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

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perturbations in the most general single-field inflation model

Xian Gao, Tsutomu Kobayashi, Maresuke Shiraishi, Masahide Yamaguchi, Jun’Ichi Yokoyama, Shuichiro Yokoyama

To cite this version:

Xian Gao, Tsutomu Kobayashi, Maresuke Shiraishi, Masahide Yamaguchi, Jun’Ichi Yokoyama, et al..

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

model. Progress of Theoretical and Experimental Physics [PTEP], Oxford University Press on behalf

of the Physical Society of Japan, 2013, 2013 (5), pp.053E03. �10.1093/ptep/ptt031�. �hal-01596056�

DOI: 10.1093/ptep/ptt031

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

Xian Gao ^{1 , 2 , 3} , Tsutomu Kobayashi ⁴ , Maresuke Shiraishi ⁵ , Masahide Yamaguchi ⁶ , Jun’ichi Yokoyama ^{7 , 8} , and Shuichiro Yokoyama ^{9 ,∗}

1

Astroparticule et Cosmologie (APC), UMR 7164-CNRS, Université Denis Diderot-Paris 7, 10 rue Alice Domon et Léonie Duquet, 75205 Paris, France

2

Laboratoire de Physique Théorique, École Normale Supérieure, 24 rue Lhomond, 75231 Paris, France

3

Institut d’Astrophysique de Paris (IAP), UMR 7095-CNRS, Université Pierre et Marie Curie-Paris 6, 98bis Boulevard Arago, 75014 Paris, France

4

Department of Physics, Rikkyo University, Tokyo 171-8501, Japan

5

Department of Physics and Astrophysics, Nagoya University, Nagoya 464-8602, Japan

6

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

7

Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan

8

Kavli Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Kashiwa, Chiba 277-8568, Japan

9

Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa, Chiba 277-8582, Japan

∗

E-mail: shu@icrr.u-tokyo.ac.jp

Received November 15, 2012; Accepted March 16, 2013; Published May 1, 2013

. . . . 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 grav- itational 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.

. . . .

Subject Index E80

1. Introduction

The non-Gaussianities of the temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation are currently receiving increasing attention because they are impor- tant tools for discriminating models of inflation [1–3]. Ongoing and near-future projects such as the Planck satellite [4], the CMBpol mission [5], and the LiteBIRD satellite [6] should reveal the proper- ties of the temperature and polarization anisotropies in detail. Such E-mode polarization anisotropies are sourced by both curvature and tensor perturbations [7–12], while only tensor (and vector) pertur- bations can generate B-mode polarization anisotropies [13,14]. ¹ Therefore, even when one estimates

1

Though vector perturbations can also generate both E-mode and B-mode polarization anisotropies, they

only have a decaying mode in linear theory and hence are suppressed in the standard inflationary cosmology

based on scalar fields.

the “auto” bispectra of the temperature and E-mode polarization fluctuations, both the auto bispectra and the cross bispectra of the primordial curvature and tensor perturbations are indispensable.

For slow-roll inflation models with the canonical kinetic term [15–17], Maldacena evaluated the full bispectra, including the cross bispectra, of the primordial curvature and tensor perturbations [18]. Inflation models are now widely generalized into more varieties such as k-inflation [19,20], DBI inflation [21], ghost inflation [22], G-inflation [23,24], and so on. However, almost all the works on the non-Gaussianities in these inflation models concentrate only on the auto bispectrum of the cur- vature perturbations [25–30], 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 [19,20], except for Ref. [31], where the primordial scalar–scalar–tensor cross bispec- trum has been calculated for inflation models with an arbitrary kinetic term. There are several related works on the primordial cross bispectra. In Refs. [32,33], 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 [34], the so-called

“graviton exchange”, and also in the context of the one-loop effects of the scalar power spectrum [35].

In Refs. [36,37], 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 [38] occupies a 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 scalar–tensor Lagrangian with the second-order equation of motion, which was proposed by Horndeski more than thirty years ago [39] and was recently rediscovered in the context of the generalized Galileon [40,41]. Indeed, it includes standard canonical inflation [1,2,15–17], non-minimally coupled inflation [42–47] including Higgs inflation [48–53], extended inflation [54], k-inflation [19,20], DBI inflation [21], R ² inflation [3,55], new Higgs inflation [56], G-inflation [23,24], and so on. Thus, once we analyze the properties of the primordial curvature and tensor perturbations in the generalized G-inflation, one can apply the result for any specific single-field inflation model.

