The four-point correlation function of graviton during inflation

We expand the Einstein-Hilbert action with a positive cosmological constant up to the fourth order in terms of gravity fluctuations, and then use the in-in formalism to calculate the four-point correlation function for gravitational waves, including both contact and exchange diagrams, generated during a period of exactly de Sitter expansion. In addition, we also present the general properties of the $n$-point function of graviton in terms of both circularly and linearly polarized states.


Introduction
The discovery of the accelerating expansion of the Universe through observations of distant supernovae [1,2], together with the theory of inflation [3,4,5,6], suggest the possibility that our universe approaches de Sitter geometries in both the far past and the far future. However, what is the nature of dark energy and inflation is still one of the deepest puzzles in modern physics. We believe that the quantum theory of gravity need to be understood before solving this puzzle.
In an asymptotically flat spacetime a unitary and analytic S-matrix can be well-defined. Even though there is no notion of an S-matrix in asymptotically anti de Sitter spacetime, one has the correlation function for the graviton which is equivalent to the correlation function of the stress tensor in the conformal field theory on the boundary [7]. Such a correspondence between gravity and conformal field theory is crucial for us to understand the quantum theory of gravity in the anti de Sitter space. On the other hand, tensor perturbations are also generated during inflation [8], whose direct detection would be taken as the evidence for inflation, and the non-Gaussianities of gravitational waves is a key feature of tensor perturbations as well as the scalar perturbations [9]. The three-point correlation function of graviton in de Sitter background has been calculated in [10]. In this paper we will investigate the general properties of n-point correlation function of graviton, and explicitly compute the four-point function of graviton in de Sitter background. Even though these gravity correlation functions appear to be outside the reach of the experiments occurring in the near future, we hope that the correlation functions of graviton in de Sitter background may also shed light on the quantum gravity theory in de Sitter space.
String theory is the most promising quantum theory of gravity. In principle, the diagram of the four-point function of graviton at tree level in string theory is unique (see Fig. 1). Unfortunately, string theory cannot be well formulated in de Sitter space. So we do not know how to calculate the four-point function of graviton in de Sitter background in string theory. Up to now, general relativity (GR) proposed by Albert Einstein one hundred years ago is still the simplest theory of gravity which is consistent with all of the experimental data, and every consistent string theory must contain a massless spin-2 state whose interactions reduce at low energy to general relativity. Here we will expand the Einstein-Hilbert action with a positive cosmological constant up to the fourth order in terms of gravity fluctuations, and utilize the in-in formalism to calculate the four-point correlation function of graviton, including both the contact and exchange diagrams, in de Sitter background. This paper is organized as follows. In section 2 we expand the Einstein-Hilbert action with a positive cosmological constant up to the fourth order in terms of gravitational waves. The parity of tensor perturbations and the general properties of n-point correlation function of graviton will be discussed in section 3. The four-point function of graviton in de Sitter background will be computed explicitly in section 4. Summary and discussion will be given in section 5. In addition, a brief review of tensor power spectrum and three-point function of graviton in de Sitter background is presented in Appendix A, and we fix the representation of polarization tensors and clarify the general four-momentum configuration into 6 parameters in Appendix B.
2 The action up to fourth order in terms of tensor fluctuations in de Sitter background In this paper we focus on the Universe whose dynamics is govern by the Einstein-Hilbert action with a positive cosmological constant Λ, namely where M pl ≡ (8πG) −1/2 is the reduced Planck mass. The background spacetime is described by nothing but de Sitter with metric where H is the Hubble constant related to Λ by H 2 = Λ/3.
During a period of exactly de Sitter expansion, there are only two physical degrees of freedom corresponding to the tensor fluctuations (gravitational waves). In order to make the dynamical degrees of freedom manifest, we call for the the ADM formalism [11], namely where N and N i are lapses and shifts. Now the action (1) reads where R (3) [g ij ] is the three-dimensional Ricci scalar associated with the metric g ij , and the symmetric tensor E ij is defined by here the dot denotes the derivative with respect to the cosmic time t, and E ≡ g ij E ij . Note that the extrinsic curvature is K ij = N −1 E ij . In this sense E ij E ij − E 2 can be taken as the"kinetic term" which governs the dynamics of the extrinsic curvature. In this form the spatial coordinate reparameterization is explicitly symmetric, while the time reparameterization is not so obvious since N i mixes up the space and time coordinates.
In the ADM form the metric g ij is dynamical, but N and N i are Lagrange multipliers. It is convenient to choose a gauge for g ij in which the time and spatial reparametrizations are fixed, and then the metric perturbation can be defined by where a = e Ht and the gravity fluctuations are transverse and traceless, i.e. ∂ i h ij = 0 and h ii = 0. Such a symmetric transverse and traceless metric g ij encodes two physical degrees of freedom of gravitational waves in GR.
From Eq. (4), the equations of motion for N and N i are given by In general, in order to work out the action up to fourth order in terms of gravitational waves, one only need to solve the above two equations up to second order in the ADM formalism because the third and fourth order perturbations multiplies the first and zeroth order constraint equation, respectively, which vanishes. See [9] for the discussion in detail. Therefore we decompose Furthermore, the shift N (n) i can be decomposed into its spin-0 scalar ψ (n) and spin-1 vector components β Here α (n) , β (n) i , ψ (n) ∼ O(h n ). Now one can solve Eqs. (7) and (8) order by order. Up to order of O(h 2 ), we get where the prime denotes the derivative with respect to the conformal time η ≡ t dt /a(t ).
Plugging the solutions in (12) and (13) into action (4) and expanding it up to the fourth order, we obtain where Here the coefficients c i (= 1) and d i (= 1) are used to denote the different interaction terms in the third and fourth order action.

