\mathit{\beta }$ vacuum:
  After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get [33]:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[\frac{3}{2}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right){cosh}^{2}2\alpha \frac{{\left(k_1{k}_2{k}_3\right)}^2}{K^3}\right.}\hfill \\ & {\displaystyle +\left(\frac{1}{c_S^2}-1\right){cosh}^{2}2\alpha \left(\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +\frac{s}{c_S^2}{cosh}^{2}2\alpha \left(-2\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +2(2ϵ-\eta ){cosh}^{2}2\alpha \sum _{i=1}^3{k}_i^3+ϵ\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle \left.+ϵ{sinh}^{2}2\alpha {cos}^2\beta \left({\displaystyle -\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+8\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\sum _{m=1}^3\frac{1}{K-2k_m}}\right)\right].}\hfill \end{array}\end{array}$$

  (311)

  Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, one can further write down the following expression for the bispectrum:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{{\left(k_1{k}_2{k}_3\right)}^3}\left[\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right){cosh}^{2}2\alpha \frac{{\left(k_1{k}_2{k}_3\right)}^2}{K^3}\right.}\hfill \\ & {\displaystyle +\left(\frac{1}{c_S^2}-1\right){cosh}^{2}2\alpha \left(\sum _{i=1}^3{k}_i^3+\frac{4}{K^2}\sum _{i,j=1,i\ne j}^3{k}_i^2{k}_j^3-\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle +2(2ϵ-\eta ){cosh}^{2}2\alpha \sum _{i=1}^3{k}_i^3+ϵ\left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+\frac{8}{K}\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\right)}\hfill \\ & {\displaystyle \left.+ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(-\sum _{i=1}^3{k}_i^3+\sum _{i,j=1,i\ne j}^3{k}_i{k}_j^2+8\sum _{i,j=1,i>j}^3{k}_i^2{k}_j^2\sum _{m=1}^3\frac{1}{K-2k_m}\right)\right],}\hfill \end{array}\end{array}$$

  (312)

Furthermore, we consider equilateral limit and squeezed limit in which we finally get:

• Equilateral limit configuration:
  Here the bispectrum for scalar perturbations in the presence of arbitrary quantum vacuum can be expressed as:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k,k,k)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\right.}\hfill \\ & {\displaystyle -\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-\frac{34}{3}\frac{s}{c_S^2}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2}\hfill \\ & {\displaystyle \left.+6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right].}\hfill \end{array}\end{array}$$

  (313)

  Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, one can further write down the following expression for the bispectrum:

  $$\begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\right.}\hfill \\ & \left.{\displaystyle +6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2}\right],\hfill \end{array}$$

  (314)

  Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$ we get:

    $$\begin{array}{cc}\hfill B_{GSF}(k,k,k)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)-\frac{34}{3}\frac{s}{c_S^2}+23ϵ-6\eta \right].}\hfill \end{array}$$

    (315)

    Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get:

    $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_1,k_2,k_3)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)+23ϵ-6\eta \right],}\hfill \end{array}\end{array}$$

    (316)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get:

    $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k,k,k)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right){cosh}^{2}2\alpha \right.}\hfill \\ & {\displaystyle -\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){cosh}^{2}2\alpha -\frac{34}{3}\frac{s}{c_S^2}{cosh}^{2}2\alpha }\hfill \\ & {\displaystyle \left.+6(2ϵ-\eta ){cosh}^{2}2\alpha +11ϵ+27ϵ{sinh}^{2}2\alpha {cos}^2\beta \right].}\hfill \end{array}\end{array}$$

    (317)

    Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get:

    $$\begin{array}{cc}\hfill B_{GSF}(k,k,k)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k^6}\left[\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right){cosh}^{2}2\alpha -\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){cosh}^{2}2\alpha \right.}\hfill \\ & \left.{\displaystyle +6(2ϵ-\eta ){cosh}^{2}2\alpha +11ϵ+27ϵ{sinh}^{2}2\alpha {cos}^2\beta }\right].\hfill \end{array}$$