So far, the power spectra of scalar and tensor fluctuations have been studied in Ref. [38] and the general formulae for them are 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. [57,58] (see also Refs. [59,60]) 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. [61] 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 the auto bispectrum of scalar pertur-

bations but also that of tensor perturbations can be large enough to be detected by cosmological

observations, e.g., the Planck satellite, as explained in Ref. [61], which suggests that cross bispectra

can be large as well. For such a case, it is not necessarily justified to consider only the auto bis-

pectrum 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 or 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 Sect. 3, 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 Sect. 4. The special limits for them in the cases of k-inflation and potential-driven inflation are taken in Sect. 5. The final section is devoted to a conclusion and discussion.

2. Generalized G-inflation—The most general single-field inflation model

The Lagrangian for 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 second-order field equations. The Lagrangian was first derived by Horndeski in 1974 in four dimensions [39], and very recently it was rediscovered in a modern form as the generalized Galileon [40], i.e., the most general extension of the Galileon [62,63], in arbitrary dimensions. Their equivalence in four dimensions has been shown in Ref. [38]. The four-dimensional generalized Galileon is described by the Lagrangian:

L = K (φ, X ) − G 3 (φ, X )φ + G 4 (φ, X ) R + G 4 X [ (φ) ² − (∇ μ ∇ ν φ) ² ] + G ₅ (φ, X ) G _μν ∇ ^μ ∇ ^ν φ − 1

6 G _5X [ (φ) ³ − 3 φ(∇ μ ∇ ν φ) ² + 2 (∇ μ ∇ ν φ) ³ ] , (1) 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 i X 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 [38].

3. 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

ds ² = − N ² dt ² + g i j ( dx ⁱ + N ⁱ dt )( dx ^j + N ^j dt ), (2) where

N = 1 + α, N i = ∂ i β, g i j = a ² ( t ) e ^{2 ζ} ( e ^h ) i j , (3)

and (e ^h ) i j = δ i j + h i j + (1/2)h i k h k j + · · · . 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 Ref. [18]. The first-order vector perturbations may be dropped. The curvature perturbation in

generalized G-inflation is shown to be conserved on large scales at nonlinear order in Ref. [64].

Substituting the above metric into the action and expanding it to third order, we obtain the action for the cosmological perturbations, which will be written, with trivial notations, as

S =

dtd ³ x (L hh + L ss + L hhh + L shh + L ssh + L sss ). (4) The first two Lagrangians are quadratic in the metric perturbations, which have already been obtained in Ref. [38]. 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. [61] and [57,58], respectively, but for completeness they are also reproduced in this section.

The mixture of the scalar and tensor perturbations, L shh and L ssh , is computed for the first time in this paper.

3.1. Quadratic Lagrangians and primordial power spectra The quadratic terms are obtained as follows [38].

3.1.1. Tensor perturbations. The most general quadratic Lagrangian for tensor perturbations is given by

L hh = a ³ 8

G T h ˙ ² _{i j} − F T

a ² h i j , k h i j , k

, (5)

where

F T : = 2[G ₄ − X ( φ ¨ G _5X + G _{5 φ} ) ] , (6) G T : = 2[G ₄ − 2 X G _4X − X ( H φ ˙ G _5X − G _5φ ) ] . (7) Here, a dot indicates a derivative with respect to t, G i φ := ∂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

E _{i j} ^h : = ∂ t ( a ³ G T h ˙ i j ) − a F T ∂ ² h i j = 0 . (8) 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:

h _{i j} ( t , x ) =

d ³ k

( 2 π) ³ h _{i j} ( t , k ) e ^{ik · x} . (9) It is convenient to use the conformal time coordinate defined by d η = dt / a. We approximate the inflationary regime by de Sitter spacetime and take F T and G T to be constant. ³

2

In cases where the graviton propagation speed is smaller than light speed, nothing special happens: the light-cone alone determines the causality. In the opposite case, it has been argued that such a theory cannot be UV-completed as a Lorentz-invariant theory [65], though others reach the opposite conclusion [66]. We need further investigation in this case.

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.

The quantized tensor perturbation is written as h i j (η, k ) =

s

[h k (η) e ^(s) _{i j} ( k ) a s ( k ) + h ^∗ _{− k} (η) e _{i j} ^∗(s) (− k ) a s ^† (− k ) ] , (10) where under these approximations the normalized mode is given by

h k (η) = i √ 2H

F T c h k ³ ( 1 + i c h k η) e ^{− i c}

^h

^{k η} . (11) Here, e ⁽ _{i j} ^{s )} is the polarization tensor with the helicity states s = ± 2, satisfying e _{i i} ^{( s )} ( k ) = 0 = k j e ⁽ _{i j} ^{s )} ( k ) . We adopt the normalization such that

e ⁽ _{i j} ^{s )} ( k ) e _{i j} ^{∗( s}

⁾ ( k ) = δ ss

, (12) and choose the phase so that the following relations hold:

e _{i j} ^{∗( s )} ( k ) = e _{i j} ^{(− s )} ( k ) = e ⁽ _{i j} ^{s )} (− k ). (13) The commutation relation for the creation and annihilation operators is

[a s ( k ), a _s ^†

( k ) ] = ( 2 π) ³ δ ss

δ( k − k ). (14) The two-point function can be written as

h i j ( k ) h kl ( k ) = ( 2 π) ³ δ( k + k )P i j , kl ( k ), (15) P i j , kl ( k ) = | h k | ² i j , kl ( k ), (16) where

i j , kl ( k ) =

s

e ⁽ _{i j} ^{s )} ( k ) e ^∗( _kl ^{s )} ( k ). (17) The power spectrum, P h = ( k ³ / 2 π ² )P i j , i j , is thus computed as

P h = 2 π ²

H ² F T c h

c

_h

k η=− 1

. (18)

3.1.2. Scalar perturbations. The quadratic Lagrangian for the scalar perturbations is given by L ss = a ³

− 3 G T ζ ˙ ² + F T

a ² ζ , i ζ , i + α ² − 2

a ² αβ , i i + 2

a ² G T ζ β ˙ , i i + 6 α ζ ˙ − 2

a ² G T αζ , i i

, (19) where

: = X K X + 2 X ² K X X + 12 H φ ˙ X G 3X + 6H φ ˙ X ² G 3X X − 2 X G 3 φ − 2X ² G 3 φ X

− 6H ² G 4 + 6[H ² ( 7 X G 4X + 16 X ² G 4X X + 4 X ³ G 4 X X X )

− H φ( ˙ G 4 φ + 5 X G 4 φ X + 2X ² G 4 φ X X ) ]

+ 30 H ³ φ ˙ X G 5X + 26H ³ φ ˙ X ² G 5X X + 4 H ³ φ ˙ X ³ G 5X X X

− 6H ² X ( 6G 5 φ + 9X G 5 φ X + 2 X ² G 5 φ X X ), (20) : = − ˙ φ X G 3X + 2 H G 4 − 8H X G 4X − 8H X ² G 4X X + ˙ φ G 4 φ + 2X φ ˙ G 4 φ X

− H ² φ(5 ˙ X G 5X + 2X ² G 5X X ) + 2 H X (3G 5 φ + 2 X G 5 φ X ). (21)

Varying Eq. (19) with respect to α and β , we get the first-order constraint equations:

α −

a ² ∂ ² β + 3 ζ ˙ − G T

a ² ∂ ² ζ = 0 , (22)

α − G T ζ ˙ = 0 , (23)

which are solved to yield

α = G T

ζ , ˙ (24)

β = 1 a G T

a ³ G S ψ − a G ² _T

ζ , (25)

with ψ : = ∂ ^{− 2} ζ ˙ . Plugging Eqs. (24) and (25) into Eq. (19), we obtain L ss = a ³

G S ζ ˙ ² − F S

a ² ζ , i ζ , i

, (26)

where we have defined

F S := 1 a

d dt

a G T ²

− F T , (27)

