On Graviton non-Gaussianities in the
Effective Field Theory of Inflation

Giovanni Cabass111gcabass@ias.edu,a, David
Stefanyszyn222david.stefanyszyn@nottingham.ac.uk,b,c, Jakub
Supeł333js2154@cam.ac.uk,d and Ayngaran Thavanesan444at735@cam.ac.uk,e

$a$ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ
08540, United States

$b$ School of Mathematical Sciences & School of Physics and Astronomy,
University of Nottingham, University Park, Nottingham, NG7 2RD, UK

$c$ Nottingham Centre of Gravity, University of Nottingham, University Park,
Nottingham, NG7 2RD, UK

$d$ Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

$e$ Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK

We derive parity-even graviton bispectra in the Effective Field Theory of
Inflation (EFToI) to all orders in derivatives. Working in perturbation
theory, we construct all cubic interactions that can contribute to tree-level
graviton bispectra, showing that they all come from EFToI operators containing
two or three powers of the extrinsic curvature and its covariant derivatives:
all other operators can be removed by field redefinitions or start at higher-
order in perturbations. For operators cubic in the extrinsic curvature, where
the single-clock consistency relations are satisfied without a correction to
the graviton two-point function, we use the Manifestly Local Test (MLT) to
efficiently extract the effects of evolving graviton fluctuations to the end
of inflation. Despite the somewhat complicated nature of the bulk
interactions, the final boundary correlators take a very compact form. For
operators quadratic in the extrinsic curvature, the leading order bispectra
are a sum of contact and single exchange diagrams, which are tied together by
spatial diffeomorphisms, and to all orders in derivatives we derive these
bispectra by computing the necessary bulk time integrals. For single exchange
diagrams we exploit factorisation properties of the bulk-bulk propagator for
massless gravitons and write the result as a finite sum over residues. Perhaps
surprisingly, we show these single exchange contributions have only total-
energy poles and also satisfy the MLT.

###### Contents

  1. 1 Introduction
  2. 2 Effective Field Theory of Inflation
    1. 2.1 Eliminating the three-dimensional Ricci tensor
  3. 3 A general action to cubic order
    1. 3.1 Two building block operators
    2. 3.2 Three building block operators
    3. 3.3 Converting to conformal time
    4. 3.4 A comment on higher-order time derivatives
  4. 4 Generalities of contact and single exchange diagrams
    1. 4.1 Contact diagrams
    2. 4.2 Single exchange diagrams
    3. 4.3 Boundary terms
    4. 4.4 Spinor-helicity formalism
  5. 5 Type-I bispectra
    1. 5.1 Single exchange diagram
    2. 5.2 Contact diagram
    3. 5.3 Putting everything together
    4. 5.4 Checking the consistency relations
  6. 6 Type-II bispectra
  7. 7 Conclusions and outlook

##  1 Introduction

Graviton self-interactions are heavily constrained in exact de Sitter space
[1]. The power of full de Sitter isometries ensures that there are only three
contributions to the cubic part of the Wavefunction of the Universe (WFU): two
of these are parity-even and arise from the Einstein-Hilbert action and a six-
derivative correction in the form of a Riemann cubed operator, while the other
is parity-odd and comes from a parity-odd Riemann cubed operator. In
perturbation theory, expectation values of graviton fields at the late-time
boundary of de Sitter space can be extracted from knowledge of the
wavefunction coefficients that appear in the WFU and upon computing the
corresponding bispectra, only the two parity-even contributions survive [2,
3]555The parity-odd interaction contributes only a pure phase to the
wavefunction so drops out when we compute expectation values.. This rigidity
of graviton interactions in de Sitter space very nicely parallels a similar
story for the $S$-matrix in Minkowski space [4, 5, 6, 7].

During inflation, however, additional shapes of graviton bispectra can arise
due to the breaking of de Sitter boosts. Indeed, in the Effective Field Theory
of Inflation (EFToI) [8] a scalar field with a time-dependent background
profile spontaneously breaks de Sitter boosts thereby allowing for a richer
structure of graviton self-interactions which only need to be invariant under
spatial diffeomorphisms. The graviton action up to cubic order in the
transverse, traceless fluctuation $\gamma_{ij}$ is fixed to be that of the
Einstein-Hilbert action to leading order in derivatives with corrections
containing at least three derivatives [9, 10]. The leading corrections to the
Einstein-Hilbert action, and the resulting bispectra, have been computed in
[11, 12] where there are two new shapes: one comes from a parity-even operator
while the second comes from a parity-odd operator. Some four-derivative
operators have also been explored in [11]. One can go even further and break
spatial diffeomorphisms as in e.g. Solid Inflation [13] to yield even more
self-interactions, but now in the presence of a (small) graviton mass. A
zoology of graviton interactions during inflation has been derived in [14]
where it is clear that even a small breaking of maximal symmetries opens up
many new possibilities.

In this work we aim to perform a more general analysis within the EFToI.
Working in perturbation theory, and assuming invariance under spatial
translations, rotations and scale invariance, we construct parity-even
graviton bispectra to all orders in derivatives. Throughout we work within the
WFU formalism and derive expectation values only at the end of our
computations, and as a consistency check we verify that our results satisfy
the leading order consistency relations of single-clock cosmologies [15, 16,
17, 18]. The highly constrained parity-odd bispectra in the EFToI have been
computed in [11, 12, 19, 20], and in this work we fill the parity-even gap.
For parity-even interactions the number of bispectra grows unbounded as we
increase the number of derivatives, which is in stark contrast to the parity-
odd situation where one can use unitarity methods to prove that only a single
tree-level shape for each helicity configuration is allowed [12]666Even if we
further break spatial diffeomorphism as in e.g. Solid inflation, only three
shapes are allowed [12] (see also [21])..

Let us explain our motivation for this work. Given that gravitational
interactions are so heavily constrained in exact de Sitter space, it is
interesting to carve out the space of consistent self-interactions during
inflation where the most powerful de Sitter symmetries are broken. The EFToI
is the natural place to start given that the unbroken spatial diffeomorphisms
are still expected to provide non-trivial relations between different
operators, and they ensure that the graviton remains massless unlike in Solid
Inflation [13] and massive gravity [22]. Our long-term objective is to gain a
solid understanding of Quantum Field Theory on inflationary backgrounds and
graviton $n$-point functions are certainly objects of interest. In flat-space,
the tree-level four-point scattering amplitude for massless gravitons at low
energies is completely fixed by symmetries, locality and consistent
factorisation (see e.g. [23, 6]) while higher-point amplitudes can be
extracted from three-point ones using BCFW momentum shifts [24, 4]. Boost-
breaking amplitudes have also been constructed in [7, 25]. Striving for a
comparable understanding of cosmological $n$-point functions is one of the
primary goals of the bootstrap approach to constructing cosmological
observables [1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 7, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69], and this endeavour requires
theoretical data from which we can learn about the allowed structures.
Ultimately we would like to distinguish between different inflationary models
directly at the level of observables and we hope that our work in this paper
will contribute to this goal.

In some sense our work builds on [12] where graviton bispectra were
constructed from very general principles. Using the aforementioned symmetries
(no de Sitter boosts), locality and unitarity, all possible shapes were
derived under the assumption that the graviton mode functions are the usual
massless de Sitter ones. Symmetries and the assumption of a Bunch-Davies
vacuum777For a discussion on graviton bispectra with non-Bunch-Davies initial
states see e.g. [70]. were used to write down a general ansatz for tree-level
wavefunction coefficients which can then be constrained by the Manifestly
Local Test (MLT) derived in [31]. This test demands that wavefunction
coefficients, when expanded for small energies888Even though in cosmology we
don’t have time translation symmetry, we follow standard convention and still
refer to the norms of spatial momenta as energies., must not contain a term
linear in any of the energies and follows from a simple property of the
massless mode functions. Unitarity, in the form of the Cosmological Optical
Theorem (COT) [27], was used to deduce which part of the wavefunction
coefficients contribute to expectation values and in particular to the
bispectra. For parity-even interactions, both rational and logarithmic
contributions to the wavefunction coefficients also contribute to the
bispectra, while for parity-odd interactions only the coefficients of
logarithms contribute. This makes parity-odd bispectra highly-constrained,
since logarithms in the wavefunction are rare, and indeed only three shapes
are allowed with only one linear combination appearing in the EFToI. In [12]
no assumption was made about how de Sitter boosts are spontaneously broken,
i.e. no assumption was made about the details of the underlying inflationary
model, and our aim in this paper is to derive the parity-even EFToI subset of
these general graviton bispectra.

To achieve this goal we use a combination of bulk and bootstrap tools. We
start with the general action of the EFToI [8] and show that all on-shell
graviton three-point functions can be derived from corrections to the
Einstein-Hilbert action that are built out of the extrinsic curvature and its
covariant derivatives only. We use geometric identities and field
redefinitions to arrive at this conclusion. Operators that can contribute to
graviton bispectra at tree-level are at most cubic in the extrinsic curvature,
and we derive all quadratic and cubic vertices for the transverse, traceless
graviton that come from these covariant operators. Operators that are
quadratic in the extrinsic curvature correct both the quadratic and cubic
action, while operators that are cubic in the extrinsic curvature correct only
the cubic vertices. In this latter case the only contributions to the
bispectra at tree-level come from WFU contact diagrams and to compute these
diagrams we use the techniques of [12]. We are able to write down closed form
expressions to all orders in derivatives with the freedom reduced to symmetric
polynomials in the external energies with their degrees fixed by scale
invariance. We refer to these bispectra as Type-II. In the former case both
contact and single exchange diagrams can contribute and for reasons that will
be explained in detail in the main body of this paper, we explicitly compute
the bulk time integrals to extract the bispectra in this case. For exchange
diagrams we use the representation of [33], which relies on a neat
factorisation property of the bulk-bulk propagator, and present the results in
a very compact form to all orders in derivatives. We refer to these bispectra
as Type-I. Such single exchange diagrams have also been very recently studied
and bootstrapped in the context of the cosmological collider in [58, 59]. For
contact and exchange diagrams alike, the largest contribution in perturbation
theory comes from diagrams that are linear in the couplings so we focus only
on these. As pointed out in [12], Type-II bispectra can be large thanks to the
weaker constraints on their size coming from the validity of perturbation
theory since they do not come with a correction to the graviton two-point
function.

Along the way we show that single exchange diagrams that contribute to the
Type-I bispectra are only singular when the total energy goes to zero. This is
somewhat surprising since in general such a diagram could have two singular
points arising when the sum of energies entering either of the two vertices
goes to zero. However, the quadratic corrections in the EFToI are such that
the additional singularities are always cancelled. In hindsight this might
have been expected since any other singularities would seem to violate the
EFToI consistency relations, and one could also apply the locality arguments
of [42] to arrive at the same conclusion. We also show that these single
exchange diagrams satisfy the MLT. This is certainly not true for general
exchange diagrams that arise due to quadratic mixing, see e.g. [58, 59], but
the vertices in the EFToI ensure that the MLT is always satisfied. These two
facts ensure that all the crucial assumptions of the analysis in [12] remain
true for both contact and exchange diagrams, so both types of bispectra that
we will derive in this paper are captured by that analysis. The easiest way to
directly distinguish between Type-I and Type-II bispectra is to take the soft
limit and check the consistency relations [15, 16, 17, 18], since for Type-I
the power spectrum needs to be modified to satisfy them, while Type-II
bispectra satisfy the consistency relations without the need for such a
correction.

The rest of the paper is organised as follows. In the following section,
Section 2, we very briefly review the EFToI and show, for the first time, that
all graviton bispectra can be derived from extrinsic curvature operators only.
In Section 3 we derive the general Lagrangian for the transverse, traceless
graviton that comes from these operators and can contribute to wavefunction
coefficients and bispectra. In Section 4 we comment further on the Feynman
diagrams that contribute to the cubic part of the wavefunction and show that
the exchange diagrams have the same singularity structure as the contact ones,
and that they satisfy the MLT. In this section we also remind the reader how
cosmological correlators are extracted from wavefunction coefficients. In
Sections 5 and 6 respectively, we construct the Type-I and Type-II bispectra,
present compact expressions to all orders in derivatives, and verify that the
leading order consistency relations are satisfied. Finally, we conclude in
Section 7.

#### Notation and conventions

Throughout we work with the mostly positive metric signature and our Fourier
transformation is defined as

$\displaystyle f(\textbf{x})$
$\displaystyle=\int\dfrac{d^{3}\textbf{k}}{(2\pi)^{3}}{f}(\textbf{k})\exp(i\textbf{k}\cdot\textbf{x})\equiv\int_{\textbf{k}}{f}(\textbf{k})\exp(i\textbf{k}\cdot\textbf{x})\,,$
(1.1) $\displaystyle{f}(\textbf{k})$ $\displaystyle=\int
d^{3}\textbf{x}\,f(\textbf{x})\exp(-i\textbf{k}\cdot\textbf{x})\equiv\int_{\textbf{x}}f(\textbf{x})\exp(-i\textbf{k}\cdot\textbf{x})\,.$
(1.2)

We parameterise the Wavefunction of the Universe $\Psi$ at conformal time
$\eta_{0}$ in terms of the graviton’s helicities as

$\displaystyle\Psi[\eta_{0},\gamma({\bf
k})]=\text{exp}\left[-\sum_{n=2}^{\infty}\frac{1}{n!}\sum_{h_{i}=\pm}\int_{{\textbf{k}}_{1},\ldots,{\bf{k}}_{n}}\psi^{h_{1}\ldots
h_{n}}_{n}({\bf k_{1}}\ldots{\bf
k_{n}})(2\pi)^{3}\delta^{3}\left(\sum\textbf{k}_{a}\right)\gamma^{h_{1}}({\bf
k}_{1})\ldots\gamma^{h_{n}}({\bf k}_{n})\right].$ (1.3)

Here the ever-present momentum conserving delta function is a consequence of
the unbroken spatial translations and we write this explicitly so it is not
included in the wavefunction coefficients $\psi_{n}$. We write the
corresponding cosmological correlators as $B_{n}$ (also with the delta
function dropped). The relations between $B_{n}$ and $\psi_{n}$ will be given
in the main body of the paper. In this WFU expression,
$\gamma^{h}(\textbf{k})$ is the spin-$h$ Fourier mode of the graviton and is
related to the position space graviton by

$\displaystyle\gamma_{ij}(x,\eta)=\int_{\textbf{k}}e^{i\textbf{k}\cdot\textbf{x}}\sum_{h=\pm}e_{ij}^{h}(\textbf{k})\gamma_{h}(\textbf{k},\eta)\,,$
(1.4)

and for us $\gamma_{h}(\textbf{k},\eta)$ is given by the usual de Sitter mode
functions. The graviton polarisation tensor satisfies the following relations:

$\displaystyle e_{ii}^{h}(\textbf{k})$
$\displaystyle=k^{i}e^{h}_{ij}(\textbf{k})=0$ $\displaystyle\text{(transverse
and traceless)}\,,$ (1.5) $\displaystyle e_{ij}^{h}(\textbf{k})$
$\displaystyle=e_{ji}^{h}(\textbf{k})$ $\displaystyle\text{(symmetric)}\,,$
(1.6) $\displaystyle e_{ij}^{h}(\textbf{k})e_{jk}^{h}(\textbf{k})$
$\displaystyle=0$ $\displaystyle\text{(lightlike)}\,,$ (1.7) $\displaystyle
e^{h}_{ij}(\textbf{k})e^{h^{\prime}}_{ij}(\textbf{k})^{\ast}$
$\displaystyle=4\delta_{hh^{\prime}}$ $\displaystyle\text{(normalization)}\,,$
(1.8) $\displaystyle e_{ij}^{h}(\textbf{k})^{\ast}$
$\displaystyle=e_{ij}^{h}(-\textbf{k})$ $\displaystyle\text{($\gamma_{ij}(x)$
is real)}\,.$ (1.9)

We will often encounter polynomials in the three external energies that are
fully symmetric and we will express these in terms of the elementary symmetric
polynomials:

$\displaystyle k_{T}$ $\displaystyle=k_{1}+k_{2}+k_{3},$ (1.10) $\displaystyle
e_{2}$ $\displaystyle=k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3},$ (1.11) $\displaystyle
e_{3}$ $\displaystyle=k_{1}k_{2}k_{3}.$ (1.12)

##  2 Effective Field Theory of Inflation

We work within the EFToI where the inflationary background is driven by a
single scalar degree of freedom, the inflaton. In the language of symmetry
breaking, this set up corresponds to the case where only time diffeomorphisms
are broken by the background i.e. we couple a superfluid to gravity [14]. Time
diffeomorphisms are nevertheless non-linearly realised by the inflationary
perturbations. The EFToI provides a formalism to capture the most general
action for the graviton and scalar fluctuations on a quasi-de Sitter
background, with operators organised in a derivative expansion and tadpole
cancellation guaranteed to all orders. We begin this section by briefly
reviewing the EFToI before showing that graviton vertices up to cubic order
are captured by operators built out of the extrinsic curvature only.