    (318)
• Squeezed limit configuration:
  Here the bispectrum for scalar perturbations in the presence of arbitrary quantum vacuum can be expressed as:

  $$\begin{array}{c}\hfill B_{GSF}(k_L,k_L,k_S)=\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}\sum _{j=-1}^3{t}_j{\left(\frac{k_S}{k_L}\right)}^j,\end{array}$$

  (319)

  where the expansion coefficients $t_j\forall j=-1,\cdots ,3$ for arbitrary vacuum are defined as:

  $$\begin{array}{ccc}\hfill t_{-1}& =& 16ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_0& =& \left(4(2ϵ-\eta )-\frac{6s}{c_S^2}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+4ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+4ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_1& =& 34ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_2& =& \left\{\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-6\left(\frac{1}{c_S^2}-1\right)-\frac{6s}{c_S^2}\right\}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\hfill \\ & & +10ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+10ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_3& =& \left\{2(2ϵ-\eta )-\frac{9}{32}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)+5\left(\frac{1}{c_S^2}-1\right)+\frac{2s}{c_S^2}\right\}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\hfill \\ & & -5ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2-ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2.\hfill \end{array}$$

  (320)

  Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get:

  $$\begin{array}{ccc}\hfill t_{-1}& =& 16ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_0& =& 4(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+4ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+4ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_1& =& 34ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_2& =& \left\{\frac{1}{8}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-6\left(\frac{1}{c_S^2}-1\right)\right\}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\hfill \\ & & +10ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+10ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2,\hfill \\ \hfill t_3& =& \left\{2(2ϵ-\eta )-\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)+5\left(\frac{1}{c_S^2}-1\right)\right\}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\hfill \\ & & -5ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2-ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2.\hfill \end{array}$$

  (321)

  Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$, we get the following expression for the expansion coefficients $t_j\forall j=-1,\cdots ,3$:

    $$\begin{array}{ccc}\hfill t_{-1}& =& 0,\hfill \\ \hfill t_0& =& 4(3ϵ-\eta )-\frac{6s}{c_S^2},\hfill \\ \hfill t_1& =& 0,\hfill \\ \hfill t_2& =& 10ϵ+\left\{\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-6\left(\frac{1}{c_S^2}-1\right)-\frac{6s}{c_S^2}\right\},\hfill \\ \hfill t_3& =& -(ϵ+2\eta )+\left\{5\left(\frac{1}{c_S^2}-1\right)+\frac{2s}{c_S^2}-\frac{9}{32}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)\right\}.\hfill \end{array}$$

    (322)

    Consequently, the bispectrum can be recast as:

    $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_L,k_L,k_S)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}\left[\left\{4(3ϵ-\eta )-\frac{6s}{c_S^2}\right\}\right.}\hfill \\ & {\displaystyle +\left(10ϵ+\left\{\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-6\left(\frac{1}{c_S^2}-1\right)-\frac{6s}{c_S^2}\right\}\right){\left(\frac{k_S}{k_L}\right)}^2}\hfill \\ & {\displaystyle +\left(-(ϵ+2\eta )+\left\{5\left(\frac{1}{c_S^2}-1\right)+\frac{2s}{c_S^2}\right.\right.}\hfill \\ & {\displaystyle \left.\left.\left.-\frac{9}{32}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)\right\}\right){\left(\frac{k_S}{k_L}\right)}^3\right].}\hfill \end{array}\end{array}$$

    (323)

    Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get:

    $$\begin{array}{ccc}\hfill t_{-1}& =& 0,\hfill \\ \hfill t_0& =& 4(3ϵ-\eta ),\hfill \\ \hfill t_1& =& 0,\hfill \\ \hfill t_2& =& \left\{\frac{1}{8}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-6\left(\frac{1}{c_S^2}-1\right)\right\}+10ϵ,\hfill \\ \hfill t_3& =& \left\{5\left(\frac{1}{c_S^2}-1\right)-\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)\right\}-(ϵ+2\eta )\hfill \end{array}$$