G S : =

² G T ² + 3 G T . (28)

The sound speed is given by c _s ² : = F S /G S . The linear equation of motion derived from the Lagrangian (26) is

E ^s : = ∂ t ( a ³ G S ζ ) ˙ − a F S ∂ ² ζ = 0 . (29) 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:

ζ( t , x ) =

d ³ k

( 2 π) ³ ζ( t , k ) e ^{i k · x} , (30) and proceed in the de Sitter approximation, assuming that F S and G S are almost constant. The quantized curvature perturbation is written as

ζ(η, k ) = ξ k (η) a ( k ) + ξ ₋ ^∗ k (η) a ^† (− k ), (31) where the normalized mode is given by

ξ k (η) = i H 2

F S c s k ³ ( 1 + i c s k η) e ^{− c}

^s

^{k η} . (32) The commutation relation for the creation and annihilation operators is

[a ( k ), a ^† ( k ) ] = ( 2 π) ³ δ( k − k ). (33) Thus, the power spectrum is calculated as

ζ( k )ζ( k ) = ( 2 π) ³ δ( k + k ) 2 π ²

k ³ P ζ , (34)

P ζ = 1 8 π ²

H ² F S c _s

c

s

kη=−1 . (35)

From Eqs. (18) and (35), the tensor-to-scalar ratio r is given by r := P h

P ζ = 16 F S c s

F T c h , (36)

where we have assumed that the relevant quantities remain practically constant between the horizon crossings of tensor and scalar perturbations that occur at different times in the case c h = c s [67].

3.2. 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 the literature [68,69].

3.2.1. Three tensors. The Lagrangian involving three tensors was derived in Ref. [61]:

L hhh = a ³ μ

12 h ˙ _{i j} h ˙ _{j k} h ˙ _ki + F T

4a ²

h _{i k} h _jl − 1 2 h _{i j} h _kl

h _{i j , kl}

, (37)

where we defined

μ : = ˙ φ X G 5 X . (38)

As discussed in Ref. [61], this cubic action for the tensor perturbation h i j is composed only of two contributions. The former has one time derivative on each h i j and newly appears in the presence of the kinetic coupling to the Einstein tensor, i.e., 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 the corresponding terms.

3.2.2. Two tensors and one scalar. The interactions involving two tensors and one scalar are given by

L shh = a ³ 3 G T

8 ζ h ˙ ² _{i j} − F T

8a ² ζ h i j , k h i j , k − μ

4 ζ ˙ h ˙ ² _{i j} −

8 α h ˙ ² _{i j} − G T

8a ² α h i j , k h i j , k − μ

2a ² α h ˙ i j h i j , kk

− a G T

4 β _, k h ˙ i j h i j , k + μ 2

h ˙ i k h ˙ j k β _, i j − 1 2 h ˙ ² _{i j} β _, kk

, (39)

where

: = 2G 4 − 8X G 4X − 8X ² G 4 X X

− 2 H φ( ˙ 5X G 5X + 2 X ² G 5X X ) + 2X ( 3G 5 φ + 2X G 5 φ X ). (40) This quantity can also be expressed in a compact form, = ∂/∂ H .

Substituting the first-order constraint equations into Eq. (39), the Lagrangian reduces to L shh = a ³

b 1 ζ h ˙ ² _{i j} + b 2

a ² ζ h i j , k h i j , k + b 3 ψ , k h ˙ i j h i j , k + b 4 ζ ˙ h ˙ ² _{i j} + b 5

a ² ∂ ² ζ h ˙ ² _{i j} + b 6 ψ , i j h ˙ i k h ˙ j k + b 7

a ² ζ , i j h ˙ i k h ˙ j k

+ E shh , (41)

where

b 1 = 3 G T

8

1 − H G ² _T F T + G T

3 d dt

G T

F T

, (42)

b ₂ = F S

8 , (43)

b 3 = − G S

4 , (44)

b 4 = G T

8 F T

G _T ² − F T

+ μ 4

G S

G T − 1 − H G _T ² F T

6 + G ˙ S

H G S + G _T ² 4

d dt

μ F T

, (45)

b 5 = μG T

4

F S G T

F T G S − 1

, (46)

b 6 = − μ 2

G S

G T , (47)

b 7 = μ 2

G T

, (48)

and

E shh = μ 4 G S

G ² _T F T

h ˙ ² _{i j} E ^s + G _T ² 2 F T

ζ 2 + μ

G T

ζ ˙

h ˙ i j E _{i j} ^h . (49)