Parity of tensor perturbations and the general properties of n-point correlation function of graviton
According to the quadratic action in Eq. (15), the linear perturbation equation can be solved in the Fourier space. So one can decompose the perturbations into momentum modes as follows e s ij (k) is the polarization tensor which has the following properties For convenience, the polarization tensor is normalized as Usually one can adopt the linear or circular polarization.

Circular polarization
In the literatures the circular polarization is widely used because they are the helicity eigenstates of gravitational waves. For the circular polarization, the polarization tensor takes the form where ±2 denote two circularly polarized states which propagate along z direction. From the above formula, the circular polarization tensors satisfy where λ = ±2. The most general solution of quantized tensor perturbation can be written by where ψ k is the mode function which satisfies and k = |k|. After choosing the standard Bunch-Davies vacuum [12], the solution of mode function is given by Now the creation and annihilation operators are normalized as From Eq. (24), the operator h λ k satisfies On the other hand, since gravity fluctuations h ij are tensor perturbations, h ij follows under parity transformation, or equivalently Since GR as well as the vacuum state is parity invariant (namely P|0 = |0 ), the n-point function of graviton satisfies Combining with Eqs. (28) and (30), we find which indicates that the n-point function for helicity state possess the symmetry that it becomes its complex conjugation when interchanging +2 ↔ −2.
When all k i lie in the same plane, without loss of generality by assuming that all k i lie in y-z plane, there is another symmetry under the mirror reflection about y-z plane.
. Thus under such a mirror reflection, or equivalently, Since k is in the y-z plane, k x = 0, k =k, and then the above equation reads For any k lies in y-z plane, e λ (k) can be expressed by e λ (k) = R 1 (θ)e λ (ẑ)R 1 (θ) T , where the definition of R 1 (θ) is given in (89). Notice that M yz commutes with R 1 (θ), and hencẽ From Eq. (22), one can easily get and thenẽ Combining with Eq. (35), we find Therefore, which indicates that the n-point function for circularly polarized states is always real when all momenta are in the same plane. This conclusion is quite nontrivial since h λ k is not a Hermitian operator for the circularly polarized state, i.e. h λ k = (h λ k ) † (see Eq. (28)). The bispectrum (three-point function) of graviton is real for the circularly polarized states because the three momentum vectors must lie in the same plane due to the energymomentum conservation. But the n(≥ 4)-point function for the circularly polarized state is complex in general.