    (324)

    for such case bispectrum is given by:

    $$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_{GSF}(k_L,k_L,k_S)& {\displaystyle =\frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}\left[4(3ϵ-\eta )\right.}\hfill \\ & {\displaystyle +\left(\left\{\frac{1}{8}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-6\left(\frac{1}{c_S^2}-1\right)\right\}+10ϵ\right){\left(\frac{k_S}{k_L}\right)}^2}\hfill \\ & {\displaystyle \left.+\left(-(ϵ+2\eta )+\left\{5\left(\frac{1}{c_S^2}-1\right)-\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)\right\}\right){\left(\frac{k_S}{k_L}\right)}^3\right].}\hfill \end{array}\end{array}$$

    (325)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$, we get the following expression for the expansion coefficients $a_j\forall j=-1,\cdots ,3$:

    $$\begin{array}{ccc}\hfill t_{-1}& =& 16ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_0& =& \left(4(2ϵ-\eta )-\frac{6s}{c_S^2}\right){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_1& =& 34ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_2& =& \left\{\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-6\left(\frac{1}{c_S^2}-1\right)-\frac{6s}{c_S^2}\right\}{cosh}^{2}2\alpha \hfill \\ & & +10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_3& =& \left\{2(2ϵ-\eta )-\frac{9}{32}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)+5\left(\frac{1}{c_S^2}-1\right)+\frac{2s}{c_S^2}\right\}{cosh}^{2}2\alpha \hfill \\ & & -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta .\hfill \end{array}$$

    (326)

    Consequently, the bispectrum can be recast as:

    $$\begin{array}{cc}\hfill B_{GSF}(k_L,k_L,k_S)& ={\displaystyle \frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}}\left[16ϵ{sinh}^{2}2\alpha {cos}^2\beta {\left(\frac{k_S}{k_L}\right)}^{-1}\right.\hfill \\ & +\left(\left(4(2ϵ-\eta )-\frac{6s}{c_S^2}\right){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta \right)\hfill \\ & +34ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(\frac{k_S}{k_L}\right)\hfill \\ & +\left(\left\{\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-6\left(\frac{1}{c_S^2}-1\right)-\frac{6s}{c_S^2}\right\}{cosh}^{2}2\alpha \right.\hfill \\ & \left.+10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^2\hfill \\ & +\left(\left\{2(2ϵ-\eta )-\frac{9}{32}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)+5\left(\frac{1}{c_S^2}-1\right)+\frac{2s}{c_S^2}\right\}{cosh}^{2}2\alpha \right.\hfill \\ & \left.\left.-5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^3\right].\hfill \end{array}$$

    (327)

    Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get:

    $$\begin{array}{ccc}\hfill t_{-1}& =& 16ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_0& =& 4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_1& =& 34ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_2& =& \left\{\frac{1}{8}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-6\left(\frac{1}{c_S^2}-1\right)\right\}{cosh}^{2}2\alpha \hfill \\ & & +10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta ,\hfill \\ \hfill t_3& =& \left\{2(2ϵ-\eta )-\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)+5\left(\frac{1}{c_S^2}-1\right)\right\}{cosh}^{2}2\alpha \hfill \\ & & -5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta .\hfill \end{array}$$

    (328)

    Consequently, the bispectrum can be recast as:

    $$\begin{array}{cc}\hfill B_{GSF}(k_L,k_L,k_S)& ={\displaystyle \frac{H^4}{32{ϵ}^2{M}_p^4}\frac{1}{k_L^3{k}_S^3}}\left[16ϵ{sinh}^{2}2\alpha {cos}^2\beta {\left(\frac{k_S}{k_L}\right)}^{-1}\right.\hfill \\ & +\left(4(2ϵ-\eta ){cosh}^{2}2\alpha +4ϵ+4ϵ{sinh}^{2}2\alpha {cos}^2\beta \right)\hfill \\ & +34ϵ{sinh}^{2}2\alpha {cos}^2\beta \left(\frac{k_S}{k_L}\right)\hfill \\ & +\left(\left\{\frac{1}{8}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-6\left(\frac{1}{c_S^2}-1\right)\right\}{cosh}^{2}2\alpha \right.\hfill \\ & \left.+10ϵ+10ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^2\hfill \\ & +\left(\left\{2(2ϵ-\eta )-\frac{3}{16}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)+5\left(\frac{1}{c_S^2}-1\right)\right\}{cosh}^{2}2\alpha \right.\hfill \\ & \left.\left.-5ϵ-ϵ{sinh}^{2}2\alpha {cos}^2\beta \right){\left(\frac{k_S}{k_L}\right)}^3\right].\hfill \end{array}$$