The last term, E _shh , can be removed by redefining the fields as h _{i j} → h _{i j} + G _T ²

F T

ζ + 2 μ

G T

ζ ˙

h ˙ _{i j} , (50)

ζ → ζ + μ 8 G S

G _T ² F T

h ˙ ² _{i j} . (51)

The contribution to the correlation function is, however, negligible, because the above field redefini- tions involve at least one time derivative of the metric perturbation, which vanishes on super-horizon scales.

3.2.3. Two scalars and one tensor. The interactions involving one tensor and two scalars are given by

L ssh = a

2 αβ _, i j h _{i j} +

2 αβ _, i j h ˙ _{i j} + μ

a ² αβ _, i j h _{i j , kk} − 3 G T

2 ζβ _, i j h ˙ _{i j} − 2 G T ζ β ˙ _, i j h _{i j} + μ ζ β ˙ _, i j h ˙ _{i j}

− F T ζ , i ζ , j h i j − 2 G T α , i ζ , j h i j + μα , i ζ , j h ˙ i j + G T

2a ² β , i j β , k h i j , k + μ

a ² β , i j β , k h ˙ i j , k

. (52) Substituting the constraint equations, we obtain the reduced Lagrangian:

L ssh = a ³ c 1

a ² h _{i j} ζ _, i ζ _, j + c 2

a ² h ˙ _{i j} ζ _, i ζ _, j + c ₃ h ˙ _{i j} ζ _, i ψ _, j + c 4

a ² ∂ ² h _{i j} ζ _, i ψ _, j

+ c 5

a ⁴ ∂ ² h _{i j} ζ ,i ζ , j + c ₆ ∂ ² h _{i j} ψ ,i ψ , j

+ E _ssh , (53)

where

c 1 = F S , (54)

c 2 =

4 (F S − F T ) + G ² _T

− 1 2 + H

4

3 + G ˙ T

H G T − 1 4

d dt

+ μF S

G T + 2 H G T μ − G T

d dt

μ

, (55)

c 3 = G S

3 2 + d

dt

2 + μ G T

−

3 H + G ˙ T

G T

2 + μ

G T

, (56)

c ₄ = G S

− G ² _T − F T

2 G T − 2H μ

+ d

dt μ

+ μ

G _T ² (F T − F S )

, (57)

c 5 = G ² _T 2

G ² _T − F T

2 G T + 2H μ − d

dt μ

− μ

G ² _T ( 3 F T − F S )

, (58)

c 6 = G ² _S 4G T

1 + 6 H μ G T − 2 G T

d dt

μ

G _T ² , (59)

and

E ssh = ¯ f i ∂ ^{− 2} ∂ i E ^s + ¯ f i j E _{i j} ^h , (60) with

f ¯ i : =

2 ζ , j h i j + μ

G T ζ , j h ˙ i j + μ

a ² ζ , j ∂ ² h i j − μG S

G ² _T ψ , j ∂ ² h i j , (61) f ¯ i j : = G S

G T

2 + μ

G T

ζ , i ψ , j − G T

a ^{2 2}

4 + μ G T

ζ , i ζ , j . (62) The field redefinition

h i j → h i j + 4 f ¯ i j , (63) ζ → ζ − 1

2 ∂ ^{− 2} ∂ i f ¯ i (64)

removes the last term E _ssh . Since all the terms involve at least one derivative of the metric per- turbation, the field redefinition does not contribute to the correlation function on super-horizon scales.