Linear polarization
The linearly polarized states can be given by a linear combination of circularly polarized states, namely and the linear polarization tensors are related to the circular ones by where + and × denote two linearly polarized states. One can easily find that the linear polarization tensors satisfy For linear polarization, the most general solution of quantized tensor perturbation becomes where s = +1, −1 for +, × polarized states respectively. Here ψ k is same as that in circular polarization, and the commutators between the creation and annihilation operators are From Eq. (46), the operator h s k satisfies Similar to the circular polarization, under the parity transformation the operator h s Therefore where Eqs. (48) and (49) are taken into account. We conclude that the n-point functions of gravitational waves in terms of linear polarized state are always real whether all k i lie in the same plane or not. Because a real n-point function is convenient in analysis and the n-point function in terms of circularly polarized state can be easily derived from n-point function in terms of linearly polarized state by using (41), we will compute the four-point correlation function of graviton in terms of linearly polarized state in this paper.
Before closing this subsection, we also want to illustrate some simple properties of n-point function of graviton when all k i are in the same plane. Similar to the former subsection, assuming that all k i lie in the y-z plane, and under a mirror reflection M yz transformation, we have and Denoting m as the number of × linear polarized modes in the n-point function, we have Thus the n-point function for linearly polarized state equals zero if m is an odd number when all the momentum vectors lie in the same plan. This property can be seen explicitly for the four-point function in the next section.

The four-point correlation function of graviton in de Sitter background
In this section we will use the in-in formalism to calculate the four-point correlation function of graviton during a period of exactly de Sitter expansion, including the contact and exchange diagrams. A brief review of tensor power spectrum and the three-point function of graviton is given in Appendix A.

In-in formalism
In this subsection we will introduce the Schwinger-Keldysh in-in formalism [13] to calculate the n-point function of gravity fluctuations during inflation. The expectation value of some product Q(t) of field operators at fixed time t is given by where |in is the initial state at very early times when the wavelength is deep inside the horizon. In the interaction picture, we have where H I is the interaction part of the Hamiltonian. Here T denotes a time-ordered product,T denotes an anti-time-ordered product, and Q I is the product Q in the interaction picture. Utilizzing the Dyson series, one can get Lagrangian for the gravity fluctuations in de Sitter background, the interaction part of Hamiltonian follows that H I = − dx 3 L I .
In general the the n-point correlation function of gravitational waves in the de Sitter background takes the form and where δ (3) ( n i=1 k i ) implies the energy-momentum conservation. In the next two subsections we will use the in-in formalism to calculate the amplitude A s 1 s 2 s 3 s 4 (k 1 , k 2 , k 3 , k 4 ) which is expected to be different for different interaction term.

Contact diagram
In this subsection we will work out the contribution to the four-point correlation function of graviton from the contact diagram (see Fig. 2). At the tree level, we only need to use d i where H I . Remind that Eq. (57) is not applicable in this case because the operator h s 1 k 1 h s 2 k 2 h s 3 k 3 h s 4 k 4 is not Hermitian. Before using the above in-in formalism to compute four-point function of graviton, we need to find the solution of ψ (2) from the constraint equation (13). Considering Eq. (18), we can easily find where k ij = k i + k j , k a ij = k a i + k a j and k ij = |k ij |. Now adopting the in-in formalism in Eq. (61), after a lengthy but straightforward computation, we get where K ≡ k 1 + k 2 + k 3 + k 4 , and e l ij ≡ e s l ij (k l ). Here the divergent parts from performing the time integration are counteracted. In order to sketch out some simple features of the four-point function of graviton from the contact diagram, we will consider some special configurations in the momentum space.
First of all, we consider the bilateral squeezed limit in which k 3 = k 4 = −k 12 /2, k 1 = k 2 =k and k 3 = k 4 = k → 0. See Fig. 3. In this limit, all k i lie in the same plane, and there are only four non-zero amplitude, namely We see that the interaction term denoted by d 1 in the fourth order action does not make a contribution to the four-point function in the bilateral squeezed limit, and all of the non-zero amplitudes have the same scaling behavior hhhh ∝ T ∝ 1/(k 3 k 6 ) which is divergent as 1/k 6 in the limit of k → 0. This is similar to the local-form four-point function of scalar perturbation from the contact diagram.