    (329)

5.2.3. Expression for EFT Coefficients for General Single-Field $P(X,\varphi )$ Inflation

Here our prime objective is to derive the analytical expressions for EFT coefficients for general single-field $P(X,\varphi )$ inflation. To serve this purpose here we start with a claim that the three-point function and the associated bispectrum for the scalar fluctuations computed from general single-field $P(X,\varphi )$ inflation is exactly same as that we have computed from EFT setup. Here we use the equilateral limit and squeezed limit configurations to extract the analytical expression for the EFT coefficients. In the two limiting cases the results are as follows:

• Equilateral limit configuration:
  For this case with arbitrary vacuum one can write:

  $$\begin{array}{ccc}\hfill B_{EFT}(k,k,k)& =& B_{GSF}(k,k,k),\hfill \end{array}$$

  (330)

  which implies that26:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\overline{M}}_1& ={\left\{\frac{\widehat{A}}{2\widehat{B}H}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]\right\}}^{\frac{1}{3}},{\overline{M}}_2\approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}=\sqrt{\frac{\widehat{A}}{8\widehat{B}{H}^2{\tilde{c}}_5}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]},\hfill \\ \hfill M_2& ={\left(-\frac{X{P}_{,XX}(X,\varphi )}{P_{,X}(X,\varphi )}\dot{H}{M}_p^2\right)}^{\frac{1}{4}},\hfill \\ \hfill M_3& ={\left\{-\frac{\widehat{A}}{2\widehat{B}}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]\right\}}^{\frac{1}{4}},\hfill \\ \hfill M_4& ={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}={\left(-\frac{\widehat{A}}{2\widehat{B}}\frac{{\tilde{c}}_3}{{\tilde{c}}_6}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]\right)}^{\frac{1}{4}}.\hfill \end{array}\end{array}$$

  (332)

  where the factors $\widehat{A}$, $\widehat{B}$ and $\widehat{C}$ are defined as:

  $$\begin{array}{ccc}\hfill \widehat{A}& =& \left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{U_1}{27}-3U_2\right]-\frac{5}{2}{U}_1+\frac{99}{98}{U}_2\hfill \\ & & -\frac{\Delta }{4}\left[\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\right.\hfill \\ & & -\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-\frac{34}{3}\frac{s}{c_S^2}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\hfill \\ & & \left.+6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right],\hfill \\ \hfill \widehat{B}& =& \left\{\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{U_1}{27}-3U_2\right]-\frac{5}{2}{U}_1+\frac{99}{98}{U}_2\right\}\frac{\Delta c_S^2}{2ϵH^2{M}_p^2},\hfill \\ \hfill \widehat{C}& =& \frac{H^2{M}_p^2ϵc_S^2}{2}\left[\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\right.\hfill \\ & & -\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-\frac{34}{3}\frac{s}{c_S^2}{\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\hfill \\ & & \left.+6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right].\hfill \end{array}$$

  (333)

  Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get the following expression for the factors $\widehat{A}$, $\widehat{B}$ and $\widehat{C}$ as:

  $$\begin{array}{ccc}\hfill \widehat{A}& =& \left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{U_1}{27}-3U_2\right]-\frac{5}{2}{U}_1+\frac{99}{98}{U}_2\hfill \\ & & -\frac{\Delta }{4}\left[\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\right.\hfill \\ & & \left.+6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right],\hfill \\ \hfill \widehat{B}& =& \left\{\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{U_1}{27}-3U_2\right]-\frac{5}{2}{U}_1+\frac{99}{98}{U}_2\right\}\frac{\Delta c_S^2}{2ϵH^2{M}_p^2},\hfill \\ \hfill \widehat{C}& =& \frac{H^2{M}_p^2ϵc_S^2}{2}\left[\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2\right.\hfill \\ & & \left.+6(2ϵ-\eta ){\left(|C_1{|}^2+{\left|C_2\right|}^2\right)}^2+11ϵ{\left(|C_1{|}^2-{\left|C_2\right|}^2\right)}^2+27ϵ{\left(C_1^{*}{C}_2+C_1{C}_2^{*}\right)}^2\right].\hfill \end{array}$$