3.2.4. Three scalars. For completeness, here we give the cubic Lagrangian for the scalar pertur- bations derived in Refs. [57,58]. The cubic Lagrangian for the scalar perturbations is given by

L sss = − a ³

3 ( + 2X X + H )α ³ + a ³

3 ζ + ζ ˙ + ( − G T ) ζ _, i i

a ² − 3a ² β , i i

α ²

− 2a αζ , i β , i + 18a ³ αζ ζ ˙ + 4a μα ζ ζ ˙ , i i −

2a α(β , i j β , i j − β , i i β , j j ) + 2 μ

a α(β ,i j ζ ,i j − β ,i i ζ , j j ) − 2a αβ ,i i ζ − 2a αβ ,i i ζ ˙ − 2a G T αζ ζ ,i i − a G T αζ ,i ζ ,i

+ 3a ³ α ζ ˙ ² + 2a ³ μ ζ ˙ ³ + a F T ζ ζ , i ζ , i − 9a ³ G T ζ ˙ ² ζ + 2a G T β , i ζ , i ζ ˙ − 2a μβ i i ζ ˙ ² + 2a G T β , i i ζζ ˙ + 1

a 3

2 G T ζ − μ ζ ˙

(β , i j β , i j − β , i i β , j j ) − 2 G T

a β , i i β , j ζ , j , (65) where

: = 12 φ ˙ X G 3X + 6 φ ˙ X ² G 3 X X − 12 H G 4

+ 6[2H ( 7X G 4X + 16 X ² G 4X X + 4X ³ G 4X X X ) − ˙ φ( G 4 φ + 5X G 4 φ X + 2X ² G 4 φ X X ) ] + 90H ² φ ˙ X G 5X + 78H ² φ ˙ X ² G 5X X + 12 H ² φ ˙ X ³ G 5X X X

− 12H X ( 6G 5 φ + 9 X G 5 φ X + 2 X ² G 5 φ X X ). (66) Using the first-order constraint equations to remove α and β from the above Lagrangian, we obtain the following reduced expression:

L sss =

dtd ³ xa ³ G S

C 1

6H ζ ˙ ³ + C 2 ζ ˙ ² ζ + C 3

2c ² _s

a ² ζ(∂ i ζ ) ² + 2 C 4 ζ∂ ˙ i ζ ∂ ⁱ ψ + 2 C 5 ∂ ² ζ(∂ i ψ) ²

, (67)

with ψ = ∂ ^{− 2} ζ ˙ . There are five independent cubic terms with coefficients:

C 1 = − 8 G _T ³

3 ³ G S + 2 H ² F S

2 G _T ³

² + 3 G _T ³

F S (G S − 2 F S ) + 36 μ(G T − G S ) + 9

G T ( 2 G T − G S )

+ 2H

6 μ 1

G S − 1 G T

+ 2 ( − X X )G _T ³

³ G S + G T

² 3G T

G S − 1

+ 3G T

G S

F S + 3G T

G S − 1

+ 3

3G T

G S − 2

− 6H ³ G S G _T ^{2 2} F _S ²

6 μ + G T

, (68)

C 2 = 3 + 3 H G S

μ

G _T ² +

2 G T − 3 G T

2 F S

+ 3 H ² G S

F S

8μ + 2 G T

− G _T ³

2 F S + 3H ³ G S G _T ^{2 2} F _S ²

3μ + G T

2

, (69)

C 3 = F T

2F S + H

( 3 G S − 2 G T )G T

4F S − μG S

2 G _T ² − G S

4G T

+ H ² G S

F S

G ³ _T

4 F S − 4 μ − G T

− H ³ G S G _T ² 2 ² F _S ²

3 μ + G T

2

, (70)

C 4 = − G S

4 G T + 3 H G S

μ

2 G ² _T +

4 G T − G T

2 F S + 3 H ² G S

F S

2 μ + G T

2

, (71)

C 5 = 3 G S

8 G T − 3 H G S

4 G T

μ G T +

2

. (72)

4. 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.

