Full bispectra from primordial scalar and tensor perturbations in the most general single-field inflation model

We compute the full bispectra, namely both auto- and cross- bispectra, of primordial curvature and tensor perturbations in the most general single-field inflation model whose scalar and gravitational equations of motion are of second order. The formulae in the limits of k-inflation and potential-driven inflation are also given. These expressions are useful for estimating the full bispectra of temperature and polarization anisotropies of the cosmic microwave background radiation.


I. INTRODUCTION
The non-Gaussianities of the temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation now receive increasing attentions because they are important tools to discriminate models of inflation [1,2]. Ongoing and near future project such as Planck satellite [3], CMBpol mission [4], LiteBIRD satellite [5] would reveal the properties of the temperature and polarization anisotropies in detail. Such E-mode polarization anisotropies are sourced by both curvature and tensor perturbations [6], while only tensor (and vector) perturbations can generate B-mode polarization anisotropies [7]. 1 Therefore, even when one estimates the "auto" bispectra of the temperature and the E-mode polarization fluctuations, not only the auto bispectra but also the cross bispectra of the primordial curvature and tensor perturbations are indispensable.
For slow-roll inflation models with the canonical kinetic term [8], Maldacena evaluated the full bispectra, including the cross bispectra, of the primordial curvature and tensor perturbations [9]. Inflation models are now widely generalized into more varieties such as k-inflation [10], DBI inflation [11], ghost inflation [12], G-inflation [13], and so on. However, almost all the works on the non-Gaussianities in these inflation models concentrate only on the auto bispectrum of the curvature perturbations [14][15][16], which is insufficient for evaluating the bispectra of the temperature and E-mode polarization anisotropies of the CMB, as explained above. To our surprise, as far as we know, the full bispectra of the primordial curvature and tensor perturbations have not yet been obtained even for k-inflation [10] except for Ref. [17] where the primordial scalar-scalar-tensor cross bispectrum has been calculated for inflation models with an arbitrary kinetic term. There are several related works on the primordial cross bispectra. In Ref. [18], the authors show the primordial tensor-scalar cross bispectra induced from a holographic model and the scalar-scalar-tensor correlation has been discussed in the calculation of the trispectrum of the scalar fluctuations [19], so-called "graviton exchange", and also in the context of one-loop effects of the scalar power spectrum [20]. In Ref. [21], the authors calculate the correlation between primordial scalar and vector (magnetic fields) fluctuations in possible inflationary models of generating primordial magnetic fields.
Among the inflation zoo, the generalized G-inflation model [22] occupies the unique position in that it includes practically all the known well-behaved single inflation models since it is based on the most general single field scalartensor Lagrangian with the second order equation of motion, which was proposed by Horndeski more than thirty years ago [23] and was recently rediscovered in the context of the generalized Galileon [24,25]. Indeed, it includes standard canonical inflation [1,8], non-minimally coupled inflation [26] including the Higgs inflation [27], extended inflation [28], k-inflation [10], DBI inflation [11], R 2 inflation [2,29], new Higgs inflation [30], G-inflation [13], and so on. Thus, once we analyze properties of the primordial curvature and tensor perturbations in the generalized G-inflation, one can apply the result for any specific single-field inflation models.
So far, the power spectra of scalar and tensor fluctuations were studied in [22] and the general formulae for them have been given there. It has been pointed out that the sound velocity squared of the tensor perturbations as well as that of the curvature perturbations can deviate from unity. Then the auto bispectrum of the curvature perturbations was estimated in Refs. [31,32] (see also [33,34]) and found to be enhanced by the inverse sound velocity squared and so on. More recently, the auto bispectrum of the tensor perturbations was investigated in Ref. [35] and found to be composed of two parts. The first is the universal one similar to that from Einstein gravity and predicts a squeezed shape, while the other comes from the presence of the kinetic coupling to the Einstein tensor and predicts an equilateral shape. What remains to be studied are the bispectra of the primordial curvature and tensor perturbations in the generic theory.
In the case of the most general single field model, not only auto bispectrum of scalar perturbations but also that of tensor perturbations can be large enough to be detected by cosmological observations, e.g., Planck satellite, as is explained in Ref. [35], which suggests that cross bispectra can be large as well. For such a case, it is not necessarily justified to consider only auto bispectrum of curvature perturbations even when you evaluate the auto bispectrum of temperature (or E-mode) fluctuations because cross ones can significantly contribute to it even if the tensor-to-scalar ratio is (relatively) small. Furthermore, when we try to evaluate the cross bispectra including B-mode fluctuations, the cross bispectra of tensor and scalar perturbations are indispensable because B-mode fluctuations are produced only from tensor perturbations. These facts are quite manifest even without any reference nor estimation.
In such a situation, in this paper, we compute the cross bispectra of the primordial curvature and tensor perturbations in the generalized G-inflation model. The formulae in the limits of k-inflation and potential-driven inflation are also given as specific examples.
The organization of this paper is given as follows. In the next section, we briefly review the most general single field scalar-tensor Lagrangian with the second order equation of motion. In Sec. III, quadratic and cubic actions for the primordial curvature and tensor perturbations are given. The full bispectra, including the cross ones, for them are discussed in the section IV. The special limits for them in the cases of k-inflation and potential driven inflation are taken in Sec. V. Final section is devoted to conclusion and discussions.