  (334)

  where for arbitrary vacuum $U_1$ and $U_2$ are defined as:

  $$\begin{array}{ccc}\hfill U_1& =& \left[{\left(C_1-C_2\right)}^3\left(C_1^{*3}+C_2^{*3}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3\left(C_1^3+C_2^3\right)\right],\hfill \end{array}$$

  (335)

  $$\begin{array}{ccc}\hfill U_2& =& \left[{\left(C_1-C_2\right)}^3{C}_1^{*}{C}_2^{*}\left(C_1^{*}-C_2^{*}\right)+{\left(C_1^{*}-C_2^{*}\right)}^3{C}_1{C}_2\left(C_1-C_2\right)\right].\hfill \end{array}$$

  (336)

  If we take $c_S=1$ and ${\tilde{c}}_S=1$ then we get then we get back all the results obtained for single-field slow-roll inflation in the previous section.
  Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$ we get the following expression for the factors $\widehat{A}$, $\widehat{B}$ and $\widehat{C}$ as:

    $$\begin{array}{ccc}\hfill \widehat{A}& =& \frac{2}{27}\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)-5-\frac{\Delta }{4}\left[\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)\right.\hfill \\ & & \left.-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)-\frac{34}{3}\frac{s}{c_S^2}+(23ϵ-6\eta )\right],\hfill \\ \hfill \widehat{B}& =& \left\{\frac{2}{27}\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)-5\right\}\frac{\Delta c_S^2}{2ϵH^2{M}_p^2},\hfill \\ \hfill \widehat{C}& =& \frac{H^2{M}_p^2ϵc_S^2}{2}\left[\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)-\frac{34}{3}\frac{s}{c_S^2}+(23ϵ-6\eta )\right].\hfill \end{array}$$

    (337)

    Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get the following expression for the factors A, B and C as:

    $$\begin{array}{ccc}\hfill \widehat{A}& =& \frac{2}{27}\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)-5-\frac{\Delta }{4}\left[\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)+(23ϵ-6\eta )\right],\hfill \\ \hfill \widehat{B}& =& \left\{\frac{2}{27}\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)-5\right\}\frac{\Delta c_S^2}{2ϵH^2{M}_p^2},\hfill \\ \hfill \widehat{C}& =& \frac{H^2{M}_p^2ϵc_S^2}{2}\left[\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)+(23ϵ-6\eta )\right].\hfill \end{array}$$

    (338)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$ we get the following expression for the factors $\widehat{A}$, $\widehat{B}$ and $\widehat{C}$ as:

    $$\begin{array}{ccc}\hfill \widehat{A}& =& \left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{J_1(\alpha ,\beta )}{27}-3J_2(\alpha ,\beta )\right]-\frac{5}{2}{J}_1(\alpha ,\beta )+\frac{99}{98}{J}_2(\alpha ,\beta )\hfill \\ & & -\frac{\Delta }{4}\left[\left\{\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)-\frac{34}{3}\frac{s}{c_S^2}\right\}{cosh}^{2}2\alpha \right.\hfill \\ & & \left.+6(2ϵ-\eta ){cosh}^{2}2\alpha +11ϵ+27ϵ{sinh}^2\alpha {cos}^2\beta \right],\hfill \\ \hfill \widehat{B}& =& \left\{\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{J_1(\alpha ,\beta )}{27}-3J_2(\alpha ,\beta )\right]-\frac{5}{2}{J}_1(\alpha ,\beta )+\frac{99}{98}{J}_2(\alpha ,\beta )\right\}\frac{\Delta c_S^2}{2ϵH^2{M}_p^2},\hfill \\ \hfill \widehat{C}& =& \frac{H^2{M}_p^2ϵc_S^2}{2}\left[\left\{\frac{1}{18}\left(\frac{1}{c_S^2}-1-\frac{2{\Sigma }_2(X,\varphi )}{{\Sigma }_1(X,\varphi )}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)-\frac{34}{3}\frac{s}{c_S^2}\right\}{cosh}^{2}2\alpha \right.\hfill \\ & & \left.+6(2ϵ-\eta ){cosh}^{2}2\alpha +11ϵ+27ϵ{sinh}^2\alpha {cos}^2\beta \right].\hfill \end{array}$$

    (339)

    Furthermore, for restricted classes of the general single-field $P(X,\varphi )$ model, which satisfies the constraint $P_{,X\varphi }(X,\varphi )=0$, we get the following expression for the factors $\widehat{A}$, $\widehat{B}$ and $\widehat{C}$ as:

    $$\begin{array}{ccc}\hfill \widehat{A}& =& \left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{J_1(\alpha ,\beta )}{27}-3J_2(\alpha ,\beta )\right]-\frac{5}{2}{J}_1(\alpha ,\beta )+\frac{99}{98}{J}_2(\alpha ,\beta )\hfill \\ & & -\frac{\Delta }{4}\left[\left\{\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)+6(2ϵ-\eta )\right\}{cosh}^{2}2\alpha \right.\hfill \\ & & \left.+11ϵ+27ϵ{sinh}^2\alpha {cos}^2\beta \right],\hfill \\ \hfill \widehat{B}& =& \left\{\left(\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right)\left[\frac{J_1(\alpha ,\beta )}{27}-3J_2(\alpha ,\beta )\right]-\frac{5}{2}{J}_1(\alpha ,\beta )+\frac{99}{98}{J}_2(\alpha ,\beta )\right\}\frac{\Delta c_S^2}{2ϵH^2{M}_p^2},\hfill \\ \hfill \widehat{C}& =& \frac{H^2{M}_p^2ϵc_S^2}{2}\left[\left\{\frac{1}{27}\left(\frac{1}{c_S^2}-1-\frac{sϵ}{3{ϵ}_X}\right)-\frac{7}{3}\left(\frac{1}{c_S^2}-1\right)+6(2ϵ-\eta )\right\}{cosh}^{2}2\alpha \right.\hfill \\ & & \left.+11ϵ+27ϵ{sinh}^2\alpha {cos}^2\beta \right].\hfill \end{array}$$

    (340)
• Squeezed limit configuration:
  For this case with arbitrary vacuum one can write:

  $$\begin{array}{ccc}\hfill B_{EFT}(k_L,k_L,k_S)& =& B_{GSF}(k_L,k_L,k_S),\hfill \end{array}$$

  (341)

  which implies that:

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\overline{M}}_1& ={\left\{\frac{\widehat{A}}{2\widehat{B}H}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]\right\}}^{\frac{1}{3}},{\overline{M}}_2\approx {\overline{M}}_3=\sqrt{\frac{{\overline{M}}_1^3}{4H{\tilde{c}}_5}}=\sqrt{\frac{\widehat{A}}{8\widehat{B}{H}^2{\tilde{c}}_5}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]},\hfill \\ \hfill M_2& ={\left(-\frac{X{P}_{,XX}(X,\varphi )}{P_{,X}(X,\varphi )}\dot{H}{M}_p^2\right)}^{\frac{1}{4}},M_3={\left\{-\frac{\widehat{A}}{2\widehat{B}}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]\right\}}^{\frac{1}{4}},\hfill \\ \hfill M_4& ={\left(-\frac{{\tilde{c}}_3}{{\tilde{c}}_6}H{\overline{M}}_1^3\right)}^{\frac{1}{4}}={\left(-\frac{\widehat{A}}{2\widehat{B}}\frac{{\tilde{c}}_3}{{\tilde{c}}_6}\left[-1+\sqrt{1+\frac{4\widehat{B}\widehat{C}}{{\widehat{A}}^2}}\right]\right)}^{\frac{1}{4}}.\hfill \end{array}\end{array}$$