4.1. Three tensors

Let us consider the three-point function of the tensor perturbations:

h i

₁

j

₁

( k 1 ) h i

₂

j

₂

( k 2 ) h i

₃

j

₃

( k 3 ) = ( 2 π) ³ δ( k 1 + k 2 + k 3 ) B _i ^(hhh)

1

j

₁

i

₂

j

₂

i

₃

j

₃

, (73) B _i ⁽

₁

^hhh _j

₁

_i

₂

⁾ _j

₂

_i

₃

_j

₃

= ( 2 π) ⁴ P _h ²

k ³ ₁ k ₂ ³ k ₃ ³

A ⁽ _i ^new

₁

_j

₁

_i

₂

⁾ _j

₂

_i

₃

_j

₃

+ A ⁽ _i

₁

^GR _j

₁

_i ⁾

₂

_j

₂

_i

₃

_j

₃

, (74)

where A _i ^(new)

₁

_j

₁

_i

₂

_j

₂

_i

₃

_j

₃

and A ^(GR) _i

₁

_j

₁

_i

₂

_j

₂

_i

₃

_j

₃

represent the contributions from the h ˙ ³ and h ² ∂ ² h terms, respectively.

Each contribution is given by A ⁽ _i ^new

₁

_j

₁

_i

₂

⁾ _j

₂

_i

₃

_j

₃

= H μ

4 G T

k ₁ ² k ₂ ² k ² ₃

K ³ i

1

j

1

,lm ( k ₁ ) i

2

j

2

,mn ( k ₂ ) i

3

j

3

,nl ( k ₃ ), (75) A ⁽ _i ^GR

₁

_j

₁

_i ⁾

₂

_j

₂

_i

₃

_j

₃

= A

i

₁

j

₁

, i k ( k 1 ) i

₂

j

₂

, jl ( k 2 )

k 3k k 3l i

₃

j

₃

, i j ( k 3 ) − 1

2 k 3i k 3k i

₃

j

₃

, jl ( k 3 )

+ 5 perms of 1 , 2 , 3

, (76)

where K = k 1 + k 2 + k 3 and

A( k ₁ , k ₂ , k ₃ ) : = − K 16

⎡

⎣1 − 1 K ³

i = j

k _i ² k _j − 4 k 1 k 2 k 3

K ³

⎤

⎦ . (77)

The first term A ⁽ _i

₁

^new _j

₁

_i

₂

⁾ _j

₂

_i

₃

_j

₃

is proportional to G 5X and hence vanishes in the case of Einstein grav- ity, while the second term A ^(GR) _i

₁

_j

₁

_i

₂

_j

₂

_i

₃

_j

₃

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 the two polarization modes as ξ ^{( s )} ( k ) : = h i j ( k ) e ^∗(s) _{i j} ( k ), (78) and their relevant amplitudes of the bispectra as

ξ ^(s

¹

⁾ ( k 1 )ξ ^(s

²

⁾ ( k 2 )ξ ^(s

³

⁾ ( k 3 ) = ( 2 π) ⁷ δ( k 1 + k 2 + k 3 ) P _h ² k ₁ ³ k ₂ ³ k ³ ₃

A ^s ₍

¹

_new ^s

²

^s ₎

³

+ A ^s ₍

¹

_GR ^s

²

^s ₎

³

. (79)

From Eqs. (75) and (76), the amplitudes A ^s ₍

¹

_new ^s

²

^s

³

_{),( GR )} are easily calculated as [61]

A ^s _{( new}

¹

^s

²

^s ₎

³

= H μ 4 G T

k ₁ ² k ₂ ² k ₃ ²

K ³ F ( s ₁ k ₁ , s ₂ k ₂ , s ₃ k ₃ ), (80) A ^s _{( GR}

¹

^s

²

^s ₎

³

= A

2 ( s ₁ k ₁ + s ₂ k ₂ + s ₃ k ₃ ) ² F ( s ₁ k ₁ , s ₂ k ₂ , s ₃ k ₃ ), (81) where

F ( x , y , z ) : = 1 64

1

x ² y ² z ² ( x + y + z ) ³ ( x − y + z )( x + y − z )( x − y − z ). (82)

As pointed out in Ref. [61], A ⁺⁺⁺ _{( new )} has a peak in the equilateral limit, while A ⁺⁺⁺ _{( GR )} has one in the

squeezed limit.