Exchange diagram
The third order perturbation Hamiltonian can also make a contribution to the four-point function of graviton through the exchange diagram (see Fig. 6). Applying the second order in-in formalism, the four-point function from the exchange diagram takes the form where the third order interaction Hamiltonian is given by It is remarkable that when doing time double integrals from −∞ to 0 we will get several divergent terms due to high negative power of conformal time.Fortunately, these divergences cancel each other exactly when we do permutations of k 1 , k 2 , k 3 and k 4 . Finally c i c j where Note that i, j = 1, 2, 3, 4 represent ith, jth momentum and need not do summation, while the indices a, b, c, d = 1, 2, 3 are the component indices.
Analogous to the analysis of contact diagram, in the bilateral squeezed limit, we find which indicates that there are only two non-zero components, i.e. T ++++ and T ××++ and only the interaction term denoted by c 2 in the third order action contributes to the four-point function in the bilateral squeezed limit. The non-zero components diverge on behavior of 1/k 6 in the limit of k → 0 as well.
Secondly, let's consider the diamond squeezed limit, namely k 1 = k 2 = k 3 = k 4 =k and k 12 → 0. In this limit, we have (77) Again we see that only the interaction term denoted by c 2 in the third order action makes a contribution to the four-point function in this limit. Unlike that from contact diagram, the amplitude of four-point function from the exchange diagram goes like 1/k 3 12 which is divergent in the limit of k 12 → 0. This behavior is similar to the local-form four-point function of scalar perturbation from exchange diagram.

Summary and discussion
In this paper we dig out some general properties of the n-point function of graviton due to the parity, and explicitly compute the four-point correlation function of graviton, including both the contact and exchange diagrams, in the de Sitter approximation in the framework of GR. One can expect that the the simplicity of the results presented in this paper is lost when we go to an inflationary background. However, our results are still the leading approximation in the slow-roll expansion. In the framework of GR, we find hhhh / hh 3 ∼ O(1). Furthermore, the four-point function of graviton in the de Sitter background has similar behavior to the local-form four-point function of scalar perturbation in some special momentum configurations.
The three-point function of graviton has been extended to the inflationary background in [14], generalized G-inflation in [15] and Horava-Lifshitz gravity in [16]. It is worthy coming back to the four-point function of graviton in more general scenario in the near future.
the permutations into account, there are only two independent non-zero components In the squeezed limit, i.e. k 1 = k 2 = k and k 3 → 0, we have which says that only the interaction term denoted by c 2 makes contributions to the threepoint function in the squeezed limit. Note that s = +1, −1 for +, × polarized states respectively. From the above formula, there are only two non-zero components, i.e.
in the squeezed limit (k 3 → 0). It is similar to the local-form three-point function of scalar perturbation during inflation.

B Polarization tensors
In this section, we will fix the representation of polarization tensors. Due to the momentum conservation k 1 + k 2 + k 3 + k 4 = 0 for the four-point function, the momentum configuration can be described by six parameters, i.e. k 12 , θ 1 , θ 2 , φ 1 , φ 2 and α, which are illustrated in Fig. 7. Here, without loss of generality, we assume that k 1 and k 2 lie in the x-z plane, and −k 12 stays alongẑ = (0, 0, 1) T direction. And then k i can be expressed by where R i (θ i ) denotes rotating θ i about x i axis: