Primordial non-Gaussianity from G-inflation

Primordial non-Gaussianity from G-inflation

Tsutomu Kobayashi Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan111Present address: Hakubi Center, Kyoto University, Kyoto 606-8302, Japan and Department of Physics, Kyoto University, Kyoto 606-8502, Japan    Masahide Yamaguchi Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan    Jun’ichi Yokoyama Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan Institute for the Physics and Mathematics of the Universe(IPMU), The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan
Abstract

We present a comprehensive study of primordial fluctuations generated from G-inflation, in which the inflaton Lagrangian is of the form with . The Lagrangian still gives rise to second-order gravitational and scalar field equations, and thus offers a more generic class of single-field inflation than ever studied, with a richer phenomenology. We compute the power spectrum and the bispectrum, and clarify how the non-Gaussian amplitude depends upon parameters such as the sound speed. In so doing we try to keep as great generality as possible, allowing for non slow-roll and deviation from the exact scale-invariance.

pacs:
98.80.Cq
preprint: RESCEU-3/11

I Introduction

Cosmological inflation inflation () is now a widely accepted paradigm explaining the flatness, homogeneity, and isotropy of the observed Universe. In the most common scenario, inflation occurs when the inflaton, a scalar field driving the accelerated expansion, rolls down a nearly flat potential slowly. During this slow-roll stage fluctuations in the inflaton field are generated quantum-mechanically and stretched outside the Hubble horizon, which eventually reenter the Hubble radius in a later epoch to be a seed for the large-scale structure of the Universe. The detailed shape of the potential can be probed by observing the power spectrum of fluctuations in terms of the cosmic microwave background (CMB) anisotropies wmap (). As to theoretical approaches, much effort has been made to determine the inflaton potential in the particle physics context. However, single-field inflation with a canonical kinetic term and a nearly flat potential is not the only option to induce the accelerated expansion and to produce almost scale-invariant perturbations with an appropriate amplitude. Liberating inflation models from the standard assumption, one may consider a variety of interesting scenarios: multiple scalar fields might participate the inflationary dynamics, the kinetic term of the inflaton(s) might be non-canonical kinflation (), and a scalar field other than the inflaton might be responsible for the density perturbation curvaton (). From a high-energy physics point of view, supersymmetric theories naturally provide many scalar fields with flat potentials susyinf (), and the Dirac-Born-Infeld (DBI)-type non-canonical kinetic term naturally arises from D3-brane motion in a warped compactification DBI ().

Different inflationary scenarios can be distinguished by future and on-going experiments such as Planck PLANCK (), aiming to obtain better constraints on the amount of non-Gaussianities in the primordial curvature perturbations as well as on the spectral index , its running, and the tensor-to-scalar ratio . The standard canonical slow-roll inflation models produce negligible non-Gaussianity Malda (), while exotic inflationary scenarios are expected to predict measurable non-Gaussian signals. In the context of single-field inflation, non-Gaussian perturbations have been computed for the Lagrangian of the form LS (); Kachru ()

 Lϕ=K(ϕ,X), (1)

where is the inflaton and . This class of models yields a sound speed different from the speed of light in general, and large non-Gaussianity is generated for . A significant non-Gaussian signal together with the confirmation of the consistency relation , where is the spectral index of primordial tensor perturbations, is a smoking gun of the inflaton Lagrangian (1).

In this paper, we consider a more general Lagrangian vikman (); GI ()

 Lϕ=K(ϕ,X)−G(ϕ,X)□ϕ, (2)