II. GENERALIZED G-INFLATION -THE MOST GENERAL SINGLE-FIELD INFLATION MODEL
The Lagrangian for the generalized G-inflation is the most general one that is composed of the metric g µν and a scalar field φ together with their arbitrary derivatives but still yields the second-order field equations. The Lagrangian was first derived by Horndeski in 1974 in four dimensions [23], and very recently it was rediscovered in a modern form as the generalized Galileon [24], i.e., the most general extension of the Galileon [36,37], in arbitrary dimensions. Their equivalence in four dimensions has been shown in Ref. [22]. The four-dimensional generalized Galileon is described by the Lagrangian: where K and G i are arbitrary functions of φ and its canonical kinetic term X := −(∂φ) 2 /2. We are using the notation G iX for ∂G i /∂X. The generalized Galileon can be used as a framework to study the most general single-field inflation model. Generalized G-inflation contains novel models, as well as previously known models of single-field inflation such as standard canonical inflation, k-inflation, extended inflation, and new Higgs inflation, and even R 2 or f (R) inflation (with an appropriate field redefinition). The above Lagrangian can also reproduce the non-minimal coupling to the Gauss-Bonnet term [22].

III. GENERAL QUADRATIC AND CUBIC ACTIONS FOR COSMOLOGICAL PERTURBATIONS
In this section, we present the quadratic and cubic actions for scalar-and tensor-type cosmological perturbations based on the most general single-field inflation model. Employing the Arnowitt-Deser-Misner formalism, we write the metric as where and (e h ) ij = δ ij + h ij + (1/2)h ik h kj + · · · . We work in the gauge in which the fluctuation of the scalar field vanishes, φ = φ(t). Concerning the perturbations of the lapse function and shift vector, α and β, it is sufficient to consider the first order quantities to compute the cubic actions, as pointed out in [9]. The first order vector perturbations may be dropped. The curvature perturbation in generalized G-inflation is shown to be conserved on large scales at non-linear order in [38]. Substituting the above metric to the action and expanding it to third order, we obtain the action for the cosmological perturbations, which will be written, with trivial notations, as The first two Lagrangians are quadratic in the metric perturbations, which have already been obtained in Ref. [22]. To define some notations used in this paper, we will begin with summarizing the quadratic results in the next subsection. The third and last cubic Lagrangians have been derived in Refs. [35] and [31,32], respectively, but for completeness they are also replicated in this section. The mixture of the scalar and tensor perturbations, L shh and L ssh , are computed for the first time in this paper.