  (342)

  where the factors $\widehat{A}$, $\widehat{B}$ and $\widehat{C}$ are defined as27:

  $$\begin{array}{ccc}\hfill \widehat{A}& =& {\widehat{P}}_1+{\widehat{P}}_2-\sum _{j=-1}^3{t}_j{\left(\frac{k_S}{k_L}\right)}^j,\widehat{B}=\frac{{\widehat{P}}_1\Delta }{2ϵH^2{M}_p^2},\widehat{C}=2ϵH^2{M}_p^2\sum _{j=-1}^3{t}_j{\left(\frac{k_S}{k_L}\right)}^j,\hfill \end{array}$$

  (344)

  where the expansion coefficients $t_j\forall j=-1,\cdots ,3$ are defined earlier for the general $P(X,\varphi )$ model and also for restricted classes of the model where $P_{,X\varphi }(X,\varphi )=0$ constraint is satisfied.
  Here the factors ${\widehat{P}}_1$ and ${\widehat{P}}_2$ are defined as:

  $$\begin{array}{ccc}\hfill {\widehat{P}}_1& =& \sum _{m=-1}^3{\widehat{e}}_m{\left(\frac{k_S}{k_L}\right)}^m,{\widehat{P}}_2=\sum _{m=-1}^3{\widehat{h}}_m{\left(\frac{k_S}{k_L}\right)}^m,\hfill \end{array}$$

  (345)

  where the expansion coefficients ${\widehat{e}}_m\forall m=-1,\cdots ,3$ and ${\widehat{h}}_m\forall m=-1,\cdots ,3$ for arbitrary vacuum are defined as:

  $$\begin{array}{ccc}\hfill {\widehat{e}}_{-1}& =& -36U_2,{\widehat{e}}_0=\left[-\frac{9}{2}{U}_1+\left(24\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}-\frac{9}{2}\right)U_2\right],\hfill \\ \hfill {\widehat{e}}_1& =& 0,{\widehat{e}}_2=\left[-\frac{27}{2}{U}_2+\left(\frac{3}{2}\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}-\frac{27}{2}\right)U_1\right],{\widehat{e}}_3=0,\hfill \end{array}$$

  (346)

  and

  $$\begin{array}{ccc}\hfill {\widehat{h}}_{-1}& =& 0,{\widehat{h}}_0=\frac{9}{c_S^2}\left(U_1+U_2\right),{\widehat{h}}_1=0,\hfill \\ \hfill {\widehat{h}}_2& =& \left[\left(15+2\left\{\frac{3}{2}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\right)U_1+\left(\frac{45}{2}+3\left\{\frac{3}{2}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\right)U_2\right],{\widehat{h}}_3=0,\hfill \end{array}$$

  (347)

  where $U_1$ and $U_2$ are already defined earlier.
  Now for Bunch–Davies and $\alpha ,\beta$ vacuum we get the following simplified expression for the bispectrum for scalar fluctuation:
  • For Bunch–Davies vacuum:
    After setting $C_1=1$ and $C_2=0$, we get $U_1=2$ and $U_2=0$. Consequently, the expansion coefficients can be recast as:

    $$\begin{array}{ccc}\hfill {\widehat{e}}_{-1}& =& 0,{\widehat{e}}_0=-9,{\widehat{e}}_1=0,{\widehat{e}}_2=-27,{\widehat{e}}_3=0,\hfill \end{array}$$

    (348)

    and

    $$\begin{array}{ccc}\hfill {\widehat{h}}_{-1}& =& 0,{\widehat{h}}_0=\frac{18}{c_S^2},{\widehat{h}}_1=0,{\widehat{h}}_2=\left(30+4\left\{\frac{3}{2}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\right),{\widehat{h}}_3=0,\hfill \end{array}$$

    (349)
  • For $\mathit{\alpha },\mathit{\beta }$ vacuum:
    After setting $C_1=cosh\alpha$ and $C_2=e^{i\beta }sinh\alpha$, we get $U_1=J_1(\alpha ,\beta )$ and $U_2=J_2(\alpha ,\beta )$. Consequently, the expansion coefficients can be recast as:

    $$\begin{array}{ccc}\hfill {\widehat{e}}_{-1}& =& -36J_2(\alpha ,\beta ),{\widehat{e}}_0=\left[-\frac{9}{2}{J}_1(\alpha ,\beta )+\left(24\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}-\frac{9}{2}\right)J_2(\alpha ,\beta )\right],\hfill \\ \hfill {\widehat{e}}_1& =& 0,{\widehat{e}}_2=\left[-\frac{27}{2}{J}_2(\alpha ,\beta )+\left(\frac{3}{2}\left\{\frac{3}{2}+\frac{4}{3}\frac{{\tilde{c}}_3}{{\tilde{c}}_4}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}-\frac{27}{2}\right)J_1(\alpha ,\beta )\right],{\widehat{e}}_3=0,\hfill \end{array}$$

    (350)

    and

    $$\begin{array}{ccc}\hfill {\widehat{h}}_{-1}& =& 0,{\widehat{h}}_0=\frac{9}{c_S^2}\left(U_1+J_2(\alpha ,\beta )U_2\right),{\widehat{h}}_1=0,\hfill \\ \hfill {\widehat{h}}_2& =& \left[\left(15+2\left\{\frac{3}{2}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\right)J_1(\alpha ,\beta )+\left(\frac{45}{2}+3\left\{\frac{3}{2}+\frac{2c_S^2}{{\tilde{c}}_4}\right\}\right)J_2(\alpha ,\beta )\right],{\widehat{h}}_3=0.\hfill \end{array}$$

    (351)

6. Conclusions

To summarize, in this paper, we have addressed the following issues:

• We have derived the analytical expressions for the two-point correlation function for scalar and tensor fluctuations and three-point correlation function for scalar fluctuations from EFT framework in quasi de Sitter background in a model-independent way. For this computation, we use an arbitrary quantum state as the initial choice of vacuum. Such a choice finally gives rise to the most general expressions for the two-point and three-point correlation functions for primordial fluctuation in EFT. Furthermore, we have simplified our results by considering the Bunch–Davies vacuum and $\alpha ,\beta$ vacuum states.
• During our computation, we have truncated the EFT action by considering the all possible two derivative terms in the metric. This allows us to derive correct expressions for the two-point and three-point correlation functions for EFT which are consistent with both the single-field slow-roll model and generalized non-canonical $P(X,\varphi )$ single-field models minimally coupled with gravity28.
• Furthermore, we have derived the analytical expressions for the coefficients of all relevant EFT operators for the single-field slow-roll model and generalized non-canonical $P(X,\varphi )$ single-field models. We have derived the results in terms of slow-roll parameters, effective sound speed parameter, and the constants which are fixed by the choice of arbitrary initial vacuum state. Next, we have simplified our results also presented the results by considering Bunch–Davies vacuum and $\alpha ,\beta$ vacuum state.
• Finally, using the CMB observation from Planck we constrain all these EFT coefficients for various single-field slow-roll models and generalized non-canonical $P(X,\varphi )$ models of inflation.

The future directions of this paper are appended below pointwise:

• One can further carry forward this work to compute four-point scalar correlation function from EFT framework using an arbitrary initial choice of the quantum vacuum state. The present work can also be extended for the computation of the three-point correlation from tensor fluctuation, and other three-point cross correlations between scalar and tensor mode fluctuation in the context of EFT with arbitrary initial vacuum.
• In the present EFT framework we have not considered the effects of any additional heavy fields ($m>>H$) in the effective action. One can redo the analysis with such additional effects in the EFT framework to study the quantum entanglement, cosmological decoherence and Bell’s inequality violation in the context of primordial cosmology. Once can also further generalize this computation for any arbitrary spin fields which are consistent with the unitarity bound.
• The analyticity property of response functions and scattering amplitudes in QFT implies significant connection between observables in IR regime and the underlying dynamics valid in the short-distance scale. Such analytic property is directly connected to the causality and unitarity of the QFT under consideration. Following this idea one can also study the analyticity property in the present version of EFT or including the effective of massive fields ($m>>H$) in the effective action.
• There are other open issues as well which one can study within the framework of EFT:
  • The role of out-of-time-ordered correlations from open quantum system [60,61,62].
  • EFT framework in a quantum dissipative system and its application to cosmology [63,64,65,66].
  • Thermalization, quantum critical quench and its application to the phenomena of reheating in early universe cosmology [67].