where and are some generic functions of the inflaton and . The new term in the Lagrangian (2) is inspired by the Galileon interaction G1 (); G2 () and reduces to the one having the Galilean shift symmetry, , in the Minkowski background in the case . One of the most important properties of the Galileon Lagrangian is that the field equations do not contain derivatives higher than two. The interaction is a generalization of the Galileon term while maintaining the second-order property. In this sense, the Lagrangian (2) defines a more generic class of single-field inflation than ever studied. Here, the Galilean shift symmetry is abandoned in exchange for generality, but one should note that the symmetry does not make sense already upon covariantization for any interaction that is Galilean invariant in the flat background.222Concerning this point, one may worry about the naturalness of G-inflation models discussed in the present paper because there is no symmetry to protect the Lagrangian. However, it should be noted that symmetry, if present, must be broken at least to end inflation. We therefore will not provide a symmetry-based argument but rather take a phenomenological approach, assuming that some UV complete theory would give the (in some sense fine-tuned) Lagrangian that leads to second-order field equations. (The name “Galileon” is therefore no longer appropriate when covariantized.) Cosmological applications of the Galileon interaction can be found in gde () with emphasis on dark energy and modified gravity. Primordial inflation based on the generic Lagrangian (2) was first proposed very recently by GI (); HGI (), and is dubbed G-inflation. Almost simultaneously the same Lagrangian was used to explain the late-time cosmic acceleration rather than the primordial one vikman (); Kimura (). In ginf1 (); ginf2 () the effective-field-theory approach eft () was employed to see the consequences of imposing the approximate Galilean shift symmetry on the Lagrangian of primordial perturbations. Interestingly, the scalar field theory with the term can violate the null energy condition stably. This fact motivates the authors of Refs. NV (); Genesis () to propose a radical scenario of the earliest Universe alternative to inflation. Some specific form of the above type of interaction arises from a probe brane action in higher dimensions dbigalileon () and from the Kaluza-Klein reduction of Lovelock gravity kkreduction (); bstring (). A supersymmetric completion of Galileons is explored in susy ().

The purpose of the present paper is to understand the nature of cosmological perturbations generated from G-inflation. We rederive the power spectrum and the tilt of the spectrum without assuming slow-roll, clarifying how the (approximate) scale-invariance is achieved in G-inflation. We then calculate the cubic action for the curvature perturbation and evaluate the full non-Gaussian amplitude, again without assuming slow-roll and the exact scale-invariance. Throughout the paper we try to make our formulas as general as possible, which we hope maximizes the usefulness of the results. Recently, non-Gaussianity from G-inflation was calculated neglecting a number of terms working in the de Sitter limit MK () and in the slow-roll limit DeFelice:2011zh (). See also a recent work by Naruko and Sasaki, in which the superhorizon evolution of the nonlinear curvature perturbation from G-inflation is addressed naruko ().

This paper is organized as follows. In the next section we review the basic properties of G-inflation and derive the power spectrum of the curvature perturbation. In Sec. III we compute the cubic action for the curvature perturbation to evaluate the three-point function in G-inflation.

Ii G-inflation

We start with a brief review on the basics of G-inflation GI (); HGI (). The scalar field Lagrangian for G-inflation is given by Eq. (2). Assuming that is minimally coupled to gravity, the total action we are going to study is

 S=∫d4x√−g[M2Pl2R+Lϕ]. (3)

In the following we will set . The energy-momentum tensor of the scalar field is given by

 Tμν=KX∇μϕ∇νϕ+Kgμν−2∇(μG∇ν)ϕ+gμν∇λG∇λϕ−GX□ϕ∇μϕ∇νϕ. (4)

Here and hereafter we use the notation for etc. Varying the action with respect to , we obtain the scalar field equation of motion,

 KX□ϕ−KXX(∇μ∇νϕ)(∇μϕ∇νϕ)−2KϕXX+Kϕ−2(Gϕ−GϕXX)□ϕ +GX[(∇μ∇νϕ)(∇μ∇νϕ)−(□ϕ)2+Rμν∇μϕ∇νϕ]+2GϕX(∇μ∇νϕ)(∇μϕ∇νϕ)+2GϕϕX −GXX(∇μ∇λϕ−gμλ□ϕ)(∇μ∇νϕ)∇νϕ∇λϕ=0, (5)

which is of course equivalent to the conservation equation . One verifies from Eqs. (4) and (5) that the gravitational and scalar field equations are indeed of second order.