A. Quadratic Lagrangians and primordial power spectra
The quadratic terms are obtained as follows [22].

Tensor perturbations
The most general quadratic Lagrangian for tensor perturbations is given by where Here, a dot indicates a derivative with respect to t, G iφ := ∂G i /∂φ and the propagation speed of gravitational waves is defined as c 2 h := F T /G T 2 . The linear equation of motion derived from the Lagrangian (5) is In deriving the above equations, we have not assumed that the background evolution is close to de Sitter. They can therefore be used for an arbitrary homogeneous and isotropic cosmological background. We now move to the Fourier space to solve this equation: It is convenient to use the conformal time coordinate defined by dη = dt/a. We approximate the inflationary regime by the de Sitter spacetime and take F T and G T to be constant 3 . The quantized tensor perturbation is written as where under these approximations the normalized mode is given by Here, e ij (k). We adopt the normalization such that and choose the phase so that the following relations hold.
The commutation relation for the creation and annihilation operators is The two-point function can be written as where The power spectrum, P h = (k 3 /2π 2 )P ij,ij , is thus computed as 3 As seen in Eqs. (27) and (28), F S and G S depend on F T and G T . The time derivatives of F S and G S affect the spectral index of the power spectrum of the scalar curvature perturbations and they are required to be small from the current cosmological observations. Hence, the assumption that the time derivatives of F T and G T are small are natural from observational perspectives, although one cannot rule out the case where F T and G T have strong time-dependence without conflicting the current cosmological observations, strictly speaking. In this exceptional case, we must say that the assumption that the time derivatives of F T and G T are small is made just for simplicity.

Scalar perturbations
The quadratic Lagrangian for the scalar perturbations is given by where Varying Eq. (19) with respect to α and β, we get the first-order constraint equations: which are solved to yield with ψ := ∂ −2ζ . Plugging Eqs. (24) and (25) to Eq. (19), we obtain where we have defined The sound speed is given by c 2 s := F S /G S . The linear equation of motion derived from the Lagrangian (26) is The scalar two-point function can be calculated in a way similar to the case of the tensor perturbations. We move to the Fourier space: and proceed in the de Sitter approximation, assuming that F S and G S are almost constant. The quantized curvature perturbation is written as where the normalized mode is given by The commutation relation for the creation and annihilation operators is Thus, the power spectrum is calculated as From Eqs. (18) and (35), tensor-to-scalar ratio r is given by where we have assumed that the relevant quantities remain practically constant between the horizon crossings of tensor and scalar perturbations that occur at different time in case c h = c s [41].

B. Cubic Lagrangians
We now present the most general cubic Lagrangians composed of the tensor and scalar perturbations. We would like to emphasize that in deriving the following Lagrangians the slow-roll approximation is not used, as discussed in literature [42].
As discussed in Ref. [35], this cubic action for the tensor perturbation h ij is composed only of two contributions. The former has one time derivative on each h ij and newly appears in the presence of the kinetic coupling to the Einstein tensor, that is, G 5X = 0. On the other hand, the latter has two spacial derivatives and is essentially identical to the cubic term that appears in Einstein gravity. Therefore, in what follows, we use the terminologies "new" and "GR" for corresponding terms.

Two tensors and one scalar
The interactions involving two tensors and one scalar are given by where This quantity can also be expressed in a compact form Γ = ∂Θ/∂H. Substituting the first-order constraint equations to Eq. (39), the Lagrangian reduces to where and The last term E shh can be removed by redefining the fields as The contribution to the correlation function is however negligible because the above field redefinitions involve at least one time derivative of the metric perturbation, which vanishes on super-horizon scales.

Two scalars and one tensor
The interactions involving one tensor and two scalars are given by Substituting the constraint equations, we obtain the reduced Lagrangian: where and The field redefinition: removes the last term E ssh . Since all the terms involve at least one derivative of the metric perturbation, the field redefinition does not contribute to the correlation function on super-horizon scales.

Three scalars
For completeness, here we give the cubic Lagrangian for the scalar perturbations derived in Refs. [31,32]. The cubic Lagrangian for the scalar perturbations is given by where Using the first-order constraint equations to remove α and β from the above Lagrangian, we obtain the following reduced expression: with ψ = ∂ −2ζ . There are five independent cubic terms with coefficients:

IV. PRIMORDIAL BISPECTRA
Having obtained the general cubic Lagrangians composed of the scalar and tensor perturbations, we now compute the bispectra in this section. Here, we use the mode functions in exact de Sitter.

A. Three tensors
Let us consider three-point function of the tensor perturbations: where A i1j1i2j2i3j3 represent the contributions from theḣ 3 term and the h 2 ∂ 2 h terms, respectively. Each contribution is given by where K = k 1 + k 2 + k 3 and The first term A (new) i1j1i2j2i3j3 is proportional to G 5X and hence vanishes in the case of Einstein gravity, while the second term A (GR) i1j1i2j2i3j3 is universal in the sense that it is independent of any model parameters and remains the same even in non-Einstein gravity.
In order to quantify the magnitude of the bispectrum, we define two polarization modes as and their relevant amplitudes of the bispectra as From Eqs. (75) and (76), the amplitudes A s1s2s3 (new),(GR) are easily calculated as [35] A s1s2s3 A s1s2s3 where F (x, y, z) := 1 64 As pointed out in Ref. [35], A +++ (new) has a peak in the equilateral limit, while A +++ (GR) in the squeezed limit. It would be convenient to introduce nonlinearity parameters defined as which are quantities analogous to the standard f NL for the curvature perturbation. We find or, more concretely,
As defined in Eq. (74), B i1j1i2j2i3j3 is normalized by P 2 h . This normalization can be justified when one concentrates on the non-Gaussianity of the B-mode polarization. Because the B-mode polarization can be generated by not curvature perturbations but tensor perturbations (except for lensing contribution), the size of the non-Gaussianity of the B-mode polarization could be directly characterized by f s1s2s3 NL(new),(GR) . However, it should be noticed that tensor perturbations can generate not only the B-mode polarization but also the temperature fluctuation and the E-mode polarization. The latter two are mainly generated by the curvature perturbations. Therefore, when one would like to quantify the auto and cross bispectra of the temperature fluctuation and the E-mode polarization, it would be better to normalize B (hhh) i1j1i2j2i3j3 by P 2 ζ , namely, i1j1i2j2i3j3 with r being the tensor-to-scalar ratio. In the same way, A s1s2s3 (new),(GR) = r 2 A s1s2s3 (new),(GR) and f s1s2s3 NL(new),(GR) = r 2 f s1s2s3 NL(new),(GR) .

B. Two tensors and one scalar
The cross bispectrum of two tensors and one scalar is given by where B

(ζhh)
ij,kl is of the form: Each contribution is given by ij,kl = k 2 1 V and , where K ′ := c s k 1 + c h (k 2 + k 3 ). Thus, it turns out that we need to evaluate only V (1) ij,kl and V (6) ij,kl . We would now like to define the amplitudes of the above cross bispectra in a similar way as the case of three tensors, for which we have adopted two different normalization conditions, (74) and (88), depending on whether we are interested in the B-mode polarization or the E-mode polarization and temperature fluctuations. The same ambiguity is present for the cases of these cross bispectra, too. Here we simply normalize them in terms of P 2 ζ taking into account the fact that these bispectra generate the auto and the cross bispectra of the temperature fluctuation and the E-mode polarization, too, which are mainly sourced by the curvature perturbation. Although this normalization may not be appropriate for those including the B-mode polarization, we do not touch the issue any further because the change of the normalization factor from P 2 ζ to P ζ P h or P 2 h can readily be done by multiplying appropriate powers of the tensor-to-scalar ratio r. Thus we adopt the following convention: where We also define the following cross bispectra: Here B (ζhh) s2,s3 and A (ζhh) s2,s3 are given by where V

C. Two scalars and one tensor
The cross bispectrum of two scalars and one tensor is given by where B (ζζh) ij is of the form Each contribution is given by and , with K ′′ := c s (k 1 + k 2 ) + c h k 3 . Thus, it turns out that we need to evaluate only V (1) ij . As in the case of two tensors and one scalar, we normalize the bispectrum by P 2 ζ as where We also define the following cross bispectra: Here B (ζζh) s and A (ζζh) s are given by where V (q) and Indeed, the above functions are independent of s due to no parity violation.

D. Three scalars
Here we give the bispectrum defined by The result is given in Ref. [31,32]: E. Let us discuss the shape of each cross bispectrum in momentum space. As shown in Ref. [35] and also mentioned in the previous subsection, for the bispectrum of the tensor mode, A +++ (new) and A +++ (GR) have respectively peaks in the equilateral and squeezed limits. The shape of the bispectrum of scalar perturbations was also discussed in Ref. [31,32] and the authors have found that it is well approximated by the equilateral shape.

Shapes of the cross bispectra in momentum space
In a similar way to the auto bispectra of tensors and scalars, we can also discuss the shapes of the cross bispectra of tensors and scalars in momentum space. However, contrary to the auto-bispectra of tensors and scalars, the shapes of cross bispectra strongly depend on the sound speeds of the tensor and scalar perturbations, as can be seen in Eqs. (92) and (101). Here, we denote a term proportional to b q in A ++,1 (k 1 k 2 k 3 ) −1 has a peak in the squeezed limit (k 1 ≪ k 2 ∼ k 3 ) for both limiting cases. However, the sharpness of the peak seems to depend on the value of c h /c s . In Fig. 2 where A (ζhh) ++,2 (k 1 k 2 k 3 ) −1 is plotted, we find that A (ζhh) ++,2 (k 1 k 2 k 3 ) −1 for the case with c h /c s = 0.01 has a sharp peak in the squeezed limit together with a non-trivial shape in wide region of the momentum space. For the case with c h /c s = 10 2 (shown in Fig. 2-(b)), A (ζhh) ++,2 (k 1 k 2 k 3 ) −1 also has a peak in the squeezed limit.
Contrary to A (ζhh) ++,1 (k 1 k 2 k 3 ) −1 and A (ζhh) ++,2 (k 1 k 2 k 3 ) −1 , both of which have a peak in the squeezed limit, A (ζhh) ++,3 (k 1 k 2 k 3 ) −1 does not have any sharp peak, but its shape strongly depends on the value of c h /c s , as shown in Fig. 3. In the case with c h /c s ≪ 1, A (ζhh) ++,3 (k 1 k 2 k 3 ) −1 becomes large at k 1 ≪ k 2 , and then its shape looks to come close to so-called orthogonal type in the limit of c h /c s ≫ 1. A (ζhh) ++,4 (k 1 k 2 k 3 ) −1 also strongly depends on the value of c h /c s . As can be seen in Fig. 4, the peak of A (ζhh) ++,4 (k 1 k 2 k 3 ) −1 shifts in the momentum space depending on c h /c s , and A (ζhh) ++,4 (k 1 k 2 k 3 ) −1 for small c h has a finite value even in the squeezed limit. Although we do not show here, we also found that A +,q (k 1 k 2 k 3 ) −1 also has strong dependence on c h /c s and there is no divergence feature in the whole region of the momentum space, unlike the socalled local shape. Since we found A (ζζh) +,q for q = 3, 4, 5, 6 have almost same shapes as A (ζζh) +,2 , we do not show the plots for these contributions.
The detailed analysis of the shapes of the cross bispectra, including a precise comparison with the standard local-, equilateral and orthogonal shapes, is an issue in progress with the detailed analysis of CMB bispectra [48].

V. EXAMPLES
In this section, we consider two representative examples of inflation to estimate the amount of non-Gaussianities from tensor and scalar perturbations. The first example is general potential-driven inflation studied in Ref. [43]. This class of inflation models includes variants of Higgs inflation enabled by enhancing the effect of Hubble friction. These potential driven models have c 2 s = O(1) and c 2 h ≃ 1. Next, to see the impact of generic c 2 s more clearly, we study k-inflation as another example.
with c h = 1 and r = 16ǫc s , which simplifies the coefficients in the cubic Lagrangians: Note that in deriving the above coefficients we have not invoked the slow-roll expansion.

VI. DISCUSSION
In this paper we have presented the full bispectra, including the cross bispectra of the primordial curvature and tensor perturbations, in the generalized G-inflation model which is the most general single-field inflation model with the second order equations of motion.
In the event full observations of these quantities could be made, we could extract many pieces of interesting information on the underlying theory. For example, by observing three-point tensor correlation function, we can in principle determine the kinetic coupling to the Einstein tensor through µ. Another interesting quantity is the cross bispectrum of two tensors and one scalar. If we could observationally identify their coefficients b 2 , b 3 and b 6 , we could in principle determine F S , G S , F T , and G T independently with the help of the three-tensor bispectrum which would provide a consistency relation of the theory for the tensor-to-scalar ratio (36).
Let us next turn to two-scalar and one-tensor bispectrum whose effective Lagrangian is given by (53). Its most interesting component is the first term proportional to c 1 = F S which could be singled out by taking k 3 small. In the standard canonical inflation as well as in k-inflation, the coefficient simply takes c 1 = F S = M 2 Pl ǫ = M 2 Pl r 16c s as derived in (119), where we have used the consistency relation in the last equality. We can also show that this feature remains valid in the case where a sizable local non-Gaussianity is generated as in the cases of the curvaton scenario [44] and the modulated reheating scenarios [45]. In such case curvature perturbation ζ is sourced by another scalar field which we denote by σ and its fluctuation by δσ. One can relate ζ and δσ as ζ = N σ (σ)δσ + 1 2 N σσ (σ)(δσ) 2 , using the δN -formalism [46]. Suppose that σ has the Lagrangian L σ = κ(Y, σ) with Y := −(∂σ) 2 /2. Since the dynamics of σ is practically frozen during inflation and it practically behaves as a massless minimally-coupled field, one can expand L σ = κ(0, σ 0 ) + κ σ (0, σ 0 )Y in this regime where σ 0 is its expectation value in the domain including our horizon today. Then the mean-square fluctuation amplitude of σ is given by the latter being an outcome of (121), and it determines the relation between δσ and ζ, too. Then the effective Lagrangian representing tensor-scalar-scalar coupling is generated from the kinetic term of σ in this case and reads Note that in this case the sound speed is equal to unity. Thus we find that if the sector responsible for the generation of curvature perturbations is minimally coupled to gravity with no extra Galileon-like terms, c 1 takes the same form whether they are generated by the inflaton or another scalar field. Thus this term can provide a test of the generalized Galileon as a source of the structure of the Universe. It is a non-trivial issue how to normalize the cross bispectra. In this paper, we have normalized them by the power spectrum of the curvature perturbation. This is mainly because these cross bispectra generate the auto-and the cross-bispectra of the temperature fluctuation and the E-mode polarization, which are mainly sourced by the curvature perturbation. However, such a normalization may be inadequate for the cross bispectra including the Bmode polarization. Therefore, we need to directly investigate the impacts on the CMB bispectra and it is interesting to see the CMB cross-bispectra between the temperature fluctuations and B-mode polarizations which are sourced directly from the primordial cross-bispectra of the scalar and the tensor modes [17,47]. Constraining the model parameters by CMB bispectra is a work in progress [48].