It is convenient to introduce nonlinearity parameters, defined as f _NL ^s

¹

^s ₍

²

_new ^s

³

_{),( GR )} = 30

A ^s ₍

¹

_new ^s

²

^s

³

_{),( GR ) k}

₁

_{= k}

₂

_{= k}

₃

K ³ , (83)

which are quantities analogous to the standard f _NL for the curvature perturbation. We find f _NL ^s

¹

^s ₍

²

_new ^s

³

₎ = − 5

10 368 [3 + 2 ( s ₁ s ₂ + s ₂ s ₃ + s ₃ s ₁ ) ] H μ

G T , (84)

or, more concretely,

f _NL ⁺⁺⁺ _{( new )} = − 5 1152

H μ

G T , f _NL ⁺⁺⁻ _{( new )} = − 5 10 368

H μ

G T , (85) with f _NL ⁺⁺⁻ _{( new )} = f _NL ⁺⁻⁻ _{( new )} and f _NL ⁻⁻⁻ _{( new )} = f _NL ⁺⁺⁺ _{( new )} . (This symmetry arises because parity is not violated.) As for f _NL ^s

¹

^s ₍

²

_GR ^s

³

₎ , we have

f _NL ^s

¹

^s ₍

²

_GR ^s

³

₎ = 85

27 648 [21 + 20 ( s 1 s 2 + s 2 s 3 + s 3 s 1 ) ] , (86) so that

f _NL ⁺⁺⁺ _{( GR )} = f _NL ⁻⁻⁻ _{( GR )} = 255

1024 , f _NL ⁺⁺⁻ _{( GR )} = f _NL ⁺⁻⁻ _{( GR )} = 85

27 648 . (87)

As defined in Eq. (74), B _i ^{( hhh )}

1

j

₁

i

₂

j

₂

i

₃

j

₃

is normalized by P _h ² . This normalization can be justified when one concentrates on the non-Gaussianity of the B-mode polarization. Because the B-mode polar- ization can be generated not by curvature perturbations but tensor perturbations (except for the lensing contribution), the size of the non-Gaussianity of the B-mode polarization can be directly characterized by f _NL ^s

¹

^s ₍

²

_new ^s

³

_{),( GR )} .

However, it should be noted 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 gener- ated 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 _i ^{( hhh )}

1

j

₁

i

₂

j

₂

i

₃

j

₃

by P _ζ ² , namely, B _i ^{( hhh )}

1

j

1

i

2

j

2

i

3

j

3

= ( 2 π) ⁴ P _ζ ² k ₁ ³ k ₂ ³ k ³ ₃

A ⁽ _i

₁

^new _j

₁

_i

₂

⁾ _j

₂

_i

₃

_j

₃

+ A ⁽ _i

₁

^GR _j

₁

_i ⁾

₂

_j

₂

_i

₃

_j

₃

, (88)

where A ⁽ _i ^new

₁

_j

₁

_i

₂

^),( _j

₂

^GR _i

₃

_j ⁾

₃

= r ² A ⁽ _i

₁

^new _j

₁

_i

₂

^),( _j

₂

^GR _i

₃

_j ⁾

₃

, with r being the tensor-to-scalar ratio. In the same way, A ^s _(new),(GR)

¹

^s

²

^s

³

= r ² A ^s _(new),(GR)

¹

^s

²

^s

³

and f NL(new),(GR) ^s

¹

^s

²

^s

³

= r ² f NL(new),(GR) ^s

¹

^s

²

^s

³

.

4.2. Two tensors and one scalar

The cross bispectrum of two tensors and one scalar is given by

ζ( k 1 ) h i j ( k 2 ) h kl ( k 3 ) = ( 2 π) ³ δ( k 1 + k 2 + k 3 ) B _{i j} ^(ζ _, ^hh _kl ⁾ ( k 1 , k 2 , k 3 ), (89) where B _{i j} ^(ζ _, ^hh _kl ⁾ is of the form

B _{i j} ^(ζ _, ^hh _kl ⁾ = 2 k ₁ ³ k ₂ ³ k ³ ₃

H ⁶ F S F _T ² c s c ² _h

7 q = 1

b q V _{i j} ^{( q} _, ⁾ _kl (k 1 , k 2 , k 3 )I ^(q) (k 1 , k 2 , k 3 ) + (k 2 , i, j ↔ k 3 , k, l). (90)