Higher order Galileon terms (with a -dependent coefficient) such as can be added to the scalar field Lagrangian while keeping the field equations of second order. Although the effect of such higher order Galileons might be interesting in the context of primordial inflation, we leave the issue for future study and concentrate on the Lagrangian of the form (2) in the present paper.

ii.1 The background equations

Let us consider homogeneous and isotropic background:

 ds2=−dt2+a2(t)dx2,ϕ=ϕ(t). (6)

Although the energy-momentum tensor (4) cannot be recast in a perfect-fluid form in general vikman (), for the above cosmological ansatz it takes the desirable form with

 ρ = 2KXX−K+3HGX˙ϕ3−2GϕX, (7) p = K−2(Gϕ+GX¨ϕ)X. (8)

The gravitational field equations are thus

 3H2 = ρ, (9) −3H2−2˙H = p, (10)

and the scalar field equation of motion is given by

 KX(¨ϕ+3H˙ϕ)+2KXXX¨ϕ+2KXϕX−Kϕ−2(Gϕ−GXϕX)(¨ϕ+3H˙ϕ) +6GX[(HX)˙+3H2X]−4GXϕX¨ϕ−2GϕϕX+6HGXXX˙X=0. (11)

If, for example, is given by the standard, canonical kinetic term with a potential, , one can consider an inflationary scenario in which the energy density is dominated by the potential as in the standard case, while the dynamics of the scalar field is modified by the term, changing the potential that effectively feels. This is the scenario proposed in HGI () and called potential driven G-inflation. Another possible scenario is that inflation is driven by ’s kinetic energy which is kept almost constant with nontrivial functional form of and . In models with the exact shift symmetry, , i.e., and , it is easy to obtain an exactly de Sitter background satisfying const and const. This may be regarded as a generalization of k-inflation kinflation (), and we call the class of models kinematically driven G-inflation GI (). Deferring the summary of these two specific classes of G-inflation to Sec. II.3, we now move on to describe the general properties of the power spectrum of primordial perturbations from G-inflation.

ii.2 Power spectrum

In this section we derive a series of general formulas for linear cosmological perturbations without assuming any specific form of and . We work in the unitary gauge, .333The unitary gauge does not coincide with the comoving gauge, , in the case of G-inflation GI (). This fact stems from the imperfect-fluid nature of the energy-momentum tensor (4). Using the remaining gauge degree of freedom the linearly perturbed metric is taken to be

 ds2=−(1+2α1)dt2+2a2∂iβ1dtdxi+a2(1+2R)dx2. (12)

Expanding the action to second order in perturbations and then varying with respect to and , we obtain the following constraint equations:

 ˙R = Θα1, (13) ∂2a2(R+a2Θβ1) = XGα1, (14)

where ,

 Θ := H−˙ϕXGX, (15) G := KX+2XKXX+6GXH˙ϕ+6G2XX2−2(Gϕ+XGϕX)+6GXXHX˙ϕ. (16)

Substituting the constraints (13) and (14) to the action, we arrive at the quadratic action for  GI (); vikman ():

 S2=∫dtd3xa3σ[1c2s˙R2−1a2(∂R)2], (17)

where

 c2s := FG, (18) σ := XFΘ2, (19)

and

 F := KX+2GX(¨ϕ+2H˙ϕ)−2G2XX2+2GXXX¨ϕ−2(Gϕ−XGϕX). (20)

One can verify that setting the quadratic action (17) reproduces the expression obtained for k-inflation per-k-inf (). It is useful to notice that can also be expressed as

 σ=−˙ΘΘ2+˙ϕXGXΘ. (21)

Let us define three parameters that characterize the rate of change of three background quantities:

 ϵ:=−˙HH2,s:=˙csHcs,δ:=˙σHσ. (22)

In this paper we assume that

 ˙ϵHϵ≃0,˙sHs≃0,˙δHδ≃0, (23)

but we do not neglect , , and . (In the next section, however, we will assume some stronger conditions to evaluate the bispectrum.) It should be noted in particular that is not necessarily small, in contrast to the usual (k-)inflation models in which is degenerate, i.e.,  per-k-inf (). Even in the slow-roll limit we may have in G-inflation.

Under the assumption that the parameters defined in (22) are constant (but not necessarily very small), it is straightforward to solve the equation of motion derived from the action (17) and compute the power spectrum of  GI (). For this purpose it is convenient to define a new time coordinate by  KP (). In terms of , the scale factor, the sound speed, and are written as

 a=cs∗(y/y∗)−1/(1−ϵ−s)(−y∗)H∗(1−ϵ−s),cs=cs∗(y/y∗)−s/(1−ϵ−s),σ=σ∗(y/y∗)−δ/(1−ϵ−s), (24)

where the quantities with are those evaluated at some reference time . Using a new variable with , the equation of motion can be written in the Fourier space as

 u′′k+(k2−~z′′~z)uk=0, (25)

where the prime denotes differentiation with respect to and we find

 ~z∝(−y)1/2−q,~z′′~z=q2−1/4y2,withq:=3−ϵ−2s+δ2(1−ϵ−s). (26)

The normalized mode solution to Eq. (25) corresponding to the Minkowski vacuum in the high frequency limit is then given in terms of the Hankel function by

 uk=√π2√−yH(1)q(−ky). (27)

We thus write the operator using the creation and annihilation modes as

 R(k,y) = ψ(k,y)^ak+ψ∗(−k,y)^a†−k, (28) ψ(k,y) = uk(y)~z, (29)

with the commutation relation . This immediately leads to the power spectrum GI (),

 PR=k32π2∣∣uk~z∣∣2=22q−3∣∣∣Γ(q)Γ(3/2)∣∣∣2(1−ϵ−s)24π2H22σcs∣∣∣ky=−1. (30)

The scalar spectral index is found to be

 ns−1=3−2q=−2ϵ+s+δ1−ϵ−s. (31)

The above formula has been derived without assuming the smallness of and , though we have assumed that they are constant. In this sense, the above expression is more general than that given in GI (); HGI (); MK (); DeFelice:2011zh (). To ensure the scale invariance we require . However, this does not force each parameter to be as small as ; each can be large, , but the three may cancel each other out to produce an almost scale-invariant spectrum. This possibility was first pointed out by KP () in the less generic context of DBI inflation, for which and consequently . We leave this interesting possibility open, and will complete the following calculation without taking the slow-roll limit. We would stress again that even if we consider the slow-roll limit, is not necessarily slow-roll suppressed.

Since the inflaton field is minimally coupled to gravity, the nature of tensor perturbations is the same as the standard one and is dependent only on the geometrical quantity . In the slow-roll limit, , the tensor power spectrum is given by . The tensor-to-scalar ratio is thus given by

 r=16σcs, (32)

where just for simplicity the scalar power spectrum is evaluated also in the slow-roll limit, .

For later convenience we introduce the following quantity:

 ν:=˙ϕXGXH, (33)

or, equivalently, . From Eq. (21) we obtain

 σ=˙νH(1−ν)2+ν1−ν+ϵ1−ν. (34)

For const, const, and const, the above equation can be integrated to yield

 HΘ=11−ν(y)=11+ϵ+σ(y)1+ϵ+δ+(11−ν∗−11+ϵ−σ∗1+ϵ+δ)(y/y∗)(1+ϵ)/(1−ϵ−s). (35)

If we assume const then we have const. In this case the two quantities are related as

 ν=σ−ϵ1+σ. (36)

Note in passing that the opposite is not in general true: for const Eq. (34) still admits time-dependent .

ii.3 G-inflation examples

ii.3.1 Kinematically driven G-inflation

Inflation can be driven by kinetic energy of . This possibility was explored in GI (). Let us consider for simplicity the Lagrangian with exact shift symmetry , i.e.,

 K=K(X),G=G(X), (37)

and look for an exact de Sitter background satisfying const and const. It follows from the field equations that

 3H2=−K, (38) KX+3GXH˙ϕ=0. (39)

For this background we have

 F = −K3Xν(1−ν), (40) G = −KXν(1+ν−2XKXXKX+2XGXXGX), (41) σ = ν1−ν, (42)

where const. In evaluating the above equations we used the background equations (38) and (39).

The concrete toy model presented in GI () is given by

 K=−X+X22M3μ,G=XM3, (43)

where and are parameters. In this case, and can be expressed in terms of . It turns out that the tensor-to-scalar ratio is an increasing function of , and is required in order for not to exceed the observationally allowed value. Explicitly, one finds  GI ().

Note, however, that is not necessary to get a stable, prolonged de Sitter phase. As already emphasized above, is made possible by a suitable choice of and , provided that remains not too large. In MK () Mizuno and Koyama have studied the case with focusing their attention on the model (43). In contrast, the analysis in the present paper can apply to more general cases with .

In the presence of exact shift symmetry the exact de Sitter solution is an attractor. Along this attractor the scalar fluctuations acquire an exactly scale-invariant spectrum. Making and/or weakly dependent on , one obtains a quasi-de Sitter attractor and thereby the spectrum can be tilted. Though we do not provide corresponding concrete examples here, more generic, possibly complicated, choices of and would lead to the interesting situation mentioned above: with .

ii.3.2 Potential driven G-inflation

In HGI () a novel class of inflation models was proposed in which the energy density is dominated by ’s potential but its dynamics is nontrivial due to the term. In particular, it was shown that slow-roll inflation can proceed even if the potential is too steep to support standard slow-roll inflation. The model examined in HGI () is described by

 K=X−V(ϕ),G=−g(ϕ)X. (44)

For , the effect of the term dominates in the slow-roll equation of motion for and the potential is effectively flattened, leading to slow-roll G-inflation. In this regime one finds

 σ≃43ϵandc2s≃23. (45)

Though could be free from the slow-roll constraint in principle, in the present case it is actually related to in a way different from standard slow-roll inflation. Since const, the scale-invariant spectrum requires that , and hence .

Iii Bispectrum

In order to evaluate the bispectrum, we compute the cubic action for working in the ADM formalism Malda (); LS (); Kachru (),

 ds2=−N2dt2+hij(dxi+Nidt)(dxj+Njdt), (46)

where

 hij=a2(t)e2Rδij,N=1+α1+α2+⋯,Ni=a2∂i(β1+β2+⋯)+~N1i+⋯, (47)

with . Here, and are . The fluctuation of the scalar field vanishes in this gauge. At linear order the above metric reduces to Eq. (12).

As pointed out in Malda (), we only need to consider first-order perturbations in and to get the cubic action. (This holds true even in the presence of the term.) Therefore, it suffices to use the first-order solution of the constraint equations, Eqs. (13) and (14), supplemented with a vanishing first-order vector perturbation, .

We plug the solution for and into the action and expand it to third order in . After cumbersome multiple integrations by parts, one ends up with

 S3 = ∫dtd3xa3[C1H˙R3+C2R˙R2+C3a4H2∂2R(∂R)2+C4a2H2˙R2∂2R+C5HR2˙R (48) +C6a4H∂2R(∂R⋅∂χ)+C7a4∂2R(∂χ)2+C8a2R(∂R)2+C9a2˙R(∂R⋅∂χ)+2a3f(R)δLδR∣∣∣1],

where with

 Λ:=a2Θ2XG˙R=a2σc2s˙R. (49)

The dimensionless coefficients are given by

 C1 = −HΘσc2s(1+2IG)−2˙ϕX(GX+XGXX)Hσc2sΘ2+H2σc4sΘ2, (50) C2 = σc2s[3−H2c2sΘ2(3+ϵ+2˙ΘHΘ)], (51) C3 = −H2˙ϕXGXΘ3, (52) C4 = 2H2˙ϕX(GX+XGXX)Θ3, (53) C5 = σ2c2sHddt(H2δc2sΘ2), (54) C6 = 2H˙ϕXGXΘ2, (55) C7 = σ4−˙ϕXGXΘ, (56) C8 = (57) C9 = σc2s(−2HΘ+σ2), (58)

where

 I := XKXX+2X23KXXX+H˙ϕGX+6X2G2X+5H˙ϕXGXX+6X3GXGXX+2H˙ϕX2GXXX (59) −2X3(2GϕX+XGϕXX).

The last term is the field equation which follows from the quadratic action,

 δLδR∣∣∣1=a[dΛdt+HΛ−σ∂2R], (60)

multiplied by

 f(R)=H˙σ4c2sΘ2σR2+Hc2sΘ2R˙R+14a2Θ2[−(∂R)2+∂−2∂i∂j(∂iR∂jR)]+12a2Θ[∂χ⋅∂R−∂−2∂i∂j(∂iR∂jχ)]. (61)

In deriving the above cubic action we have not performed any slow-roll expansion, so that we have kept full generality up to here. Taking the limit , , and , we can verify that the above equations reproduce the previous result derived for generic k-inflation models,  LS (); Kachru (). In particular, the , , and terms are absent in that case. The term is clearly a higher order term so that we will neglect it in the following.

Employing the in-in formalism, the 3-point function can be computed from the following formula:

 ⟨Rk1Rk2Rk3⟩=−i∫tt0dt′⟨[R(k1,t)R(k2,t)R(k3,t),Hint(t′)]⟩, (62)

where is some early time when the fluctuation is well inside the horizon, is a time several e-foldings after the horizon exit, and the interaction Hamiltonian is given by

 Hint(t)=−∫d3xa3[C1H˙R3+C2R˙R2+⋯]. (63)

We use Eqs. (27) and (28) to evaluate each contribution, which can be conventionally expressed as

 ⟨Rk1Rk2Rk3⟩ = (2π)7δ(3)(k1+k2+k3)P2RAk31k32k33, (64) A = ∑MAM. (65)

The power spectrum here is to be calculated for the mode with .

To proceed, we assume that const, which holds in a wide class of G-inflation models as described in Sec. II.3. We then immediately see that const, and , , and are all constant in time as well. The coefficients are explicitly given by

 C3=−(1+σ)2(σ−ϵ)(1+ϵ)3,C6=2(1+σ)(σ−ϵ)(1+ϵ)2,C7=4ϵ−σ(3−ϵ)4(1+ϵ). (66)

In order to evaluate the contributions from the () and () terms, we further assume that and are of the form

 IG = J1+J2c2s, (67) ˙ϕX2GXXH = ϱ1+ϱ2c2s, (68)

where , and are constants. In kinematically driven G-inflation GI () we indeed have const and const in the de Sitter limit. In potential driven G-inflation HGI () const and . Therefore, the assumptions made here are sufficiently general and reasonable. It then follows that and take the form

 C1 = D1c2s+E1c4s, (69) C4 = D4+E4c2s, (70)

where , and are constant and are given by

 D1 = −σ(1+σ)1+ϵ[1+2J1+2σ−ϵ+(1+σ)ϱ11+ϵ], (71) E1 = −σ(1+σ)1+ϵ[2J2−1+σ1+ϵ(1−2ϱ2)], (72) D4 = 2(1+σ)3(1+ϵ)3[σ−ϵ1+σ+ϱ1], (73) E4 = 2(1+σ)3(1+ϵ)3ϱ2. (74)

Each contribution can now be evaluated as

 A1 = 32σ(1−ϵ−s)∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1[D1I1(ns−1)+E1c2s∗I1(q′)], (75) A2 = 14∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1[3I2(ns−1)−3−ϵc2s∗(1+σ1+ϵ)2I2(q′)], (76) A3 = 12C3σc2s∗(1−ϵ−s)2∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1I3(q′), (77) A4 = 3σ(1−ϵ−s)2∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1[D4I4(ns−1)+E4c2s∗I4(q′)], (78) A6 = C68c2s∗(1−ϵ−s)∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1I6(q′) (79) A7 = C74σc2s∗∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1I7(q′), (80) A8 = 18∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1[−I8(ns−1)+1+ϵ−2sc2s∗(1+σ1+ϵ)2I8(q′)], (81) A9 = C9∗8∣∣∣Γ(q)Γ(3/2)∣∣∣2(k1k2k32k3t)ns−1I9(q′), (82)

where and are evaluated at sound horizon crossing, , and

 q′:=s−2ϵ1−ϵ−s. (83)

The -dependent functions are given by

 I1(z) := k21k22k23k3tcos(πz2)Γ(3+z)2, (84) I2(z) := (85) I3(z) := (k1⋅k2)k23ktcos(πz2)2+z2{Γ(1+z)+Γ(2+z)[k1k2+k2k3+k3k1k2t+(3+z)k1k2k3k3t]}+sym., (86) I4(z) := k21k22k23k3tcos(πz2)(6+z)Γ(3+z)12, (87) I6(z) := (k1⋅k2)k23ktcos(πz2)[(3+z)Γ(1+z)+(3+z)Γ(2+z)k3kt−Γ(3+z)k23k2t]+sym., (88) I7(z) := (k1⋅k