Since we are interested in graviton interactions, we work with the following
line element

$ds^{2}=-dt^{2}+a^{2}(t)(e^{\gamma})_{ij}dx^{i}dx^{j},$ (2.1)

where $a=a(t)$ is the scale factor and we define the Hubble parameter
$H=\dot{a}/a$. For now we work in cosmological time but later on we will
convert to conformal time which is more suitable for computing late-time
cosmological correlators. Relative to the usual ADM formalism, we have set the
lapse variable to unity and the shift together with the curvature perturbation
$\zeta$ to zero. Usually we would integrate out the non-dynamical parts of the
metric which would introduce additional interactions for $\gamma$ and $\zeta$,
but up to cubic order this procedure does not alter the $\gamma$ interactions
so we can safely work with (2.1)999Since the graviton does not mix with the
non-dynamical modes at linear order, it must appear at least quadratically in
the solutions to the constraint equations. Plugging these solutions back into
the action can therefore only affect the graviton interactions at quartic
order and higher.. We fix the gauge completely by taking the graviton to be
transverse and traceless: $\gamma_{ii}=0=\partial_{i}\gamma_{ij}$101010Note
that in this gauge the metric determinant is independent of $\gamma_{ij}$ and
is fixed by its background value.. This gauge is usually referred to as
unitary gauge where all degrees of freedom live in the metric i.e. the
inflaton perturbation has been eaten by the metric. In this gauge the most
general action that we can write down is the one consistent with spatial
diffeomorphisms [8]:

$\displaystyle S=\int
d^{4}x\sqrt{-g}\,F(R_{\mu\nu\rho\sigma},g^{00},K_{\mu\nu},\nabla_{\mu},t),$
(2.2)

where all free indices are, in general, upper $0$’s. For our purposes,
however, there are simplifications. All $t$ dependence will be fixed by scale
invariance, since we are working in the scale invariant approximation, i.e. we
assume that $H$ and $\dot{H}$ vary slowly and restrict all $t$ dependence to
be that coming from the metric. In practice this means that we work with a
fixed de Sitter background metric, so all correlators we compute are valid up
to small slow-roll corrections [16]. Scale invariance will be most transparent
when we convert to conformal time. This is a technically natural set-up since
in the $\phi$ language it corresponds to an approximate shift symmetry for the
inflaton. We also take $g^{00}=-1$, since we don’t include the lapse, and this
means we can write all temporal indices downstairs. The extrinsic curvature of
constant-$t$ hyper-surfaces, for our metric, is given by
$K_{\mu\nu}=\Gamma^{0}_{\mu\nu}$ with non-zero components
$K_{ij}=\Gamma_{ij}^{0}=\dot{g}_{ij}/2$. Throughout we will actually work with
the perturbed extrinsic curvature defined by $\delta
K_{\mu\nu}=K_{\mu\nu}-Hh_{\mu\nu}$, where
$h_{\mu\nu}=g_{\mu\nu}+\delta^{0}_{\mu}\delta^{0}_{\nu}$.

Now it is well-known that the full 20 components of the Riemann tensor are not
independent of the extrinsic curvature. Indeed, only the three-dimensional
part of the full four-dimensional object is independent and we denote this
object by $\tilde{R}_{ijkl}$111111Throughout we use a tilde to represent
three-dimensional objects. (see e.g. [71]). A further simplification comes
from the fact that the Weyl tensor is identically zero in three dimensions, so
the three-dimensional Riemann tensor is completely determined by the three-
dimensional Ricci tensor $\tilde{R}_{ij}$. This is most easily seen by
counting the number of degrees of freedom of the Weyl tensor. The Riemann
tensor in $D$ dimensions has $\frac{1}{12}D^{2}(D^{2}-1)$ degrees of freedom,
while its trace has $\frac{1}{2}D(D+1)$ degrees of freedom. The Weyl tensor is
the traceless part of the Riemann tensor, so in three dimensions it has
$6-6=0$. We can also use only the spatial components of $\delta K_{\mu\nu}$
without loss of generality. Our general action is therefore

$\displaystyle S=S_{0}+\int d^{4}x\sqrt{-g}\,F(\tilde{R}_{ij},\delta
K_{ij},\nabla_{0},\nabla_{i}),$ (2.3)

where we have separated out $S_{0}$ which includes the Einstein-Hilbert action
plus the terms required to make the unperturbed metric a consistent solution.
This part of the action describes the minimal set-up of slow-roll inflation
with all other operators describing higher derivative corrections. Any
operators not contained in $S_{0}$ start at quadratic order in $\gamma$ so do
not affect the tadpole cancellation [8]: they capture all different theories
of cosmological perturbations on the same FRW background. Indeed, $\delta
K_{ij}$ is a perturbed object by construction and $\tilde{R}_{ij}$ vanishes on
the background. We have [8]

$\displaystyle S_{0}=\frac{M_{\text{pl}}^{2}}{2}\int
d^{4}x\sqrt{-g}\,\big{[}R-2(\dot{H}+3H^{2})+2\dot{H}g^{00}\big{]},$ (2.4)

and we remind the reader that we are working in the limit where $H$ and
$\dot{H}$ do not vary significantly in one Hubble time: all time dependence of
the action is slow-roll suppressed. We note that the perturbed action
$S-S_{0}$ is derivatively coupled: in the general EFToI the only terms without
derivatives are polynomials in $g^{00}$ and for us these are all trivial.

There is one final simplification we can make before moving onto field
redefinitions: since $\tilde{R}_{ij}$ and $\delta K_{ij}$ are three-
dimensional objects we only need to use the three-dimensional covariant
derivative $\tilde{\nabla}_{i}$. Indeed, out of the non-vanishing Christoffel
symbols only the three-dimensional one $\tilde{\Gamma}^{i}_{jk}$ cannot be
expressed in terms of the extrinsic curvature. We can also treat all temporal
covariant derivatives as partial ones $\partial_{t}$ for the same reason. We
therefore have

$\displaystyle S=S_{0}+\int d^{4}x\sqrt{-g}\,F(\tilde{R}_{ij},\delta
K_{ij},\partial_{t},\tilde{\nabla}_{i}).$ (2.5)

Of course in some cases it will be wise to use covariant derivatives and only
make this final simplification when we come to expand the action.

###  2.1 Eliminating the three-dimensional Ricci tensor

We will now show that to construct graviton bispectra it is sufficient to work
with the restricted action that does not depend on $\tilde{R}_{ij}$. Both
$\tilde{R}_{ij}$ and $\delta K_{ij}$ start at linear order in perturbations so
the graviton action up to cubic order, that comes in addition to the Einstein-
Hilbert part, comes from EFToI operators that are at most cubic in these
building blocks.

First consider operators that are constructed out of three building blocks
i.e. are of the schematic form: $\tilde{R}^{3}$, $\tilde{R}^{2}\delta K$,
$\tilde{R}\delta K^{2}$ and $\delta K^{3}$ where we have suppressed indices
and derivatives. Above we have argued that we can always use partial time
derivatives, while for spatial derivatives the difference between using a
partial one and a covariant one is captured by $\tilde{\Gamma}^{i}_{jk}$ which
starts at linear order in perturbations. It follows that the difference
between using a partial spatial derivative and a three-dimensional covariant
one is $\mathcal{O}(\gamma^{2})$, which for three building block operators
will only introduce differences at $\mathcal{O}(\gamma^{4})$. For our
interests we can therefore treat all derivatives as partial ones for three
building block operators. Now such operators do not contribute to the
quadratic action for the graviton so to compute the cubic wavefunction
coefficient we only need to consider contact diagrams where all external lines
are on-shell in which case any appearances of $\partial^{2}\gamma_{ij}$ are
degenerate with $\dot{\gamma}_{ij}$, and its time derivatives, by the graviton
equation of motion:

$\displaystyle\ddot{\gamma}_{ij}+3H\dot{\gamma}_{ij}-a^{-2}\partial^{2}\gamma_{ij}=0.$
(2.6)

As we will discuss below, the two-derivative quadratic action can be brought
into the Einstein-Hilbert form without loss of generality [9], with higher-
derivative corrections that we treat perturbatively such that (2.6) is always
the on-shell relation. Once we remove all copies of $\partial^{2}\gamma_{ij}$,
all remaining interactions can be derived from $\delta K_{ij}$ and its
derivatives since at linear order we have $\delta
K_{ij}\sim\dot{\gamma}_{ij}$. So for three building blocks operators we can
safely ignore operators containing
$\tilde{R}_{ij}\sim\partial^{2}\gamma_{ij}$, at least when computing the on-
shell cubic vertices. We can also make the redundancy of any operators
containing $\tilde{R}_{ij}$ manifest with field redefinitions, as we will show
below. We provide more information about the contact diagrams we will be
computing in Section 4, and provide the general on-shell action coming from
three building block operators in Section 3.

Two and single building block operators are slightly more complicated but
nevertheless we can still reduce the action to one constructed from $\delta
K_{ij}$ only using field redefinitions. Let’s start by considering which
operators can contribute to the action up to cubic order. Since the graviton
is transverse all vector quantities start at quadratic order in perturbations:
$\nabla_{i}\tilde{R}_{ij}\sim\mathcal{O}(\gamma^{2})$ and $\nabla_{i}\delta
K_{ij}\sim\mathcal{O}(\gamma^{2})$. It follows that all spatial covariant
derivatives must be contracted with other spatial covariant derivatives since
any other scalar quantities will start at quartic order in perturbations. The
action for two building block operators is therefore

$\displaystyle S=S_{0}+M_{\text{pl}}^{2}\int d^{4}x\sqrt{-g}\,\left(\delta
K^{ij}\mathcal{O}_{(0)}\delta K_{ij}+\tilde{R}^{ij}\mathcal{O}_{(1)}\delta
K_{ij}+\tilde{R}^{ij}\mathcal{O}_{(2)}\tilde{R}_{ij}\right),$ (2.7)

where $\mathcal{O}_{(i)}$ are derivative operators constructed out of
$\nabla_{0}$ and $\tilde{\nabla}^{2}=\tilde{\nabla}^{i}\tilde{\nabla}_{i}$,
and we have integrated by parts to move all derivatives onto a single building
block. One might worry about boundary terms that could affect the
wavefunction, but in Section 4 we will show that there are always enough
derivatives for the boundary terms to vanish. For $\mathcal{O}_{(i)}$, $i$
counts its negative mass dimension which we can fix using $M_{\text{pl}}$:

$\displaystyle\mathcal{O}_{(i)}=\frac{1}{M_{\text{pl}}^{i}}\sum_{m,n}b_{m,n}\nabla_{0}^{m}\tilde{\nabla}^{2n},$
(2.8)

where $b_{m,n}$ are constant couplings with mass dimension $-(m+2n)$ (recall
that we are working with a dimensionless $\gamma_{ij}$). Note that we have not
included any terms that depend on the trace of the extrinsic curvature since
this object vanishes to all orders in $\gamma$121212This can be most easily
seen by writing $2K=g^{ij}\dot{g}_{ij}=g^{-1}\dot{g}$ where $g$ is the
determinant of the spatial metric $g_{ij}$. Since the graviton is traceless,
it drops out of $g$ and therefore drops out of the trace of the extrinsic
curvature., and we have not included any terms that depend on the three-
dimensional Ricci scalar $\tilde{R}$ since a linear term can be eliminated by
Gauss-Codazzi relations [9]131313Eliminating a linear term in $\tilde{R}$ is
more involved if we allow for time-dependent couplings but it can still be
done [9, 10]., while non-linear terms start at $\mathcal{O}(\gamma^{4})$ since
$\tilde{R}\sim\mathcal{O}(\gamma^{2})$.

Now it was shown in [9] that we can set the coefficient of $\delta
K^{ij}\delta K_{ij}$ to any value we like such that that the quadratic action
takes the canonical form. This is achieved thanks to conformal and disformal
transformations of the metric. We can also go further by considering
additional field redefinitions of the spatial components of the metric of the
form

$\displaystyle\delta
g^{ij}=\mathcal{O}_{(2)}\tilde{R}^{ij}+\mathcal{O}_{(1)}\delta K^{ij}.$ (2.9)

Note that different copies of $\mathcal{O}_{i}$ with the same $i$ are not
necessarily the same derivative operators. The linear variation of the minimal
action can be written in terms of the EFToI operators $\tilde{R}_{ij}$ and
$\delta K_{ij}$. We find

$\displaystyle\delta S_{0}=\frac{M_{\text{pl}}^{2}}{2}\int
d^{4}x\sqrt{-g}\,\big{[}\tilde{G}_{ij}+\nabla_{0}\delta K_{ij}+3H\delta
K_{ij}\big{]}\delta g^{ij},$ (2.10)

where in computing the variation of the Einstein-Hilbert action we have
dropped contributions that depend on $g^{ij}\delta K_{ij}$ since as we
mentioned above this object vanishes to all orders in $\gamma$, and we have
dropped two building block terms since under our chosen field redefinition
these will only correct three building block operators which we have already
considered. The three-dimensional Einstein tensor is given by
$\tilde{G}_{ij}=\tilde{R}_{ij}-\frac{1}{2}g_{ij}\tilde{R}$. One might wonder
why we have only transformed $\delta S_{0}$: this is simply because we are
treating all two building block operators as small corrections to the
Einstein-Hilbert action. We have done this such that we can use the usual on-
shell condition (2.6). This means that performing the field redefinition on
two building block operators can only generate even further suppressed
operators. We will discuss perturbation theory in more detail in Section 4.

Now from these expressions it is clear that, up to three building block
operators which we have already considered, we can eliminate all
$\tilde{R}^{ij}\mathcal{O}_{(1)}\delta K_{ij}$ and
$\tilde{R}^{ij}\mathcal{O}_{(2)}\tilde{R}_{ij}$ operators from (2.7). Indeed
we can first use the product $\tilde{G}_{ij}\mathcal{O}_{(2)}\tilde{R}^{ij}$
in (2.10) to eliminate $\tilde{R}^{ij}\mathcal{O}_{(2)}\tilde{R}_{ij}$
operators while renormalising $\tilde{R}^{ij}\mathcal{O}_{(1)}\delta K_{ij}$
and introducing non-linear terms in $\tilde{R}$ which don’t contribute to the
action up to cubic order. We can then use the product
$\tilde{G}_{ij}\mathcal{O}_{(1)}\delta K^{ij}$ to eliminate all
$\tilde{R}^{ij}\mathcal{O}_{(1)}\delta K_{ij}$ operators while renormalising
$\delta K^{ij}\mathcal{O}_{(0)}\delta K_{ij}$ and introducing other operators
that don’t contribute to the cubic action. We cannot also eliminate the
$\delta K^{ij}\mathcal{O}_{(0)}\delta K_{ij}$ operators since there is not
enough freedom in (2.9). Note that in terms of $\gamma_{ij}$, these field
redefinitions shift $\gamma_{ij}$ by terms with derivatives and then scale
invariance ensures that the redefinitions vanish at late-times. So
wavefunction coefficients and cosmological correlators evaluated at the end of
inflation are not affected by these field redefinitions.

Now it is clear how we can generalise this discussion to manifestly remove all
copies of $\tilde{R}_{ij}$ from three building block operators. We simply need
to use field redefinitions of the schematic form $\delta
g=\tilde{R}^{2}+\tilde{R}\delta K+\delta K\delta K$ where we have suppressed
indices and derivatives. Acting on the Einstein-Hilbert action with these
field redefinitions allows us to reduce three building block operators to ones
cubic in $\delta K_{ij}$. It follows that the most general action that can
contribute to graviton bispectra at tree-level is

$\displaystyle S=S_{0}+M_{\text{pl}}^{2}\int d^{4}x\sqrt{-g}\,\big{[}\delta
K^{ij}\mathcal{O}_{(0)}\delta K_{ij}+\mathcal{O}(\delta K^{3})\big{]}.$ (2.11)

For reasons that will become clear a bit later on we keep the coefficient of
$\delta K^{ij}\delta K_{ij}$ non-zero for now.

We have therefore shown in this section that to compute the bispectra for
gravitons, in addition to the minimal action, we only need to consider EFToI
actions constructed out of the extrinsic curvature. Time derivatives can
always be taken to be partial ones and spatial derivatives only need to be
covariant for two building block operators and they only need to be contracted
amongst themselves.

##  3 A general action to cubic order

We can now compute the general action up to cubic order in $\gamma_{ij}$. The
Einstein-Hilbert part of the action yields

$\displaystyle S_{\gamma,\text{GR}}=\frac{M_{\text{pl}}^{2}}{8}\int
dtd^{3}xa^{3}(t)[\dot{\gamma}_{ij}\dot{\gamma}_{ij}-a^{-2}\partial_{k}\gamma_{ij}\partial_{k}\gamma_{ij}+a^{-2}(2\gamma_{ik}\gamma_{jl}-\gamma_{ij}\gamma_{kl})\partial_{k}\partial_{l}\gamma_{ij}]+\mathcal{O}(\gamma^{4}),$
(3.1)

and we remind the reader that for $\gamma_{ij}$ operators, spatial indices are
raised and lowered with $\delta_{ij}$ so we will not be so careful about the
placement of indices. For the corrections to the minimal action we consider
the two and three building block operators separately.

###  3.1 Two building block operators

Let’s start with operators quadratic in $\delta K_{ij}$ which take the form

$\displaystyle S$ $\displaystyle=M_{\text{pl}}^{2}\int d^{4}x\sqrt{-g}\,\delta
K^{ij}\mathcal{O}_{(0)}\delta K_{ij}$ $\displaystyle=M_{\text{pl}}^{2}\int
d^{4}x\sqrt{-g}\,\sum_{m,n}g_{n,m}\nabla_{0}^{2n}\delta
K^{ij}\tilde{\nabla}^{2m}\delta K_{ij},$ (3.2)

where we have taken the number of time derivatives to be even such that this
operator is not a total derivative, and we integrated by parts to separate the
time and space derivatives. One might worry about the ordering of the
covariant derivatives, but we can guarantee this ordering up to curvature
corrections which we have already dealt with above. The couplings $g_{m,n}$
are in general dimensionful. Again, in Section 4 we will show that any
boundary terms that might arise when we integrate by parts will not contribute
to the late-time wavefunction. As we explained above we can also safely take
time derivatives to be partial ones - the difference can be accounted for by
adding appropriate three building block operators - and we will denote the
$n^{\text{th}}$ partial time derivative by $(\delta K_{ij})^{n}$. Note that
this procedure does not affect the couplings $g_{n,m}$ and only changes the
couplings of three building block operators. We will construct the most
general set of bispectra so we can do this without loss of generality. We
would now like to expand the redefined action

$S=M_{\text{pl}}^{2}\int
d^{4}x\sqrt{-g}\,\sum_{m,n}g_{n,m}\partial_{0}^{2n}\delta
K^{ij}\tilde{\nabla}^{2m}\delta K_{ij},$ (3.3)

to find the graviton action up to cubic order which for these two building
block operators we denote by $S_{\gamma,\text{2BB}}$. Recall that the metric
determinant is independent of $\gamma$, while $\delta K$ vanishes on the
background. So to find the action up to cubic order we always need to expand
one of the $\delta K$ to linear order and the other to quadratic order.

First consider $m=0$ where all derivatives are temporal ones. In this case we
see that by expanding the action there are no corrections at
$\mathcal{O}(\gamma^{3})$. For $n=0$ this was noticed in [16] and it holds
true for all $n$. This can be most easily seen by writing the operators as
$(\delta K^{i}{}_{j})^{2n}\delta K^{j}{}_{i}$ and by noticing that, for all
$n$, $(\delta K^{i}{}_{j})^{2n}$ is symmetric at linear order while anti-
symmetric at quadratic order141414Of course we should raise and lower these
indices before we replace covariant derivatives with partial ones. . We have

$\displaystyle\delta
K^{i}{}_{j}=\frac{1}{2}a^{2}(t)g^{il}\partial_{t}(e^{\gamma})_{lj}=\frac{1}{2}\dot{\gamma}^{i}{}_{j}+\frac{1}{4}(\dot{\gamma}^{i}{}_{l}\gamma_{lj}-\gamma^{i}{}_{l}\dot{\gamma}_{lj})+\mathcal{O}(\gamma^{3}),$
(3.4)

and the symmetry properties of this object are not altered when we take the
partial derivative. The action for the graviton perturbation, for $m=0$, is
then simply the quadratic one given by

$\displaystyle S_{\gamma,\text{2BB}}\supset\frac{M_{\text{pl}}^{2}}{4}\int
dtd^{3}xa^{3}(t)\sum_{n}g_{n,0}(\gamma_{ij})^{2n+1}\dot{\gamma}_{ij}.$ (3.5)

As we mentioned above, for $n=0$ it looks like the two-derivative action is
altered by this operator but we can always set $g_{0,0}=0$ using field
redefinitions [9] (we will comment on this further below).

Now for $m\neq 0$ we need to compute the spatial covariant derivatives of the
extrinsic curvature, and in this case we cannot simply treat them as partial
derivatives since $\tilde{\Gamma}^{i}_{jk}$ cannot be written in terms of the
extrinsic curvature. As we showed above, $(\delta K^{i}{}_{j})^{2n}$ is anti-
symmetric at quadratic order, while at linear order we can treat covariant
derivatives as partial ones and we have

$\displaystyle\tilde{\nabla}^{2m}\delta
K^{i}{}_{j}=\frac{1}{2}a^{-2m}\partial^{2m}\dot{\gamma}^{i}{}_{j}+\mathcal{O}(\gamma^{2}),$
(3.6)

which is symmetric. It follows that the only contribution to the cubic action
comes from taking $(\delta K^{i}{}_{j})^{2n}$ at linear order and
$\tilde{\nabla}^{2m}\delta K^{i}{}_{j}$ at quadratic order, and only keeping
the symmetric contributions. Let’s now focus on computing
$\tilde{\nabla}^{2m}\delta K^{i}{}_{j}$. For $m=1$ we find, by direct
computation, that up to quadratic order in $\gamma_{ij}$ we have

$\displaystyle\tilde{\nabla}^{2}\delta
K^{i}{}_{j}=\frac{1}{2}a^{-2}\left(\partial^{2}\dot{\gamma}^{i}{}_{j}-\gamma_{lm}\partial_{l}\partial_{m}\dot{\gamma}^{i}{}_{j}+Q_{s}[\gamma,\dot{\gamma}]^{i}{}_{j}\right)+\text{anti-
symmetric},$ (3.7)

where the symmetric function $Q_{s}$ is given by

$\displaystyle
Q_{s}[\gamma,\dot{\gamma}]_{ij}=-\partial_{i}\gamma_{kl}\partial_{k}\dot{\gamma}_{lj}+\partial_{l}\gamma_{jk}\partial_{k}\dot{\gamma}_{il}+(i\leftrightarrow
j),$ (3.8)

and we leave implicit the $\mathcal{O}(\gamma^{2})$ anti-symmetric
contributions to $\tilde{\nabla}^{2}\delta K^{i}{}_{j}$ since when contracted
with the linear expansion of $(\delta K^{i}{}_{j})^{2n}$ they will sum to
zero. The first term in (3.7) comes from treating both covariant derivatives
as partial ones, the second comes from expanding the metric that contracts the
two spatial derivatives, while the final structure comes from the Christoffel
symbols. With this expression at hand we can now recursively generalise to any
$m\neq 0$. Again working up to quadratic order, we find

$\displaystyle\tilde{\nabla}^{2m}\delta
K^{i}{}_{j}=\frac{1}{2}a^{-2m}\left(\partial^{2m}\dot{\gamma}^{i}{}_{j}-m\gamma_{lk}\partial^{2m-2}\partial_{l}\partial_{k}\dot{\gamma}^{i}{}_{j}+\sum_{p}^{m-1}\partial^{2p}Q_{s}[\gamma,\partial^{2m-2-2p}\dot{\gamma}]^{i}{}_{j}\right),$
(3.9)

where again we have dropped anti-symmetric terms. The action up to cubic
order, now with $m=0$ included, is therefore

$\displaystyle S_{\gamma,\text{2BB}}=\frac{M_{\text{pl}}^{2}}{4}\int
dtd^{3}xa^{3}(t)\sum_{n}g_{n,0}(\gamma_{ij})^{2n+1}\dot{\gamma}_{ij}~{}+$
$\displaystyle\frac{M_{\text{pl}}^{2}}{4}\int dtd^{3}xa^{3-2m}(t)$
$\displaystyle\sum_{n,m=1}g_{n,m}(\gamma_{ij})^{2n+1}\left(\partial^{2m}\dot{\gamma}_{ij}-m\gamma^{lk}\partial^{2m-2}\partial_{l}\partial_{k}\dot{\gamma}_{ij}+\sum_{p}^{m-1}\partial^{2p}Q_{s}[\gamma,\partial^{2m-2-2p}\dot{\gamma}]_{ij}\right).$
(3.10)

This is the most general action, from two building block operators, that can
contribute to the graviton bispectra in the EFToI. It contains a sum of
quadratic and cubic terms with their relative coefficients tied together by
the linear realization of spatial diffeomorphisms and nonlinear realization of
time diffeomorphisms. Note that in this section we have freely added and
subtracted three building block operators. We were allowed to do this, because
we are going to consider these operators in their full generality in the next
section.

###  3.2 Three building block operators

Now consider three building block operators which all start at
$\mathcal{O}(\gamma^{3})$. As we discussed above we can treat all covariant
derivatives as partial ones and since $\delta K_{ij}\sim\dot{\gamma}_{ij}$ at
linear order, we simply need to write down all independent contractions of
three copies of $\dot{\gamma}_{ij}$, and spatial derivatives. This problem was
tackled in [12] at the level of polarisation tensors and spatial momenta but
it is simple to convert it into a Lagrangian statement. There are five
independent tensor structures once we use the fact that the graviton is
transverse and traceless, and integrate by parts (which is equivalent to
momentum conservation). These structures are organised by the number of
spatial derivatives that are contracted with a graviton and we denote this
number by $\alpha$. For parity-even interactions this is an even number and
for $\alpha=0,4,6$ there is a single structure, while for $\alpha=2$ there are
two. We cannot have $\alpha\geq 8$ since there are only $6$ free indices
across the three gravitons. For each $\gamma_{ij}$ we can add additional time
derivatives, while additional spatial derivatives can be restricted to two
fields only by integration by parts, and two derivatives that are contracted
with each other should act on different fields. Any other contractions can be
removed by the graviton’s equation of motion in favour of time derivatives
which we are already adding. We have

$\displaystyle S_{\gamma,\text{3BB}}=\sum_{\alpha}\int
dtd^{3}x\mathcal{L}_{\gamma,\alpha},$ (3.11)

where

$\displaystyle\mathcal{L}_{\gamma,\alpha=0}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{q}(t)h^{0}_{n_{1},n_{2},n_{3},p}(\gamma_{ij})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}(\gamma_{jk})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}(\gamma_{ki})^{1+n_{3}},$ (3.12)
$\displaystyle\mathcal{L}_{\gamma,\alpha=2}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{q}(t)h^{2}_{n_{1},n_{2},n_{3},p}(\gamma_{ij})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}\partial_{i}(\gamma_{lm})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{j}(\gamma_{lm})^{1+n_{3}}$ (3.13)
$\displaystyle+\sum_{n_{1},n_{2},n_{3},p}a^{q}(t)\hat{h}^{2}_{n_{1},n_{2},n_{3},p}(\gamma_{ij})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}(\gamma_{lm})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{i}\partial_{l}(\gamma_{jm})^{1+n_{3}},$ (3.14)
$\displaystyle\mathcal{L}_{\gamma,\alpha=4}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{q}(t)h^{4}_{n_{1},n_{2},n_{3},p}\partial_{i}(\gamma_{lk})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}\partial_{j}(\gamma_{mk})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{l}\partial_{m}(\gamma_{ij})^{1+n_{3}},$ (3.15)
$\displaystyle\mathcal{L}_{\gamma,\alpha=6}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{q}(t)h^{6}_{n_{1},n_{2},n_{3},p}\partial_{m}\partial_{k}(\gamma_{il})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}\partial_{i}\partial_{n}(\gamma_{jm})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{j}\partial_{l}(\gamma_{kn})^{1+n_{3}},$ (3.16)

with constant couplings $h^{\alpha}_{n_{1},n_{2},n_{3},p}$ and $q=3-2p-\alpha$
which is fixed by scale invariance. Despite the complicated looking nature of
these interactions, in Section 6 we will show that the resulting bispectra
take a very compact form151515When computing the bispectra we will not
actually use these explicit Lagrangian expressions, instead for three-building
block operators we will use the MLT [31] to efficiently write down all
possibilities..

We have therefore written down the most general actions that can contribute to
graviton bispectra, $S_{\gamma,\text{2BB}}$ and $S_{\gamma,\text{3BB}}$. In
the following sections we discuss the diagrams that contribute to the cubic
wavefunction and present compact expressions for the bispectra, but first we
close this section by converting our actions to conformal time and addressing
the fact that these general actions can contain higher-order time derivatives.

###  3.3 Converting to conformal time

Throughout we have been working in cosmological time but when we come to
compute the late-time wavefunction we would like to do so in conformal time
$\eta$ where we evolve perturbations from the far past at $\eta=-\infty$ to
the boundary of quasi-de Sitter space, or the end of inflation, at $\eta=0$.
We therefore need to change coordinates in the actions we have just derived so
that the background metric is

$\displaystyle ds^{2}=a^{2}(\eta)[-d\eta^{2}+dx^{2}],$ (3.17)

and we will approximate the scale factor as $a(\eta)=-1/(H\eta)$ which is the
de Sitter one in Poincaré or flat-slicing coordinates.

First consider two building block operators and the action (3.1). When we
convert an object of the form $(\gamma_{ij})^{n}$ to conformal time we will
generate a sum of terms with all derivatives from $1$ to $n$. For $m=0$, we
can use the fact that the couplings $g_{n,0}$ are arbitrary, to simply replace
all cosmological time derivatives with conformal ones without loss of
generality, while introducing the required scale factors. We can also now make
use our freedom to fix the coefficient of $\delta K^{ij}\delta K_{ij}$ to
anything we like to eliminate any quadratic terms with two derivatives. This
is the statement that the graviton speed of sound can always be fixed to unity
[9], and as expected this statement is true in both cosmological and conformal
time161616In fact, we can also bring the two derivative action to its
canonical GR form by redefining time, then redefining $\gamma_{ij}$, then
finally redefining all couplings. We can always do this since we only have one
field in the theory.. This procedure will generate terms with an overall odd
number of time derivatives but these can always be integrated by parts in
favour of ones with an even number of derivatives, and as always we can safely
drop any boundary terms that may arise (see Section 4). We can therefore write
the first line of (3.1) as

$\displaystyle S_{\gamma,\text{2BB}}\supset\frac{M_{\text{pl}}^{2}}{4}\int
d\eta
d^{3}x\sum_{n=1}g_{n,0}a^{2-2n}(\eta)(\gamma_{ij})^{2n+1}\gamma^{\prime}_{ij},$
(3.18)

where we keep the same labels for the arbitrary couplings and still use
brackets to denote higher-order time derivatives, with the integration
variables informing the reader which coordinate it refers to. We note that the
couplings are now not exactly as they were defined in the covariant action,
instead these new couplings are linear sums of the old ones. We now start the
sum from $n=1$ since we have at least four time derivatives, without loss of
generality, and importantly adding $\delta K^{ij}\delta K_{ij}$ does not
introduce any additional terms since it does not contribute to the cubic
action for $\gamma_{ij}$. The scale symmetry of the action is now crystal
clear in these coordinates and offers a good consistency check that we have
the correct number of scale factors: we need four scale factors to cancel the
scaling of $d\eta d^{3}x$ under a scale transformation, then we need to remove
a scale factor for each derivative. For an action like the one above with
$2n+2$ derivatives we therefore need $4-(2n+2)=2-2n$ scale factors.

We can deal with the second line of (3.1) in a similar way. The structure of
the time derivatives is the same for each term so we can simply replace all
cosmological time derivatives with conformal ones, while adding the
appropriate number of scale factors, and redefining the couplings $g_{m,n}$.
Here there is no need to add operators to remove certain terms so the sums
still run from $n=0$ and $m=1$. However, now that we have cubic interactions
we cannot simply drop all terms with an overall odd number of time
derivatives, or equivalently those with an even number that come from
transforming $(\gamma_{ij})^{2n+1}$ to conformal time. So for $m\neq 0$ we
must replace $2n$ with $n$. In conformal time the full (3.1) then becomes

$\displaystyle S_{\gamma,\text{2BB}}=\frac{M_{\text{pl}}^{2}}{4}\int d\eta
d^{3}x\sum_{n=1}g_{n,0}a^{2-2n}(\eta)(\gamma_{ij})^{2n+1}\gamma^{\prime}_{ij}~{}+$
$\displaystyle\frac{M_{\text{pl}}^{2}}{4}\int d\eta d^{3}x$
$\displaystyle\sum_{n=0,m=1}g_{n,m}a^{q}(\eta)(\gamma_{ij})^{n+1}\left(\partial^{2m}\gamma^{\prime}_{ij}-m\gamma^{lk}\partial^{2m-2}\partial_{l}\partial_{k}\gamma^{\prime}_{ij}+\sum_{p}^{m-1}\partial^{2p}Q_{s}[\gamma,\partial^{2m-2-2p}\gamma^{\prime}]_{ij}\right),$
(3.19)

where $q=2-2m-n$. Now each operator has at least four derivatives meaning that
it is impossible to integrate by parts to generate operators with fewer
derivatives. This can only happen when a time derivative acts on a scale
factor, but when there are exactly four derivatives there are no scale
factors. This will have important consequences for the structure of the
corresponding bispectra.

For the three building block operators we use the same logic such that the
vertices take exactly the same structure as before, but now with a different
number of scale factors. We now have

$\displaystyle S_{\gamma,\text{3BB}}=\sum_{\alpha}\int d\eta
d^{3}x\mathcal{L}_{\gamma,\alpha},$ (3.20)

where

$\displaystyle\mathcal{L}_{\gamma,\alpha=0}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{r}(\eta)h^{0}_{n_{1},n_{2},n_{3},p}(\gamma_{ij})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}(\gamma_{jk})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}(\gamma_{ki})^{1+n_{3}},$ (3.21)
$\displaystyle\mathcal{L}_{\gamma,\alpha=2}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{r}(\eta)h^{2}_{n_{1},n_{2},n_{3},p}(\gamma_{ij})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}\partial_{i}(\gamma_{lm})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{j}(\gamma_{lm})^{1+n_{3}}$ (3.22)
$\displaystyle+\sum_{n_{1},n_{2},n_{3},p}a^{r}(\eta)\hat{h}^{2}_{n_{1},n_{2},n_{3},p}(\gamma_{ij})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}(\gamma_{lm})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{i}\partial_{l}(\gamma_{jm})^{1+n_{3}},$ (3.23)
$\displaystyle\mathcal{L}_{\gamma,\alpha=4}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{r}(\eta)h^{4}_{n_{1},n_{2},n_{3},p}\partial_{i}(\gamma_{lk})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}\partial_{j}(\gamma_{mk})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{l}\partial_{m}(\gamma_{ij})^{1+n_{3}},$ (3.24)
$\displaystyle\mathcal{L}_{\gamma,\alpha=6}$
$\displaystyle=\sum_{n_{1},n_{2},n_{3},p}a^{r}(\eta)h^{6}_{n_{1},n_{2},n_{3},p}\partial_{m}\partial_{k}(\gamma_{il})^{1+n_{1}}\partial_{i_{1}\ldots
i_{p}}\partial_{i}\partial_{n}(\gamma_{jm})^{1+n_{2}}\partial_{i_{1}\ldots
i_{p}}\partial_{j}\partial_{l}(\gamma_{kn})^{1+n_{3}},$ (3.25)

and we have defined $r=1-2p-n_{1}-n_{2}-n_{3}-\alpha$, and have redefined the
couplings. These are the actions we will use to compute the graviton
bispectra.

###  3.4 A comment on higher-order time derivatives

Our actions for both two and three building block operators can contain
higher-order time derivatives which may concern the reader. When treated
perturbatively such interactions should be harmless, however there are other
actions that give exactly the same bispectra as the ones we are going to
compute which have at most one time derivative per field. This should reassure
a concerned reader. We use the actions above since they have a very simple
covariant form, and the properties of $\delta K^{i}{}_{j}$ made expanding the
action a relatively simple task.

To see how we could remove higher-order time derivatives lets go back to the
field redefinitions we used to eliminate the $\tilde{R}^{2}$ and
$\tilde{R}\delta K$ operators (again we are dropping indices and derivatives).
We have three different structures in the action and two different structures
in the field redefinition which is schematically $\delta g=\tilde{R}+\delta
K$. So instead of removing the $\tilde{R}^{2}$ and $\tilde{R}\delta K$
operators leaving us with only $\delta K^{2}$ ones, we could remove the
$\delta K^{2}$ ones and use the remaining freedom to fix the tuning between
$\tilde{R}^{2}$ and $\tilde{R}\delta K$. Such tunings can be fixed to remove
all higher-order time derivatives in the quadratic corrections to the graviton
action. For example, consider the following operator:

$\displaystyle\tilde{R}^{ij}\tilde{\nabla}^{2m}\nabla_{0}^{2n}\tilde{R}_{ij}\supset\partial^{2}\gamma_{ij}\partial^{2m+2}(\gamma_{ij})^{2n},$
(3.26)

which contains higher-order time derivatives that cannot be integrated by
parts for $n\geq 2$. We can cancel these using the operators:

$\displaystyle\tilde{R}^{ij}\tilde{\nabla}^{2m+2}\nabla_{0}^{2n-1}\delta
K_{ij}\supset\partial^{2}\gamma_{ij}\partial^{2m+2}(\gamma_{ij})^{2n},$ (3.27)

since their coefficients can be chosen freely thanks to the field
redefinitions at our disposal. The resulting action would still contain
quadratic corrections but these would only come from
$\tilde{R}^{ij}\tilde{\nabla}^{2m}\tilde{R}_{ij}$ and so would only contain
spatial derivatives. The cubic action is likely to take a more complicated
form than the one we have above, however. Let us emphasise again that field
redefinitions of this form do not change the wavefunction coefficients, and
therefore do not change correlators. For these two building block operators it
could still be the case that the resulting cubic vertices have higher-order
time derivatives even though the quadratic ones do not. This is not an issue
for the on-shell action as we can again use the equation of motion to
eliminate them, but off-shell we would not be able to do this. At this stage
we have exhausted all possible field redefinitions at the level of $g_{ij}$,
which is the most sensible way to formulate these field redefinitions. So if
the off-shell action was to also not have higher-order time derivatives, the
tuning between the $\tilde{R}^{2}$ and $\tilde{R}\delta K$ operators that
removes higher-order time derivatives at quadratic order must also be enough
to remove them at cubic order.

We can play the same game for three building block operators. At the level of
the expanded action it is clear that we can use the equation of motion to
remove all higher-order time derivatives but it can also be made manifest
using covariant field redefinitions. For three building block operators we
have $\tilde{R}^{3}$, $\tilde{R}^{2}\delta K$, $\tilde{R}\delta K^{2}$ and
$\delta K^{3}$. We also have three different field redefinitions due to the
three different bi-products of $\tilde{R}$ and $\delta K$. We can use this
freedom to eliminate the $\tilde{R}\delta K^{2}$ and $\delta K^{3}$ operators
while tuning the coefficients between the $\tilde{R}^{3}$ and
$\tilde{R}^{2}\delta K$ ones such that there are no higher-order time
derivatives. We choose to not eliminate these higher-order time derivatives
since, as we will discuss in Section 6, we can very easily extract the
bispectra from (3.21) using the MLT. The bispectra are invariant under such
field redefinitions so it makes sense to work with the action that lends
itself to extracting correlators in the most efficient way.

##  4 Generalities of contact and single exchange diagrams

Before constructing EFToI bispectra, in this section we will provide more
details on the diagrams that we need to compute. We will work within the WFU
approach then extract expectation values using the usual formula. The WFU has
been reviewed in many places in the recent literature e.g [31, 27], so here we
skip such a review and concentrate on the diagrams of interest which are cubic
contact diagrams and single exchange diagrams. The Feynman rules for computing
these diagrams is reviewed in [31, 27], but we will briefly remind the reader
of them in this section too.

###  4.1 Contact diagrams

Contact diagrams, see Figure 1, are the easiest to compute. For gravitons they
factorise into a polarisation factor which contains the three polarisation
tensors for the external gravitons which can be contracted with each other or
with spatial momenta, and a trimmed wavefunction that carries information
about the time evolution. The general form of a cubic contact diagram is
therefore [12]171717Here and throughout this work we are dropping the momentum
conserving delta function that always appears thanks to the spatial
translation symmetry of the theory.

$\displaystyle\psi_{3}(\\{k\\},\\{\textbf{k}\\})=\sum_{\text{contractions}}[e^{h_{1}}(\textbf{k}_{1})e^{h_{2}}(\textbf{k}_{2})e^{h_{3}}(\textbf{k}_{3})\textbf{k}_{1}^{\alpha_{1}}\textbf{k}_{2}^{\alpha_{2}}\textbf{k}_{3}^{\alpha_{3}}]\psi_{3}^{\text{trimmed}}(\\{k\\}),$
(4.1)

where $h_{i}$ are the helicities of the external fields, we define
$\alpha=\alpha_{1}+\alpha_{2}+\alpha_{3}$, and $\\{k\\}$ and
$\\{\textbf{k}\\}$ collectively denote the external energies and momenta,
respectively. Since we are only interested in massless spin-$2$ modes, we have
$h_{i}=\pm 2$, and we will concentrate on the $+++$ and $++-$ configurations
as the others can be extracted from these by parity transformations [12]. The
polarisation factors are easily read off from the Lagrangian and by converting
to momentum space, while traditionally the trimmed wavefunction would be
computed by integrating the three bulk-boundary propagators, which can be
differentiated with respect to time, from the far past at $\eta=-\infty$ to
the future boundary at $\eta=0$. In the far past we Wick rotate such that
there is some evolution in Euclidean time which has the effect of projecting
onto the vacuum and damping these early time contributions [16].

$\displaystyle\upgamma$$\displaystyle\upgamma$$\displaystyle\upgamma$ Figure
1: Cubic contact diagram.

Another way of computing $\psi_{3}^{\text{trimmed}}$ was derived in [31] which
doesn’t require any time evolution, which is anyway completely unobservable in
the final answer where it has been integrated out, and is referred to as the
Manifestly Local Test (MLT). The test requires trimmed wavefunction
coefficients to satisfy

$\displaystyle\frac{\partial}{\partial
k_{c}}\psi_{3}^{\text{trimmed}}\Big{|}_{k_{c}=0}=0\,,\qquad\forall\,c=1,2,3\,,$
(4.2)

which is simply the statement that the trimmed wavefunction does not contain
any terms linear in any of the three external energies (locality also forces
there to be no inverse powers of the energies: at cubic order there are no
inverse Laplacians to worry about). Note that as we send one of the external
energies to zero we do so while holding the others fixed which distinguishes
the MLT from soft theorems. The MLT can be used to construct cosmological
correlators as was done in [31, 12, 72], but also to verify that the often
complicated final results have the correct structure as was done in [33]. It
was shown in [12] that general solutions to these equations pick out all
possible trimmed wavefunctions, and clearly solving these equations is simpler
than having to evaluate the necessary bulk time integrals. The MLT follows
from the fact that the bulk-boundary propagator for massless gravitons (and
scalars) satisfies

$\displaystyle K_{\gamma}(k,\eta)=(1-ik\eta)e^{ik\eta},\qquad\frac{\partial
K_{\gamma}}{\partial k}\Big{|}_{k=0}=0,$ (4.3)

and this property is inherited by the trimmed wavefunction since it holds for
all $\eta$. The MLT can also be derived by demanding the absence of spurious
poles in four-point functions that arise from gluing together two three-point
functions. Indeed, such an exchange diagram should be regular as the energy of
the exchanged field is taken to zero, and the Cosmological Optical Theorem
[27] implies that this is only the case if the constituent three-point
functions satisfy the MLT. We refer the reader to [31] for further details on
these two derivations.

Now the assumption of a Bunch-Davies vacuum, which we impose throughout, tells
us that $\psi_{3}^{\text{trimmed}}$ is a rational function with poles only
occurring at $k_{T}=k_{1}+k_{2}+k_{3}=0$ which is the kinematical limit where
energy is conserved. Interestingly, the residue of the leading total-energy
pole contains the flat-space scattering amplitude for the same process [73, 1,
27] (in some cases the relationship between correlators and amplitudes is not
so straightforward due to enhanced symmetry in the flat-space limit. This is
the case for DBI as was shown in [74] and confirmed in [75] at the level of
boost-breaking amplitudes.). There can also be logs in the trimmed
wavefunction, for low derivative operators, but no other kinematic
singularities unless there is some form of non-locality or different vacuum
conditions [42]. In this paper we will not encounter logs since in the EFToI
there are too many derivatives in the graviton interactions. Indeed the
absence of logs in contact diagrams requires
$2n_{\partial_{\eta}}+n_{\partial_{i}}>3$ [42], where $n_{\partial_{\eta}}$
and $n_{\partial_{i}}$ are respectively the number of conformal time and space
derivatives, and we have seen in Section 3 that this condition is always
satisfied in the EFToI.

Now to solve the MLT we simply write down an ansatz and organise the solutions
in terms of the order of the leading total-energy pole which we denote by $p$
and which counts the number of derivatives in the corresponding vertex [42].
Guidance comes from Bose symmetry: we fix $\psi_{3}^{\text{trimmed}}$ to have
the same symmetry as the polarisation part and sum over the remaining
permutations once we have constructed $\psi_{3}^{\text{trimmed}}$. This is a
consistent thing to do since the MLT is satisfied for all permutations. With
the cubic wavefunction coefficient in hand we can compute the corresponding
bispectrum $B_{3}$. The relationship is given by [12]181818The COT for cubic
contact diagrams is
$\psi_{3}(\\{k\\},\\{\textbf{k}\\})+\psi_{3}^{\star}(\\{-k\\},\\{-\textbf{k}\\}=0$
[27] and this dictates which part of the wavefunction contributes to
expectation values [12].

$\displaystyle
B_{3}(\\{k\\},\\{\textbf{k}\\})=-\frac{\psi_{3}(\\{k\\},\\{\textbf{k}\\})+\psi_{3}^{\star}(\\{k\\},\\{-\textbf{k}\\})}{\prod_{a=1}^{3}2\text{Re}~{}\psi_{2}(k_{a})},$
(4.4)

where $\psi_{2}$ is the quadratic wavefunction coefficient which is
perturbatively fixed by the two-derivative quadratic action coming from GR. We
have

$\displaystyle\psi_{2,\text{GR}}^{hh^{\prime}}=\frac{M_{\text{pl}}^{2}}{H^{2}}k^{3}\delta_{hh^{\prime}}.$
(4.5)

We have dropped imaginary terms in this expression, which are actually
divergent at late-times, since they never contribute to correlators. For
parity-even interactions, which are the ones of interest here, the numerator
is $2\text{Re}~{}\psi_{3}$ which follows from having an even number of spatial
momenta. We can also use
$e_{ij}^{h}(\textbf{k})^{\ast}=e_{ij}^{h}(-\textbf{k})$, which follows from
the reality of $\gamma_{ij}(x)$, to see that the polarisation factor is a
common factor on both the left and right hand side of this equation. We refer
the reader to [12] for more details on deriving this relationship.

We have to compute such contact diagrams for both Type-I and Type-II
bispectra. For Type-II we will explain in Section 6 that we can use the MLT to
very efficiently write down all allowed wavefunction coefficients by taking
into account the fact that each graviton is differentiated with respect to
time at least once, while for Type-I, in Section 5 we will directly compute
the necessary bulk time integrals. As we can see from (3.3), the time
dependence for two building block operators is quite specific so computing the
integrals is more straightforward than finding the necessary subset of MLT
solutions. For this reason let us briefly review the Feynman rules for
computing contact diagrams.

It is easiest to illustrate the Feynman rules with an example. Consider the
graviton interaction

$\displaystyle S_{\text{int}}=g_{\text{int}}\int
d^{3}xd\eta~{}a(\eta)\gamma_{ij}^{\prime}\gamma_{jk}^{\prime}\gamma_{ki}^{\prime},$
(4.6)

which we take in addition to the free theory coming from GR. This is the
leading parity-even correction to the Einstein-Hilbert action in the EFToI. To
compute the cubic wavefunction coefficient we convert to momentum space and
extract the tensor structure which in this case is simply
$e_{ij}(\textbf{k}_{1})e_{jk}(\textbf{k}_{2})e_{ki}(\textbf{k}_{3})$. For each
graviton we evolve it from time $\eta$ to the late-time boundary at $\eta_{0}$
by inserting a bulk-boundary propagator $K_{\gamma}$ for each field. We insert
an overall $(-i)$ and sum over permutations. This overall factor of $(-i)$
follows from our definition of the wavefunction coefficients in (1.3). We do
this for the two helicity configurations $+++$ and $++-$. For this example,
the $+++$ configuration wavefunction coefficient is

$\displaystyle\psi_{3}^{+++}=\frac{6ig_{\text{int}}}{H}e^{+}_{ij}(\textbf{k}_{1})e^{+}_{jk}(\textbf{k}_{2})e^{+}_{ki}(\textbf{k}_{3})\int^{\eta_{0}}_{-\infty(1-i\bm{\epsilon})}\frac{d\eta}{\eta}~{}K_{\gamma}^{\prime}(k_{1},\eta)K_{\gamma}^{\prime}(k_{2},\eta)K_{\gamma}^{\prime}(k_{3},\eta),$
(4.7)

where the minus sign coming from the Feynman rules has been cancelled by the
one coming from the scale factor (which appears linearly), and we remind the
reader that we don’t include the momentum conserving delta function in the
wavefunction coefficient c.f. (1.3). We integrate from the far past to the
future boundary and we project onto the vacuum at early times. In the
following we will suppress the $i\epsilon$ prescription. Computing the time
integral we find

$\displaystyle\psi_{3}^{+++}=-\frac{12g_{\text{int}}}{H}e^{+}_{ij}(\textbf{k}_{1})e^{+}_{jk}(\textbf{k}_{2})e^{+}_{ki}(\textbf{k}_{3})\frac{e_{3}^{2}}{k_{T}^{3}},$
(4.8)

where we have written the fully symmetric time integral in terms of the
symmetric polynomials given in (1.12). This can be easily generalised to any
other contact diagram, and as we will explain in Section 4.4, other helicity
configurations can be extracted from the $+++$ one. We nicely see from this
example that the degree of the leading order total-energy pole is counting the
number of derivatives in the bulk vertex.

###  4.2 Single exchange diagrams

Since we have corrections to the quadratic action, there are other diagrams in
addition to contact ones that can contribute to the cubic wavefunction
coefficient at tree-level. In principle there are infinitely many new diagrams
which would ultimately resum into a single contact diagram with a new
propagator which takes into account the quadratic corrections. However, we are
treating these corrections perturbatively in which case only one diagram
contributes at leading order. This diagram is shown in Figure 2 and
corresponds to a single exchange process with a single cubic interaction
connected to a quadratic correction (QC) vertex by a bulk-bulk propagator. For
two building block operators the largest of these diagrams comes from taking
the cubic vertex to be the GR one [12]. Indeed, we are treating the
corrections to the quadratic action perturbatively so the largest contribution
comes at linear order in these new couplings for which we need to take the
cubic vertex to be independent of this small coupling: the GR vertex scales
like $1/P_{\gamma}$, where $P_{\gamma}$ is the power spectrum of the graviton,
while the bulk-bulk propagator scales as $P_{\gamma}$, so the product of the
two is $\mathcal{O}(1)$. At this order, this exchange diagram is of a
comparable size to the contact diagram that comes from these two building
block operators, and cancellations are required between these two diagrams for
the EFToI consistency relations to be satisfied. This makes sense since the
quadratic and cubic operators in the action are tied together by spatial
diffeomorphisms which is where these consistency relations come from [15, 16,
17]. This was checked explicitly for the parity-odd Chern-Simons term in the
EFToI in [12], and we will check it in Section 5 for these parity-even
bispectra.

$\displaystyle\upgamma$$\displaystyle\upgamma$$\displaystyle\upgamma$GRQC
Figure 2: Single exchange diagram.

Single exchange diagrams are more complicated to compute compared to contact
ones since there are now nested time integrals. Let us illustrate the Feynman
rules for computing Figure 2 by taking the cubic vertex to be GR, and the
quadratic mixing vertex to be one of the terms in (3.18) where we have $m=0$.
As always we introduce an overall factor of $(-i)$, a bulk-boundary propagator
for each external line, and a bulk-bulk propagator for the internal line. We
act on these propagators with derivatives according to the bulk vertices, add
the appropriate polarisation tensors, and sum over permutations. For our
example of interest here we have

$\displaystyle\psi^{(n,m=0)}_{3,\text{exchange}}$
$\displaystyle=-i\times\frac{M_{\text{pl}}^{2}}{8}(-2e_{ik}^{h_{1}}e_{jl}^{h_{2}}k^{3}_{k}k^{3}_{l}e_{ij}^{h_{3}}+e_{ij}^{h_{1}}e_{kl}^{h_{2}}k^{3}_{k}k^{3}_{l}e_{ij}^{h_{3}}+5~{}\text{perms})\times
4\times\frac{M_{\text{pl}}^{2}g_{n,0}}{4}$ $\displaystyle\times\int d\eta
d\eta^{\prime}a^{2}(\eta)a^{2-2n}(\eta^{\prime})K_{\gamma}(k_{1},\eta)K_{\gamma}(k_{2},\eta)$
$\displaystyle\times\left[K^{(2n+1)}_{\gamma}(k_{3},\eta^{\prime})\frac{\partial}{\partial\eta^{\prime}}G(k_{3},\eta,\eta^{\prime})+K^{\prime}_{\gamma}(k_{3},\eta^{\prime})\frac{\partial^{2n+1}}{\partial\eta^{\prime
2n+1}}G(k_{3},\eta,\eta^{\prime})\right]+2~{}\text{perms},$ (4.9)

where the factor of $4$ in the first line comes from applying (1.8) to the
quadratic mixing, and the bulk-bulk propagator is given by (see e.g. [27])

$\displaystyle
G(k,\eta,\eta^{\prime})=2P_{\gamma}(k)[\theta(\eta-\eta^{\prime})K(k,\eta^{\prime})\text{Im}K(k,\eta)+(\eta\leftrightarrow\eta^{\prime})],$
(4.10)

where $\text{Im}K(k,\eta)$ is the imaginary part of the bulk-boundary
propagator which takes the form

$\displaystyle\text{Im}K(k,\eta)=-\frac{i}{2}[(1-ik\eta)e^{ik\eta}-(1+ik\eta)e^{-ik\eta}].$
(4.11)

Computing and analysing such time integrals is far simpler when there are no
time derivatives on the bulk-bulk propagator. We can guarantee this by
integrating by parts and, as always, we can drop all boundary terms. We then
have

$\displaystyle\psi^{(n,m=0)}_{3,\text{exchange}}=-\frac{iM_{\text{pl}}^{4}g_{n,0}}{8}(2e_{ik}^{h_{1}}e_{jl}^{h_{2}}k^{3}_{k}k^{3}_{l}e_{ij}^{h_{3}}-e_{ij}^{h_{1}}e_{kl}^{h_{2}}k^{3}_{k}k^{3}_{l}e_{ij}^{h_{3}}+5~{}\text{perms})\mathcal{I}_{n,0}(k_{1},k_{2},k_{3})+2~{}\text{perms},$
(4.12)

where we have defined

$\displaystyle\mathcal{I}_{n,0}(k_{1},k_{2},k_{3})=$ $\displaystyle\int d\eta
d\eta^{\prime}a^{2}(\eta)K_{\gamma}(k_{1},\eta)K_{\gamma}(k_{2},\eta)G(k_{3},\eta,\eta^{\prime})$
$\displaystyle\times\left[\frac{\partial}{\partial\eta^{\prime}}(a^{2-2n}(\eta^{\prime})K^{(2n+1)}_{\gamma}(k_{3},\eta^{\prime}))+\frac{\partial^{2n+1}}{\partial\eta^{\prime
2n+1}}(a^{2-2n}(\eta^{\prime})K^{\prime}_{\gamma}(k_{3},\eta^{\prime}))\right].$
(4.13)

A very similar expression also applies when we take quadratic mixing vertices
with $m\neq 0$ from (3.3), and is given by

$\displaystyle\psi^{(n,m\geq 1)}_{3,\text{exchange}}=$
$\displaystyle\frac{(-1)^{m+1}iM_{\text{pl}}^{4}g_{n,m}}{8}(2e_{ik}^{h_{1}}e_{jl}^{h_{2}}k^{3}_{k}k^{3}_{l}e_{ij}^{h_{3}}-e_{ij}^{h_{1}}e_{kl}^{h_{2}}k^{3}_{k}k^{3}_{l}e_{ij}^{h_{3}}+5~{}\text{perms})k_{3}^{2m}\mathcal{I}_{n,m}(k_{1},k_{2},k_{3})$
$\displaystyle+2~{}\text{perms},$ (4.14)

where

$\displaystyle\mathcal{I}_{n,m}(k_{1},k_{2},k_{3})=$ $\displaystyle\int d\eta
d\eta^{\prime}a^{2}(\eta)K_{\gamma}(k_{1},\eta)K_{\gamma}(k_{2},\eta)G(k_{3},\eta,\eta^{\prime})$
$\displaystyle\times\left[\frac{\partial}{\partial\eta^{\prime}}(a^{q}(\eta^{\prime})K^{(n+1)}_{\gamma}(k_{3},\eta^{\prime}))+(-1)^{n}\frac{\partial^{n+1}}{\partial\eta^{\prime
n+1}}(a^{q}(\eta^{\prime})K^{\prime}_{\gamma}(k_{3},\eta^{\prime}))\right].$
(4.15)

We remind the reader that $q=2-2m-n\leq 0$. In all cases it is easy to see
that every time integral we need to compute for these single exchange diagrams
is of the form

$\displaystyle\mathcal{M}(\alpha,\beta)=\int d\eta
d\eta^{\prime}~{}a^{2}(\eta)K_{\gamma}(k_{1},\eta)K_{\gamma}(k_{2},\eta)G(k_{3},\eta,\eta^{\prime})a^{\alpha}(\eta^{\prime})K_{\gamma}^{(\beta)}(k_{3},\eta^{\prime}),$
(4.16)

with $\alpha\leq 0$ and $\beta\geq 1$. We will refer to this integral as the
master time integral. For $m=0$ we have

$\displaystyle\mathcal{I}_{n,0}(k_{1},k_{2},k_{3})=\int d\eta
d\eta^{\prime}a^{2}(\eta)K_{\gamma}(k_{1},\eta)K_{\gamma}(k_{2},\eta)G(k_{3},\eta,\eta^{\prime})a^{2-2n}(\eta^{\prime})\times$
$\displaystyle\left[K^{(2n+2)}_{\gamma}(k_{3},\eta^{\prime})+(2-2n)Ha(\eta^{\prime})K^{(2n+1)}_{\gamma}(k_{3},\eta^{\prime})+\sum_{k=0}^{2n-2}c_{n,k}H^{k}a^{k}(\eta^{\prime})K^{(2+2n-k)}_{\gamma}(k_{3},\eta^{\prime})\right],$
(4.17)

where we have defined

$\displaystyle c_{n,k}=(-1)^{k}k!{{2n+1}\choose{k}}{{2n-2}\choose{k}},$ (4.18)

and from this expression we can write

$\displaystyle\mathcal{I}_{n,0}(k_{1},k_{2},k_{3})$
$\displaystyle=\mathcal{M}(2-2n,2n+2)+(2-2n)H\mathcal{M}(3-2n,2n+1)$
$\displaystyle+\sum_{k=0}^{2n-2}c_{n,k}H^{k}\mathcal{M}(k+2-2n,2+2n-k).$
(4.19)

Similarly, for $m\neq 0$ we can write

$\displaystyle\mathcal{I}_{n,1}(k_{1},k_{2},k_{3})$
$\displaystyle=\mathcal{M}(-n,n+2)-nH\mathcal{M}(1-n,n+1)+\sum_{k=0}^{n}d_{n,1,k}H^{k}\mathcal{M}(k-n,n+2-k),$
(4.20) $\displaystyle\mathcal{I}_{n,m}(k_{1},k_{2},k_{3})$
$\displaystyle=\mathcal{M}(2-2m-n,n+2)+(2-2m-n)H\mathcal{M}(3-2m-n,n+1)$
$\displaystyle+\sum_{k=0}^{n+1}d_{n,m,k}H^{k}\mathcal{M}(k+2-2m-n,n+2-k),\qquad
m\geq 2,$ (4.21)

where we have defined

$\displaystyle d_{n,m,k}=(-1)^{n+k}k!{{n+1}\choose{k}}{{n+2m-2}\choose{k}}.$
(4.22)

We will essentially dedicate Section 5.1 to computing
$\mathcal{M}(\alpha,\beta)$ and then finding the final form of these single
exchange diagrams, but let us first derive some general properties. We will
show that $\mathcal{M}(\alpha,\beta)$ is only singular at $k_{T}=0$, and
satisfies the MLT for each external leg. These two properties are enough for
us to conclude that single exchange diagrams in the EFToI have the same
structure as cubic contact diagrams and are therefore captured by the general
analysis of [12]. It would be interesting to investigate if this holds for
other cubic diagrams too i.e. those with more bulk-bulk propagators.

To study this master time integral we need expressions for the derivatives of
the bulk-boundary propagator. We have

$\displaystyle
K^{(\beta)}(k,\eta)=\frac{\partial^{\beta}}{\partial\eta^{\beta}}K(k,\eta)=(ik)^{\beta}(1-\beta-
ik\eta)e^{ik\eta}.$ (4.23)

Let’s first ask what singularities $\mathcal{M}(\alpha,\beta)$ can have. As
has been discussed in the literature in several places, see e.g. [40],
exchange diagrams at tree-level have a restricted set of singularities: they
can be singular when the energy of all external legs sums to zero, and when
the energy of an individual vertex sums to zero. This is a consequence of
having local vertices and Bunch-Davies initial conditions. For a diagram like
ours, in general the allowed singularities are therefore at $k_{T}=0$ and
$k_{3}=0$. How can $k_{3}=0$ singularities arise? If we first perform the
$\eta$ integral, meaning that we take the $\theta(\eta^{\prime}-\eta)$ part of
the bulk-bulk propagator, then no $k_{3}=0$ singularities can arise. Indeed,
we would integrate from the far past at $\eta=-\infty$ up to $\eta^{\prime}$
and in the process singularities can only arise from exponential factors, and
their arguments will always contain a sum of energies. The subsequent
$\eta^{\prime}$ integral also cannot yield $k_{3}=0$ poles for the same
reason. If we do the $\eta^{\prime}$ integral first, however, then it does
look like inverse powers of $k_{3}$ are possible. The relevant part of the
integral is

$\displaystyle\frac{1}{k_{3}^{3}}\int_{-\infty}^{\eta}d\eta^{\prime}~{}k_{3}^{\beta}\eta^{\prime-\alpha}(1-ik_{3}\eta^{\prime})(1-\beta-
ik_{3}\eta^{\prime})e^{2ik_{3}\eta^{\prime}},$ (4.24)

where the factor of $k_{3}^{-3}$ comes from the power spectrum in the bulk-
bulk propagator. This integral contributes various powers of $k_{3}$ with the
smallest power given by $(\beta+\alpha-4)$. For $m=0$ we have
$\beta+\alpha-4=0$ so there are no inverse powers of $k_{3}$, while for $m\neq
0$ we have $\beta+\alpha-4=-2m$. However, the spatial derivatives in this case
introduce an additional factor of $k_{3}^{2m}$ so again there are no inverse
powers of $k_{3}$. No further inverse powers of $k_{3}$ can be generated when
we come to now do the $\eta$ integral since now all exponents contain sums of
energies. Any folded singularities drop out when we compute the full integral,
so only total-energy poles are allowed. We therefore conclude that in the
EFToI there are too many derivatives for singularities to occur as
$k_{3}\rightarrow 0$.

What about logs? We showed in the previous section that contact diagrams in
the EFToI can never have logs since there are too many derivatives in the
corresponding vertices. This analysis does not apply to single exchange
diagrams, but again we don’t expect logs to arise from the corresponding time
integrals. If such a log did arise from one of these single exchange diagrams,
it could not be cancelled by a contact diagram and since we are computing
parity-even correlators, it would contribute to the bispectra. This was proven
in generality in [12] where it was shown that only for parity-even
interactions can bispectra have a log. Furthermore, the log would also need to
depend on a sum of energies. Indeed, if it was to just depend on $k_{3}$ it
would need to come from first integrating with respect to $\eta^{\prime}$ but
because we have at least four derivatives, and therefore non-negative powers
of $\eta^{\prime}$, that part of the integral cannot give rise to a log. The
presence of a log would be felt in the leading soft theorem which is given by

$\displaystyle\langle\gamma^{h_{s}}_{\textbf{k}-\textbf{q}/2}\gamma^{h_{s}}_{-\textbf{k}-\textbf{q}/2}\gamma^{h_{l}}_{\textbf{q}}\rangle\sim\frac{3}{2}P^{h_{l}}(q)P^{h_{s}}(k)e_{ij}^{h_{l}}\hat{k}_{i}\hat{k}_{j},$
(4.25)

for some soft momentum q. The right hand side of this soft theorem does not
have a log dependence so there also cannot be a log on the left191919A similar
argument can be used to show that there cannot be any singularities at
$k_{3}=0$.. This shows that the cubic wavefunction coefficient, and therefore
single exchange diagrams in the EFToI, cannot have log singularities. We note
that the absence of logs might be highly non-trivial, and require
cancellations between different parts of the full integral in (4.16). As an
example, consider the case where $m=2$ and $n=0$ i.e. the quadratic correction
only has spatial derivatives. If we first compute the
$\theta(\eta-\eta^{\prime})$ part of the integral then we find that at late
times there is a log dependence of the form $\log[(k_{T}+2k_{3})/k_{T}]$,
while the $\theta(\eta^{\prime}-\eta)$ part of the integral contributes an
equal and opposite term such that no logs appear in the final result.

What about the MLT? It is easy to see that the MLT is satisfied for $k_{1}$
and $k_{2}$ since they only appear through the bulk-boundary propagators which
as we explained above ensures that the MLT is satisfied for those energies.
Now for $k_{3}$, the only dependence that we didn’t take into account above
was from $\text{Im}K(k_{3},\eta)$ which when expanded for small $k_{3}$ starts
at $\mathcal{O}(k_{3}^{3})$. It follows that the result of this integral
(multiplied by any factors coming from spatial derivatives) always contains a
factor of $k_{3}^{3}$, from both theta functions, which in turn ensures that
it satisfies the MLT for this leg too. So we conclude that the leading single
exchange diagrams in the EFToI only have poles as $k_{T}\rightarrow 0$ and
they always satisfy the MLT for all three legs. It follows that the final
result of these diagrams is captured by the analysis of [12].

Finally, let us outline how one goes from such an exchange contribution to the
wavefunction to expectation values. This was worked out in detail in [12]
where it was shown that the correction to the two-point function yields a
slightly more complicated relation compared to the case for contact diagrams.
Up to linear order in the correction to the two-point function, the
relationship is

$\displaystyle
B_{3}=\frac{1}{\Pi_{i=1}^{3}\mathcal{P}_{2}^{(0)}(k_{i})}\left(-\mathcal{P}_{3}^{\\{\lambda_{i}\\}}(\\{\textbf{k}\\},\\{k\\})+\mathcal{P}_{3}^{\\{\lambda_{i}\\}}(\\{\textbf{k}\\},\\{k\\})\left(\frac{\delta\mathcal{P}_{2}^{\lambda_{1}}(k_{1})}{\mathcal{P}_{2}^{(0)}(k_{1})}+2~{}\text{perms}\right)\right),$
(4.26)

where the permutations are of both momenta and helicity labels, and we have
defined the combination

$\displaystyle\mathcal{P}_{n}^{\\{\mu_{i}\\}}(\\{\textbf{k}\\},\\{k\\})=\psi_{n}^{\\{\mu_{i}\\}}(\\{\textbf{k}\\},\\{k\\})+\psi_{n}^{\\{\mu_{i}\\}}(\\{-\textbf{k}\\},\\{k\\})^{\star}.$
(4.27)

Here $\mathcal{P}_{2}^{(0)}$ is the GR contribution, with the two-point
function given by (4.5), while $\delta\mathcal{P}_{2}$ comes from the small
quadratic corrections. Note that this formula includes all tree-level
contributions to $\psi_{3}$, so both contact and exchange diagrams are added.
In Section 5 we will use this formula to convert our calculations of
wavefunction coefficients into bispectra.

###  4.3 Boundary terms

Before computing the bispectra, let us first verify that boundary terms
evaluated at $\eta=0$ cannot contribute to the late-time wavefunction. Such
terms could arise either independently, or as a consequence of the integration
by parts we have performed to arrive at (2.7). The general forms of boundary
contributions from two and three building block boundary operators are,
schematically,

$\lim\limits_{\eta\to 0^{-}}\left(\eta^{a+b-3}{\cal O}^{(a)}A\cdot{\cal
O}^{(b)}B\right),\quad\lim\limits_{\eta\to 0^{-}}\left(\eta^{a+b+c-3}{\cal
O}^{(a)}A\cdot{\cal O}^{(b)}B\cdot{\cal O}^{(c)}C\right)\,,$ (4.28)

where $a,b,c$ denote the number of derivatives in the corresponding operators,
$A,B,C\in\\{\delta K^{i}_{\ j},\eta^{2}\tilde{R}^{i}_{\ j}\\}$ are scale-
invariant objects, and indices in (4.28) have been suppressed. The powers of
$\eta$ are fixed by scale invariance. In conformal time, the extrinsic
curvature is given by

$\delta K^{i}_{\
j}=-\frac{1}{2}a^{2}g^{il}H\eta\gamma^{\prime}_{mn}\frac{\delta\left(e^{\gamma}\right)_{lj}}{\delta\gamma_{mn}}\,,$
(4.29)

and given that the graviton equation of motion dictates that asymptotically
$\gamma^{\prime}_{ij}\sim\mathcal{O}(\eta)$ as $\eta\to 0^{-}$, we have

$\delta K^{i}_{\ j}\sim\mathcal{O}(\eta^{2})\quad\text{as}\quad\eta\to
0^{-}\,.$ (4.30)

Here we have used the fact that $a^{2}g^{il}$ is constant at late-times.
Similarly, for the scale-invariant Ricci tensor we have

$\eta^{2}\tilde{R}^{i}_{\
j}\sim\mathcal{O}(\eta^{2})\quad\text{as}\quad\eta\to 0^{-}\,.$ (4.31)

This late-time behaviour of the building blocks ensures that the boundary
terms in (4.28) approach zero at the future boundary. Indeed, close to
$\eta=0$ we have

$\displaystyle\eta^{a+b-3}\prod\limits_{a,b}{\cal O}^{(x)}A$
$\displaystyle\sim$ $\displaystyle{\cal
O}\left(\eta^{a+b-3+\sum_{a,b}\max\\{2-x,0\\}}\right)\sim{\cal
O}\left(\eta^{1+\sum_{a,b}\max\\{0,x-2\\}}\right)\sim{\cal O}(\eta),$
$\displaystyle\eta^{a+b+c-3}\prod\limits_{a,b,c}{\cal O}^{(x)}A$
$\displaystyle\sim$ $\displaystyle{\cal
O}\left(\eta^{a+b+c-3+\sum_{a,b,c}\max\\{2-x,0\\}}\right)\sim{\cal
O}\left(\eta^{3+\sum_{a,b,c}\max\\{0,x-2\\}}\right)\sim{\cal O}(\eta^{3})\,.$

This shows that we can freely integrate by parts at the level of these
covariant building blocks, as we have done in Section 2, without having to
worry about boundary contributions to the wavefunction.

A question distinct from the one we have just discussed, is whether we could
have generated any boundary terms when we integrated by parts in Section 4.2
to remove all time derivatives from the bulk-bulk propagator to the bulk-
boundary one, c.f. (4.16). For clarity, recall that the nested time integral
of interest is

$\int\limits_{-\infty}^{0}d\eta^{\prime}\alpha(\\{k_{i}\\},\eta^{\prime})\int\limits_{-\infty}^{0}d\eta~{}k^{n}G^{(a)}(k;\eta,\eta^{\prime})K^{(b)}(k;\eta)\eta^{n+a+b-4}\,,$
(4.32)

where $a,b\geqslant 1$, because $\delta K_{ij}\propto\gamma^{\prime}_{ij}$ at
linear order so each propagator must include at least one time derivative.
Again the power of $\eta$ is fixed by scale invariance, and the factor of
$k^{n}$ originates from $n$ spatial derivatives acting on either line attached
to the quadratic vertex. The $\eta^{\prime}$ vertex is the one coming from GR
which won’t affect the below discussion since it doesn’t contain any time
derivatives. Our goal is to use integration by parts to eliminate all
conformal time derivatives from $G$. We have

$\displaystyle\int\limits_{-\infty}^{0}d\eta~{}G^{(a)}(\eta)K^{(b)}(\eta)\eta^{n_{s}+a+b-4}=(-1)^{a}\int\limits_{-\infty}^{0}d\eta~{}G(\eta)\partial^{a}_{\eta}\left(K^{(b)}(\eta)\eta^{n+a+b-4}\right)$
$\displaystyle+\sum\limits_{j=0}^{j=a-1}(-1)^{j}\left[G^{(a-1-j)}(\eta)\partial^{j}_{\eta}\left(K^{(b)}(\eta)\eta^{n+a+b-4}\right)\right]_{\eta\to
0}.$ (4.33)

What are the necessary and sufficient conditions for the boundary terms in the
second line to vanish? A useful limit is that as $\eta\to 0$, the bulk-bulk
propagator decays as $\eta^{3}$:

$G(k;\eta,\eta^{\prime})\sim\frac{1}{2}K(k;\eta^{\prime})\eta^{3}+\eta^{3}\cdot\mathcal{O}(k\eta)\quad\text{as}\quad\eta\to
0^{-}.$ (4.34)

First consider the case $a-1-j\leqslant 3$. We have

$\displaystyle G^{(a-1-j)}(k;\eta,\eta^{\prime})$ $\displaystyle\sim$
$\displaystyle{\cal O}\left(\eta^{4+j-a}\right),$ (4.35)
$\displaystyle\partial^{j}_{\eta}\left(K^{(b)}(k;\eta)\eta^{n_{s}+a+b-4}\right)$
$\displaystyle=$ $\displaystyle\mathcal{O}(\eta^{n_{s}+a+b-4-j}),$ (4.36)

so we conclude that

$G^{(a-1-j)}(k;\eta,\eta^{\prime})\partial^{j}_{\eta}\left(K^{(b)}(k;\eta)\eta^{n+a+b-4}\right)=\mathcal{O}(\eta^{n+b}).$
(4.37)

Thus, the boundary term decays at least as fast as $\eta^{n+b}$. Since
$b\geqslant 1$ (which must hold for operators constructed out of $\delta
K^{i}_{\ j}$), this is sufficient for the boundary term to vanish.

Now consider $a-1-j>3$, which implies $a>4+j\geqslant 4$. Then for the
boundary term to vanish it is necessary and sufficient that

$\lim\limits_{\eta\to 0}K^{(b)}(k;\eta)\eta^{n+a+b-4}=0.$ (4.38)

Since $a\geq 4$, the above relation holds for any $b$.

In conclusion, for quadratic operators that appear in the EFToI we can proceed
as we did in Section 4.2 and move all time derivatives in the bulk integrals
onto the bulk-boundary propagators, which significantly simplifies the
calculation we will perform in Section 5.

###  4.4 Spinor-helicity formalism

Once we have the final form of the various time integrals, the last step is to
account for the tensor structure. For the single exchange diagram of interest
in this paper, the nontrivial tensor structure comes from the GR vertex, while
for contact diagrams this structure originates from the EFToI cubic
interactions beyond GR. The computation and the final result are much
simplified if we work with the de Sitter spinor-helicity formalism developed
in [1]. Here we discuss the subject only very briefly and refer the reader to
[1, 12, 40, 76] for further details, with many useful relations given in [40].

A null four-vector $k_{\mu}$ can be represented by a pair of two-component,
non-Grassmanian spinors $(\lambda,\tilde{\lambda})$ as follows:

$k_{\alpha\dot{\alpha}}=\sigma^{\mu}_{\alpha\dot{\alpha}}k_{\mu}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}\,,$
(4.39)

where we defined $k_{\mu}=(k,\textbf{k})$ and
$\sigma^{\mu}=(\mathds{1},\bm{\sigma})$ are the Pauli matrices. Here we have
constructed a null four-vector from the spatial momenta with $k=|\textbf{k}|$.
In contrast to flat-space, here energy is not conserved so a number of spinor
helicity identities in flat-space are altered in de Sitter. For example,
conservation of spatial momenta for three particles yields

$\displaystyle\langle ab\rangle[ab]$
$\displaystyle=k_{T}I_{c},\qquad\text{for}\quad a\neq b\neq c,$ (4.40)
$\displaystyle\sum_{a=1}^{3}\lambda^{(a)}_{\alpha}\tilde{\lambda}^{(a)}_{\dot{\alpha}}$
$\displaystyle=k_{T}(\sigma^{0})_{\alpha\dot{\alpha}},$ (4.41)

where we have defined the spinor brackets

$\displaystyle\langle ab\rangle$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha\beta}\lambda^{(a)}_{\alpha}\lambda^{(b)}_{\beta},$
(4.42) $\displaystyle{}[ab]$ $\displaystyle=$
$\displaystyle\epsilon^{\dot{\alpha}\dot{\beta}}\tilde{\lambda}^{(a)}_{\dot{\alpha}}\tilde{\lambda}^{(b)}_{\dot{\beta}}\,,$
(4.43)

and $I_{a}=k_{b}+k_{c}-k_{a}$. The object
$(\bar{\sigma}^{0})^{\dot{\alpha}\alpha}$ can be used to pick out of the time
component of a vector e.g.
$(\bar{\sigma}^{0})^{\dot{\alpha}\alpha}k_{\alpha\dot{\alpha}}=2k$. This would
not be allowed in a Lorentz invariant theory but is perfectly fine in our
cosmological setting. This allows us to define polarisation vectors which have
a vanishing time component. If we write the polarization tensors for
$\gamma_{ij}$ as
$e^{\pm}_{ij}(\textbf{k})=e^{\pm}_{i}(\textbf{k})e^{\pm}_{j}(\textbf{k})$,
then we have

$\displaystyle e^{+}_{\alpha\dot{\alpha}}(\textbf{k})$ $\displaystyle=$
$\displaystyle\frac{\left(\sigma^{0}\right)_{\alpha\dot{\beta}}\tilde{\lambda}^{\dot{\beta}}\tilde{\lambda}_{\dot{\alpha}}}{k}\,,$
(4.44) $\displaystyle e^{-}_{\alpha\dot{\alpha}}(\textbf{k})$ $\displaystyle=$
$\displaystyle\frac{\left(\sigma^{0}\right)_{\beta\dot{\alpha}}\lambda^{\beta}\lambda_{\alpha}}{k}\,.$
(4.45)

The numerical factors in these expressions follow from
$e^{h}_{ij}(\textbf{k})e^{h^{\prime}}_{ij}(\textbf{k})^{\ast}=4\delta_{hh^{\prime}}$,
and the simplicity of these expressions motivates this particular
normalisation. One can check that these polarisation tensors are transverse to
the momentum and have a vanishing time component. For parity-even interactions
at three-points, useful formulae for going from polarisations to spinors are

$\displaystyle e^{a+}\cdot e^{b+}=-\frac{[ab]^{2}}{2k_{a}k_{b}},\qquad
e^{a-}\cdot e^{b-}=-\frac{\langle ab\rangle^{2}}{2k_{a}k_{b}},\qquad
e^{a+}\cdot e^{b-}=\frac{I_{b}^{2}}{2k_{a}k_{b}}\frac{\langle
cb\rangle^{2}}{\langle
ca\rangle^{2}}=\frac{I_{a}^{2}}{2k_{a}k_{b}}\frac{[ca]^{2}}{[cb]^{2}},$ (4.46)
$\displaystyle k^{a}\cdot
e^{b+}=\frac{I_{b}}{2k_{b}}\frac{[ab][bc]}{[ac]},\qquad k^{a}\cdot
e^{b-}=\frac{I_{b}}{2k_{b}}\frac{\langle ab\rangle\langle bc\rangle}{\langle
ac\rangle}.$ (4.47)

The wavefunction coefficients can then always be expressed in terms of
coupling constants, the square and angle brackets, and the energies $k_{a}$,
with the linear combinations $I_{a}$ playing a special role. Let us therefore
convert all tensor structures into bracket expressions. The tensor structure
part of $\psi_{3,+++}$ always takes the following form [12]

$\left(e^{+}(\textbf{k}_{1})e^{+}(\textbf{k}_{2})e^{+}(\textbf{k}_{3})\textbf{k}_{1}^{\alpha_{1}}\textbf{k}_{2}^{\alpha_{2}}\textbf{k}_{3}^{\alpha_{3}}\right)=\frac{[12]^{2}[23]^{2}[31]^{2}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}}h_{\alpha}(k_{a})\equiv{\rm
SH}_{+++}h_{\alpha}(k_{a})\,,$ (4.48)

where $h_{\alpha}$ is a polynomial in the energies, of dimension
$\alpha=\alpha_{1}+\alpha_{2}+\alpha_{3}$. For parity even interactions, which
we are considering here, possible values are $\alpha=0,2,4,6$. For each
interaction, $h_{\alpha}$ must be found by an explicit calculation. For
example, the GR vertex can be represented by $h_{\alpha=2}$. The simplicity of
the method lies in the fact that once the $+++$ computation is performed, all
other helicity configurations come almost automatically. The time integral
part of the wavefunction coefficient is the same as for the $+++$, while the
tensor structure part can be found as follows:

$\displaystyle\left(e^{+}(\textbf{k}_{1})e^{+}(\textbf{k}_{2})e^{-}(\textbf{k}_{3})\textbf{k}_{1}^{\alpha_{1}}\textbf{k}_{2}^{\alpha_{2}}\textbf{k}_{3}^{\alpha_{3}}\right)$
$\displaystyle=$
$\displaystyle\frac{[12]^{6}}{[23]^{2}[31]^{2}}\frac{I_{1}^{2}I_{2}^{2}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}}h_{\alpha}(k_{1},k_{2},-k_{3})\equiv{\rm
SH}_{++-}h_{\alpha}(k_{1},k_{2},-k_{3})\,,$
$\displaystyle\left(e^{-}(\textbf{k}_{1})e^{-}(\textbf{k}_{2})e^{+}(\textbf{k}_{3})\textbf{k}_{1}^{\alpha_{1}}\textbf{k}_{2}^{\alpha_{2}}\textbf{k}_{3}^{\alpha_{3}}\right)$
$\displaystyle=$ $\displaystyle\frac{\langle 12\rangle^{6}}{\langle
23\rangle^{2}\langle
31\rangle^{2}}\frac{I_{1}^{2}I_{2}^{2}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}}h_{\alpha}(k_{1},k_{2},-k_{3})\equiv{\rm
SH}_{--+}h_{\alpha}(k_{1},k_{2},-k_{3})\,,$
$\displaystyle\left(e^{-}(\textbf{k}_{1})e^{-}(\textbf{k}_{2})e^{-}(\textbf{k}_{3})\textbf{k}_{1}^{\alpha_{1}}\textbf{k}_{2}^{\alpha_{2}}\textbf{k}_{3}^{\alpha_{3}}\right)$
$\displaystyle=$ $\displaystyle\frac{\langle 12\rangle^{2}\langle
23\rangle^{2}\langle
31\rangle^{2}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}}h_{\alpha}(k_{1},k_{2},k_{3})\equiv{\rm
SH}_{---}h_{\alpha}(k_{a})\,,$ (4.49)

We refer the reader to [12] for a derivation and a more in-depth discussion of
this construction. We will present the final form of graviton bispectra using
these spinor variables.

##  5 Type-I bispectra

We now start computing the graviton bispectra in the EFToI. We begin with the
Type-I bispectra that arise from two building block operators. In this case
both single exchange and contact diagrams contribute and we deal with them in
turn. We then write down the final form of the bispectra which requires us to
convert our wavefunction expressions into expectation values. Finally we check
that our results satisfy the consistency relations which relate the soft limit
of a three-point function to the two-point function[15, 16, 17], which now
receives corrections. This provides a non-trivial check of our final result.

###  5.1 Single exchange diagram

We begin with the exchange contributions to the wavefunction which arise due
to the quadratic corrections to the free theory in (3.3). We have computed
these contributions when we gave examples of the Feynman rules for exchange
diagrams, and they are given by (4.12) and (4.2). If we write the polarisation
tensors in such a way that we have full symmetry in particles $1$ and $2$,
then for the $+++$ configuration we have

$\displaystyle
4e_{ik}^{h_{1}}e_{jl}^{h_{2}}k_{k}^{3}k_{l}^{3}e_{ij}^{h_{3}}-e_{ij}^{h_{1}}e_{kl}^{h_{2}}k_{k}^{3}k_{l}^{3}e_{ij}^{h_{3}}-e_{ij}^{h_{2}}e_{kl}^{h_{1}}k_{k}^{3}k_{l}^{3}e_{ij}^{h_{3}}=-\frac{1}{16}\text{SH}_{+++}\left(4I_{1}I_{2}+I_{1}^{2}+I_{2}^{2}\right).$
(5.1)

Using this expression in (4.12) and (4.2), for the $+++$ configuration, we
find

$\displaystyle\psi^{+++}_{\text{2BB, exchange}}$
$\displaystyle=\sum_{n=1}\frac{iM_{\text{pl}}^{4}g_{n,0}}{64}\text{SH}_{+++}k_{T}^{2}\mathcal{I}_{n,0}(k_{1},k_{2},k_{3})+2~{}\text{perms},$
(5.2)
$\displaystyle+\sum_{n=0,m=1}\frac{(-1)^{m}iM_{\text{pl}}^{4}g_{n,m}}{64}\text{SH}_{+++}k_{T}^{2}k_{3}^{2m}\mathcal{I}_{n,m}(k_{1},k_{2},k_{3})+2~{}\text{perms}.$
(5.3)

Notice how once we summed over the remaining two permutations in the tensor
structure, the contribution reduced simply to $k_{T}^{2}$. Recall that here we
are using the tensor structure of GR where this factor of $k_{T}^{2}$ is
familiar [1] and cancels the $1/k_{T}^{2}$ that comes from the time integral
when computed in pure gravity. This means that the $+++$ configuration for
pure gravity in de Sitter space does not have a total-energy pole which can be
traced back to the fact that the corresponding amplitude for this
configuration is zero. We will see that this behaviour is unique to pure
gravity: the EFToI corrections to the bispectrum do indeed have total-energy
poles for the $+++$ configuration. As we explained above we can now easily
extract the $++-$ configuration which is given by

$\displaystyle\psi^{++-}_{\text{2BB, exchange}}$
$\displaystyle=\sum_{n=1}\frac{iM_{\text{pl}}^{4}g_{n,0}}{64}\text{SH}_{++-}I_{3}^{2}\left(\mathcal{I}_{n,0}(k_{1},k_{2},k_{3})+2~{}\text{perms}\right),$
(5.4)
$\displaystyle+\sum_{n=0,m=1}\frac{(-1)^{m}iM_{\text{pl}}^{4}g_{n,m}}{64}\text{SH}_{++-}I_{3}^{2}\left(k_{3}^{2m}\mathcal{I}_{n,m}(k_{1},k_{2},k_{3})+2~{}\text{perms}\right),$
(5.5)

where we have used the fact that $k_{T}^{2}\rightarrow I_{3}^{2}$ as we send
$k_{3}\rightarrow-k_{3}$. We remind the reader that $\mathcal{I}_{n,0}$ and
$\mathcal{I}_{n,m}$ are defined in (4.2) and (4.2), which can be compactly
written as (4.2),(4.20) and (4.21) in terms of the master time integral
(4.16). Our focus now is on computing this master time integral.

To compute this integral we use the formalism developed in [33] to write down
a dispersion formula in terms of discontinuities of the bulk-boundary
propagator. In order to ensure a Bunch-Davies vacuum in the infinite past
requires the use of the $i\epsilon$ prescription, $k\to k-i\epsilon$, where
the norm of $k$ is given a negative imaginary part, i.e. $-k\to-k+i\epsilon$.
In polar coordinates, $k^{2}=e^{i\theta}$, this becomes the condition
$\theta\in(-2\pi,0)$. If $\theta$ is in this interval, then the Feynman
integrals converge. It is therefore natural to place the $k^{2}$ branch cut on
the positive real axis. We then define the following monodromy operation

${\rm
disc}_{p^{2}}f(k^{2})=f((e^{-i\pi}k)^{2})-f(k^{2})=f((-k+i\epsilon)^{2})-f((k-i\epsilon)^{2})=f((-k^{*})^{2})-f(k^{2}),$
(5.6)

i.e. the argument of the term on the left hand side comes from rotating $k$ in
the complex plane by $\theta=-\pi$. By utilising the Hermitian analyticity of
the bulk-boundary propagator, we can express the discontinuity of the bulk-
boundary propagator as202020A function is described to be Hermitian analytic
provided it satisfies the relation $f^{*}(-k^{*})=f(k)$. We refer the reader
to [29, 27] for further discussion on this topic.

${\rm
disc}_{p^{2}}K(p,\eta)=K(e^{-i\pi}p,\eta)-K(p,\eta)=K^{*}(p,\eta)-K(p,\eta)=-2i\,{\rm
Im}K(p,\eta).$ (5.7)

We can write a general bulk-bulk propagator $G_{\nu}(k,\eta,\eta^{\prime})$ in
terms of a dispersion formula[33]

$G_{\nu}(k,\eta,\eta^{\prime})=\frac{1}{2\pi
i}\int_{0}^{+\infty}\frac{dp^{2}}{p^{2}-k^{2}+i\epsilon}{\rm
disc}_{p^{2}}G_{\nu}(p,\eta,\eta^{\prime}),$ (5.8)

where $\nu$ is the usual order of the Hankel function related to the mass of
the bulk field by

$\nu=\sqrt{\frac{9}{4}-\frac{m^{2}}{H^{2}}}\,.$ (5.9)

The discontinuity of the bulk-bulk propagator can then be expressed in terms
of the discontinuities of the bulk-boundary propagator as

${\rm disc}_{p^{2}}G_{\nu}(p,\eta,\eta^{\prime})=iP_{\nu}(p){\rm
disc}_{p^{2}}K(p,\eta){\rm disc}_{p^{2}}K(p,\eta^{\prime})\,,$ (5.10)

where $P_{\nu}$ is the power spectrum. Given that we are working with massless
gravitons in the EFToI, we can take $\nu=3/2$, allowing us to rewrite (4.16)
as

$\displaystyle\mathcal{M}(\alpha,\beta)=\frac{1}{2\pi i}$
$\displaystyle\int_{-\infty}^{+\infty}\frac{dp}{p^{2}-k_{3}^{2}+i\epsilon}\frac{iH^{2}}{2M_{\text{pl}}^{2}\,p^{2}}\int_{-\infty}^{0}d\eta\,(-H\eta)^{-2}K(k_{1},\eta)K(k_{2},\eta)[K^{*}(p,\eta)-K(p,\eta)]$
$\displaystyle\int_{-\infty}^{0}d\eta^{\prime}\,(-H\eta^{\prime})^{-\alpha}K^{(\beta)}(k_{3},\eta^{\prime})[K^{*}(p,\eta^{\prime})-K(p,\eta^{\prime})]\,.$
(5.11)

Two comments are in order. First, we have switched the order of the integrals.
This is typically not allowed: the behavior of ${\rm disc}_{p^{2}}K(p,\eta)$
in the infinite past is such that the $\eta$ integral converges only for
particular values of $p^{2}$. One can carry out the integral for these values
and then analytically continue: the final result is that the nested integral
becomes an integral in $p^{2}$ of the product of the discontinuities of single
integrals. The second observation is that the product of these discontinuities
is an even function of $p$. We can then change variables from $p^{2}$ to $p$,
and take advantage of the fact that the whole integrand is even in $p$ to
extend the range of integration from $-\infty$ to $+\infty$ [33]. Carrying out
the integrals in $d\eta$ and $d\eta^{\prime}$ we find

$\mathcal{M}(\alpha,\beta)=\frac{1}{2\pi i}\int_{-\infty}^{+\infty}dp\;{\cal
N}(k_{1},k_{2},k_{3},p)\,,$ (5.12)

where we have dropped the $i\epsilon$ for simplicity of notation. The function
$\cal N$ is given by

$\begin{split}{\cal
N}(k_{1},k_{2},k_{3},p)=&{-\frac{iH^{-\alpha}}{M_{\text{pl}}^{2}}}\frac{p(k_{1}^{2}+4k_{1}k_{2}+k_{2}^{2}-p^{2})}{(p^{2}-k_{3}^{2})(k_{1}+k_{2}-p)^{2}(k_{1}+k_{2}+p)^{2}}\\\
&\;\;\;\;\times(ik_{3})^{\beta}(-\alpha)!i^{\alpha}\bigg{[}\frac{\left((k_{3}-p)(k_{3}+p)^{\alpha}-(k_{3}+p)(k_{3}-p)^{\alpha}\right)(\beta-1)}{k^{2}_{3}-p^{2}}\\\
&\hphantom{\;\;\;\;\times(ik_{3})^{\beta}(-\alpha)!i^{\alpha}\bigg{[}}+({-\alpha}+1)\big{(}(k_{3}-p)^{\alpha-2}(k_{3}+\beta
p-p)-(k_{3}+p)^{\alpha-2}(k_{3}-\beta p+p)\big{)}\\\
&\hphantom{\;\;\;\;\times(ik_{3})^{\beta}(-\alpha)!i^{\alpha}\bigg{[}}-({-\alpha}+2)({-\alpha}+1)k_{3}p\big{(}(k_{3}-p)^{\alpha-3}+(k_{3}+p)^{\alpha-3}\big{)}\bigg{]}\,,\end{split}$
(5.13)

which is manifestly symmetric under $p\to{-p}$. We also see that the integrand
in (5.13) vanishes as $p\to\infty$ in the complex plane, and therefore we can
close the $p$-contour in either the upper or lower half-plane.

The poles in $p$ are located at

$\displaystyle p$ $\displaystyle=\pm k_{3}$ $\displaystyle p$
$\displaystyle=\pm(k_{1}+k_{2}).$ (5.14)

When evaluating the contour integral, we should recall that the Bunch-Davies
boundary condition implies that the $k_{a}$ have a small, negative imaginary
part. Finally, we can close the contour in the lower half-plane to find

$\mathcal{M}(\alpha,\beta)={-\left({\rm Res}_{p=(k_{1}+k_{2})}\bigg{[}{\cal
N}(k_{1},k_{2},k_{3},p)\bigg{]}+{\rm Res}_{p=k_{3}}\bigg{[}{\cal
N}(k_{1},k_{2},k_{3},s)\bigg{]}\right)}\,.$ (5.15)

The residue at $(k_{1}+k_{2})$ can be computed for generic $n,m$ but the same
is not true for the residue at $k_{3}$, for which we cannot find a closed-form
expression. In any case we notice that, taken separately, the residues at
$k_{3}$ and $(k_{1}+k_{2})$ both present a divergence at $k_{1}+k_{2}=k_{3}$.
It is only when combined that such divergence cancels, as it was expected from
the analysis of Section 4.2.

Let us end this section by writing down some expressions for the time
integrals of interest so we can see the expected properties explicitly. Since
in all cases the result must be symmetric in the exchange of $k_{1}$ and
$k_{2}$, we write the integrals in terms of the symmetric polynomials in two
variables: $\hat{e}_{1}=k_{1}+k_{2}$, $\hat{e}_{2}=k_{1}k_{2}$. We still use
$k_{T}=\hat{e}_{1}+k_{3}$. We have:

$\displaystyle\mathcal{I}_{1,0}$
$\displaystyle=\frac{ik_{3}^{3}}{2M_{\text{pl}}^{2}k_{T}^{5}}(7\hat{e}_{1}^{3}+14\hat{e}_{1}\hat{e}_{2}+11\hat{e}_{1}^{2}k_{3}-2\hat{e}_{2}k_{3}+5\hat{e}_{1}k_{3}^{2}+k_{3}^{3}),$
(5.16) $\displaystyle\mathcal{I}_{2,0}$
$\displaystyle=\frac{3iH^{2}k_{3}^{3}}{2M_{\text{pl}}^{2}k_{T}^{7}}[15\hat{e}_{1}^{5}+30\hat{e}_{1}^{3}\hat{e}_{2}+k_{3}(25\hat{e}_{1}^{4}-30\hat{e}_{1}^{2}\hat{e}_{2})$
$\displaystyle\;\;\;\;+k_{3}^{2}(26\hat{e}_{1}^{3}+34\hat{e}_{1}\hat{e_{2}})+k_{3}^{3}(22\hat{e}_{1}^{2}-2\hat{e}_{2})+7k_{3}^{4}\hat{e}_{1}+k_{3}^{5}],$
(5.17) $\displaystyle\mathcal{I}_{0,1}$
$\displaystyle=\frac{ik_{3}}{2M_{\text{pl}}^{2}k_{T}^{5}}(-\hat{e}_{1}^{3}-2\hat{e}_{1}\hat{e}_{2}+3\hat{e}_{1}^{2}k_{3}+14\hat{e}_{2}k_{3}+5\hat{e}_{1}k_{3}^{2}+k_{3}^{3}),$
(5.18) $\displaystyle\mathcal{I}_{1,1}$
$\displaystyle=\frac{iHk_{3}}{4M_{\text{pl}}^{2}k_{T}^{5}}(-\hat{e}_{1}^{3}-2\hat{e}_{1}\hat{e}_{2}+3\hat{e}_{1}^{2}k_{3}+14\hat{e}_{2}k_{3}+5\hat{e}_{1}k_{3}^{2}+k_{3}^{3}),$
(5.19) $\displaystyle\mathcal{I}_{2,1}$
$\displaystyle=\frac{12iH^{2}k_{3}^{2}}{M_{\text{pl}}^{2}k_{T}^{7}}(\hat{e}_{1}^{4}+3\hat{e}_{1}^{2}\hat{e}_{2}-7\hat{e}_{1}\hat{e}_{2}k_{3}-\hat{e}_{1}^{2}k_{3}^{2}+2\hat{e}_{2}k_{3}^{2}).$
(5.20)

We note that the expressions for $\mathcal{I}_{0,1}$ and $\mathcal{I}_{1,1}$
are proportional which was to be expected. This is a consequence of the fact
that for $m\neq 0$ and odd $n$, we can integrate by parts to write the
quadratic mixing in terms of operators with lower values of $n$. As we
explained before, it is still useful to keep our current labelling and
definition of $n$ however, since we expect the contact contributions to be
different for all $n$. For larger values of $n$, it will remain the case that
for odd $n$ we can write the result in terms of expressions with lower and
even $n$. As another example, we have
$\mathcal{I}_{3,1}=\frac{9H}{2}\mathcal{I}_{2,1}-3H^{2}\mathcal{I}_{1,1}$. As
expected, we see that only total-energy poles arise, and once we include the
factors of $k_{3}^{2m}$ the MLT will be satisfied for each external leg.

###  5.2 Contact diagram

In addition to the exchange contributions we have just computed, for Type-I
bispectra there are also contact diagrams that contribute to the bispectra at
the same order in perturbation theory. These arise due to the cubic
interactions in (3.3) which take the form

$\displaystyle S_{\gamma,\text{2BB}}\supset\frac{M_{\text{pl}}^{2}}{4}\int
d\eta
d^{3}x\sum_{n,m=1}g_{n,m}a^{q}(\eta)(\gamma_{ij})^{n+1}\left(-m\gamma^{lk}\partial^{2m-2}\partial_{l}\partial_{k}\gamma^{\prime}_{ij}+\sum_{p=0}^{m-1}\partial^{2p}Q_{s}[\gamma,\partial^{2m-2-2p}\gamma^{\prime}]_{ij}\right),$
(5.21)

where $q=2-2m-n$ and $m\geq 1$. Each contribution contains one $\gamma_{ij}$
with $n+1$ time derivatives, one with a single time derivative, and one
without any. Up to permutations and overall factors, the time integral that we
therefore need to compute is

$\displaystyle\mathcal{J}_{n,m}(k_{1},k_{2},k_{3})=\int
d\eta~{}\eta^{n+2m-2}K^{(n+1)}(k_{1},\eta)K^{\prime}(k_{2},\eta)K(k_{3},\eta).$
(5.22)

With the result of this integral we can multiply it by the tensor structure
dictated by the cubic interactions, then sum over permutations. Given that
$n+2m-2\geq 0$, the result of this integral will only have poles at $k_{T}=0$,
and no logs. Using (4.23) and

$\displaystyle\lim_{\eta\rightarrow 0}\int
d\eta~{}\eta^{z}e^{ik_{T}\eta}=\frac{z!(-i)^{1-z}}{k_{T}^{1+z}}+\mathcal{O}(\eta^{z+1}),$
(5.23)

for $z\geq 0$, we find that at late times the bulk time integral yields

$\displaystyle\mathcal{J}_{n,m}^{123}$
$\displaystyle=(-1)^{m-n}in(n+2m-1)!\frac{k_{1}^{n+1}k_{2}^{2}}{k_{T}^{n+2m}}$
$\displaystyle+(-1)^{1+m-n}i(n+2m)!\frac{k_{1}^{n+1}k_{2}^{2}(k_{1}-nk_{3})}{k_{T}^{n+2m+1}}$
$\displaystyle+(-1)^{1+m-n}i(n+2m+1)!\frac{k_{1}^{n+2}k_{2}^{2}k_{3}}{k_{T}^{n+2m+2}},$
(5.24)

where we have introduced a more compact notation with
$\mathcal{J}_{n,m}^{123}\equiv\mathcal{J}_{n,m}(k_{1},k_{2},k_{3})$. We see
that the degree of the leading total-energy pole is equal to the number of
derivatives in the cubic vertices, as expected [42], and one can check that
this expression satisfies the MLT for each external energy.

Now for the tensor structure we have

$\displaystyle
Q_{s}[\gamma,\gamma^{\prime}]_{ij}=-\partial_{i}\gamma_{kl}\partial_{k}\gamma^{\prime}_{lj}+\partial_{l}\gamma_{jk}\partial_{k}\gamma^{\prime}_{il}+(i\leftrightarrow
j),$ (5.25)

and in the final term in (5.21) we can integrate by parts to move all
$\partial^{2p}$ terms onto the first $\gamma_{ij}$ such that

$\displaystyle S_{\gamma,\text{2BB}}\supset-$
$\displaystyle\frac{M_{\text{pl}}^{2}}{4}\int d\eta
d^{3}x\sum_{n,m=1}a^{q}(\eta)mg_{n,m}(\gamma_{ij})^{n+1}\gamma^{lk}\partial^{2m-2}\partial_{l}\partial_{k}\gamma^{\prime}_{ij}$
$\displaystyle+$ $\displaystyle\frac{M_{\text{pl}}^{2}}{4}\int d\eta
d^{3}x\sum_{n,m=1}a^{q}(\eta)\sum_{p=0}^{m-1}g_{n,m}\partial^{2p}(\gamma_{ij})^{n+1}Q_{s}[\gamma,\partial^{2m-2-2p}\gamma^{\prime}]_{ij}.$
(5.26)

From the first line, once we convert to momentum space using (1.4), we will
find tensor structures of the form
$k_{2}^{2m-2}e_{ij}^{h_{1}}(\textbf{k}_{1})e_{ij}^{h_{2}}(\textbf{k}_{2})k_{l}^{2}k_{k}^{2}e_{lk}^{h_{3}}(\textbf{k}_{3})$,
and if we convert this expression into one with spinors, for the $+++$
configuration we find

$\displaystyle
k_{2}^{2m-2}e_{ij}^{+}(\textbf{k}_{1})e_{ij}^{+}(\textbf{k}_{2})k_{l}^{2}k_{k}^{2}e_{lk}^{+}(\textbf{k}_{3})=\frac{1}{16}k_{2}^{2m-2}I_{3}^{2}\text{SH}_{+++}.$
(5.27)

We remind the reader that the $++-$ configuration can be extracted from this
expression, as we explained above. For the second line, the tensor structures
are of the form
$k_{1}^{2p}k_{2}^{2m-2-2p}e_{ij}^{h_{1}}(\textbf{k}_{1})[e_{il}^{h_{2}}(\textbf{k}_{2})e_{jk}^{h_{3}}(\textbf{k}_{3})k_{k}^{2}k_{l}^{3}-e_{lj}^{h_{2}}(\textbf{k}_{2})e_{kl}^{h_{3}}(\textbf{k}_{3})k_{k}^{2}k_{i}^{3}]$,
which when converted to spinors for the $+++$ configuration yields

$\displaystyle
k_{1}^{2p}k_{2}^{2m-2-2p}e_{ij}^{+}(\textbf{k}_{1})[e_{il}^{+}(\textbf{k}_{2})e_{jk}^{+}(\textbf{k}_{3})k_{k}^{2}k_{l}^{3}-e_{lj}^{+}(\textbf{k}_{2})e_{kl}^{+}(\textbf{k}_{3})k_{k}^{2}k_{i}^{3}]=-\frac{1}{8}k_{1}^{2p}k_{2}^{2m-2-2p}k_{3}I_{3}\text{SH}_{+++}.$
(5.28)

We can now collect everything together. By multiplying these tensor structures
by the result of the time integral, and including all constant factors as
dictated by the Feynman rules we reviewed in Section 4, for the $+++$ helicity
configuration we find

$\displaystyle\psi_{\text{2BB,contact}}^{+++}=\frac{iM^{2}_{\rm
pl}}{64}\text{SH}_{+++}\sum_{n,m=1}(-1)^{n+m}g_{n,m}H^{n+2m-2}m\left[k_{2}^{2m-2}I_{3}^{2}\mathcal{J}_{n,m}^{123}+5~{}\text{perms}\right]$
$\displaystyle+\frac{iM^{2}_{\rm
pl}}{16}\text{SH}_{+++}\sum_{n,m=1}(-1)^{n+m}g_{n,m}H^{n+2m-2}\sum_{p=0}^{m-1}\left[k_{1}^{2p}k_{2}^{2m-2p-2}k_{3}I_{3}\mathcal{J}_{n,m}^{123}+5~{}\text{perms}\right].$
(5.29)

We have summed over the remaining permutations to find an object with the
correct symmetry. The symmetrisation in the expression for $Q_{s}$ simply
yields a factor of $2$. Now to extract the wavefunction coefficient for the
$++-$ helicity configuration we need to send $k_{3}\rightarrow-k_{3}$ and
$\text{SH}_{+++}\rightarrow\text{SH}_{++-}$ for each term, while keeping
$\mathcal{J}_{n,m}$ fixed. This yields

$\displaystyle\psi_{\text{2BB,contact}}^{++-}=\frac{iM^{2}_{\rm
pl}}{64}\text{SH}_{++-}\sum_{n,m=1}(-1)^{n+m}g_{n,m}H^{n+2m-2}m[k_{2}^{2m-2}k_{T}^{2}\mathcal{J}_{n,m}^{123}+k_{1}^{2m-2}k_{T}^{2}\mathcal{J}_{n,m}^{213}$
$\displaystyle+k_{1}^{2m-2}I_{1}^{2}\mathcal{J}_{n,m}^{312}+k_{3}^{2m-2}I_{1}^{2}\mathcal{J}_{n,m}^{132}+k_{2}^{2m-2}I_{2}^{2}\mathcal{J}_{n,m}^{321}+k_{3}^{2m-2}I_{2}^{2}\mathcal{J}_{n,m}^{231}]$
$\displaystyle+\frac{iM^{2}_{\rm
pl}}{16}\text{SH}_{++-}\sum_{n,m=1}(-1)^{n+m}g_{n,m}H^{n+2m-2}\sum_{p=0}^{m-1}[-k_{1}^{2p}k_{2}^{2m-2p-2}k_{3}k_{T}\mathcal{J}_{n,m}^{123}$
$\displaystyle-
k_{2}^{2p}k_{1}^{2m-2p-2}k_{3}k_{T}\mathcal{J}_{n,m}^{213}-k_{1}^{2p}k_{3}^{2m-2p-2}k_{2}I_{1}\mathcal{J}^{132}_{n,m}$
$\displaystyle-
k_{3}^{2p}k_{1}^{2m-2p-2}k_{2}I_{1}\mathcal{J}^{312}_{n,m}-k_{2}^{2p}k_{3}^{2m-2p-2}k_{1}I_{2}\mathcal{J}_{n,m}^{231}-k_{3}^{2p}k_{2}^{2m-2p-2}k_{1}I_{2}\mathcal{J}_{n,m}^{321}].$
(5.30)

We note that for both helicity configurations these contributions to the
wavefunction are real since the overall factor of $i$ is cancelled by the $i$
in $\mathcal{J}_{n,m}$. This ensures that these contributions do indeed
contribute to the bispectra since for parity-even interactions only the real
part contributes (c.f. (4.4)).

###  5.3 Putting everything together

Now that we have all contributions to the cubic wavefunction coefficient, to
leading order in new couplings, we can now convert these into expressions for
the bispectra. In perturbation theory expectation values are algebraically
related to wavefunction coefficients with the relations derived in a number of
places e.g. [12, 22, 77]. When there is a small correction to the two-point
function, as is the case here, the general expression is given by (4.26) which
for parity-even interactions becomes

$\displaystyle
B_{3}^{\\{\lambda_{i}\\}}=\frac{1}{\Pi_{i=1}^{3}2\text{Re}\left(\psi_{2,\text{GR}}\right)}\left[-2\text{Re}\left(\psi_{3,\text{total}}^{\\{\lambda_{i}\\}}\right)+2\text{Re}\left(\psi_{3,\text{GR}}^{\\{\lambda_{i}\\}}\right)\left(\frac{\text{Re}\left(\delta\psi_{2}^{\lambda_{1}}\right)}{\text{Re}\left(\psi_{2,\text{GR}}\right)}+\text{2
perms}\right)\right],$ (5.31)

where $\psi_{2,\text{GR}}$ is the GR contribution to the two-point function,
while $\delta\psi_{2}$ is a small correction due to our higher-derivative
corrections to the quadratic action. Since we are working up to linear order
in new couplings, we take the first term in the square brackets to be all
contributions we have computed in this section i.e.
$\psi_{3,\text{total}}=\psi_{\text{2BB,exchange}}+\psi_{\text{2BB,contact}}$,
while the $\psi_{3}$ in the second term must be the GR contribution since
$\delta\psi_{2}$ is already linear the new couplings. In addition to
$\psi_{3,\text{total}}$, we now also need the GR wavefunction up to cubic
order, and the small corrections to $\psi_{2}$.

We have essentially already computed the GR cubic wavefunction coefficient
when we computed the Type-I exchange diagrams so let us simply write the
result here. We have212121As we did with the quadratic wavefunction, here we
are dropping any imaginary contributions to the GR cubic wavefunction
coefficient.

$\displaystyle\psi^{+++}_{3,\text{GR}}=\frac{M_{\text{pl}}^{2}}{64H^{2}}\text{SH}_{+++}\frac{k_{T}^{2}}{k_{T}^{2}}(e_{3}+k_{T}e_{2}-k_{T}^{3}),$
(5.32)
$\displaystyle\psi^{++-}_{3,\text{GR}}=\frac{M_{\text{pl}}^{2}}{64H^{2}}\text{SH}_{++-}\frac{I_{3}^{2}}{k_{T}^{2}}(e_{3}+k_{T}e_{2}-k_{T}^{3}).$
(5.33)

For the corrections to the two-point function, we need to compute a Feynman
diagram that is analogous to the one in Figure 1 but with two rather than
three external legs. Since this is a small correction to the quadratic
wavefunction, we compute it in the way we compute any contact diagram: we
insert a bulk-boundary propagator for each external line, add tensor
structures and time derivatives as dictated by corresponding the bulk vertex,
and use the Feynman rules we discussed above. For example, for $m=0$ we have

$\displaystyle\delta\psi_{2}^{hh^{\prime}}$
$\displaystyle=-i\sum_{n=1}\frac{g_{n,0}M_{\text{pl}}^{2}H^{2n-2}}{4}\int
d\eta~{}\eta^{2n-2}K^{(2n+1)}_{\gamma}(k,\eta)K^{\prime}_{\gamma}(k,\eta)\times
4\delta_{hh^{\prime}}\times 2$ (5.34)
$\displaystyle=\sum_{n=1}\frac{(2n)!}{4^{n}}g_{n,0}M_{\text{pl}}^{2}H^{2n-2}k^{3}\delta_{hh^{\prime}},$
(5.35)

where we have used momentum conservation and summed over the two possible
permutations. The computation for $m\neq 0$ is very similar and in total we
have

$\displaystyle\delta\psi_{2}^{hh^{\prime}}=$
$\displaystyle\sum_{n=1}\frac{(2n)!}{4^{n}}g_{n,0}M_{\text{pl}}^{2}H^{2n-2}k^{3}\delta_{hh^{\prime}}$
$\displaystyle+\sum_{n=0,m=1}\frac{(2m+n-1)!(n-2m)}{2^{n+2m}}g_{n,m}M_{\text{pl}}^{2}H^{n+2m-2}k^{3}\delta_{hh^{\prime}}.$
(5.36)

We now have all the ingredients to compute the bispectra. Below we write down
their final forms at linear order in the new couplings for a few choices of
$n$ and $m$. We define

$\displaystyle\delta
P=-\frac{P_{\text{GR}}\delta\psi_{2}}{\psi_{2,\text{GR}}}=-\frac{H^{4}}{2M_{\text{pl}}^{4}k^{6}}\delta\psi_{2}.$
(5.37)

We have, for the examples with the lowest degree leading total-energy poles,

$\displaystyle\text{GR}:$ $\displaystyle\quad
B_{3}^{+++}=\frac{H^{4}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{+++}(k_{T}^{3}-k_{T}e_{2}-e_{3})$
(5.38) $\displaystyle\quad
B_{3}^{++-}=\frac{H^{4}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{++-}\frac{I_{3}^{2}}{k_{T}^{2}}(k_{T}^{3}-k_{T}e_{2}-e_{3}),$
(5.39) $\displaystyle\quad
P_{\text{GR}}^{hh^{\prime}}=\frac{H^{2}}{2M_{\text{pl}}^{2}k^{3}}\delta_{hh^{\prime}}.$
(5.40) $\displaystyle n=1,m=0:$ $\displaystyle\quad\delta
B_{3}^{+++}=-\frac{H^{6}g_{1,0}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{+++}\frac{k_{T}^{6}-k_{T}^{4}e_{2}-k_{T}^{3}e_{3}+24e_{3}^{2}}{k_{T}^{3}},$
(5.41) $\displaystyle\quad\delta
B_{3}^{++-}=-\frac{H^{6}g_{1,0}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{++-}\frac{I_{3}^{2}}{k_{T}^{2}}\frac{k_{T}^{6}-k_{T}^{4}e_{2}-k_{T}^{3}e_{3}+24e_{3}^{2}}{k_{T}^{3}},$
(5.42) $\displaystyle\quad\delta
P^{hh^{\prime}}=-\frac{H^{4}g_{1,0}}{4M_{\text{pl}}^{2}k^{3}}\delta_{hh^{\prime}}.$
(5.43) $\displaystyle n=2,m=0:$ $\displaystyle\quad\delta
B_{3}^{+++}=-\frac{3H^{8}g_{2,0}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{+++}\frac{k_{T}^{8}-e_{2}k_{T}^{6}-e_{3}k_{T}^{5}+80e_{3}^{2}k_{T}^{2}-240e_{2}e_{3}^{2}}{k_{T}^{5}},$
(5.44) $\displaystyle\quad\delta
B_{3}^{++-}=-\frac{3H^{8}g_{2,0}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{++-}\frac{I_{3}^{2}}{k_{T}^{2}}\frac{k_{T}^{8}-e_{2}k_{T}^{6}-e_{3}k_{T}^{5}+80e_{3}^{2}k_{T}^{2}-240e_{2}e_{3}^{2}}{k_{T}^{5}},$
(5.45) $\displaystyle\quad\delta
P^{hh^{\prime}}=-\frac{3H^{6}g_{2,0}}{4M_{\text{pl}}^{2}k^{3}}\delta_{hh^{\prime}}.$
(5.46) $\displaystyle n=0,m=1:$ $\displaystyle\quad\delta
B_{3}^{+++}=\frac{H^{6}g_{0,1}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{+++}\frac{k_{T}^{6}-k_{T}^{4}e_{2}-k_{T}^{3}e_{3}+72e_{3}^{2}}{k_{T}^{3}}$
(5.47) $\displaystyle\quad\delta
B_{3}^{++-}=\frac{H^{6}g_{0,1}}{256M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{++-}\frac{\text{Poly}_{8a}(\hat{e}_{1},\hat{e}_{2},k_{3})}{k_{T}^{5}}$
(5.48) $\displaystyle\quad\delta
P^{hh^{\prime}}=\frac{H^{4}g_{0,1}}{4M_{\text{pl}}^{2}k^{3}}\delta_{hh^{\prime}}.$
(5.49) $\displaystyle n=1,m=1:$ $\displaystyle\quad\delta
B_{3}^{+++}=\frac{H^{7}g_{1,1}}{512M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{+++}\frac{k_{T}^{8}-k_{T}^{6}e_{2}-k_{T}^{5}e_{3}+24k_{T}^{2}e_{3}^{2}+384e_{2}e_{3}^{2}}{k_{T}^{5}}$
(5.50) $\displaystyle\quad\delta
B_{3}^{++-}=\frac{H^{7}g_{1,1}}{512M_{\text{pl}}^{4}e_{3}^{3}}\text{SH}_{++-}\frac{\text{Poly}_{8b}(\hat{e}_{1},\hat{e}_{2},k_{3})}{k_{T}^{5}}$
(5.51) $\displaystyle\quad\delta
P^{hh^{\prime}}=\frac{H^{5}g_{1,1}}{8M_{\text{pl}}^{2}k^{3}}\delta_{hh^{\prime}},$
(5.52)

where we have defined the polynomials

$\displaystyle\text{Poly}_{8a}(\hat{e}_{1},\hat{e}_{2},k_{3})$
$\displaystyle=\hat{e}_{1}^{8}+3\hat{e}_{1}^{7}k_{3}-\hat{e}_{1}^{6}(\hat{e}_{2}-2k_{3}^{2})-3\hat{e}_{1}^{5}k_{3}(\hat{e}_{2}+k_{3}^{2})-6\hat{e}_{1}^{4}k_{3}^{4}+\hat{e}_{1}^{3}(6\hat{e}_{2}k_{3}^{3}-3k_{3}^{5})$
$\displaystyle+\hat{e}_{1}^{2}(72\hat{e}_{2}^{2}k_{3}^{2}+3\hat{e}_{2}k_{3}^{4}+2k_{3}^{6})+3\hat{e}_{1}(80\hat{e}_{2}^{2}k_{3}^{3}-\hat{e}_{2}k_{3}^{5}+k_{3}^{7})+(72\hat{e}_{2}^{2}k_{3}^{4}-2\hat{e}_{2}k_{3}^{6}+k_{3}^{8}),$
(5.53) $\displaystyle\text{Poly}_{8b}(\hat{e}_{1},\hat{e}_{2},k_{3})$
$\displaystyle=\text{Poly}_{8a}(\hat{e}_{1},\hat{e}_{2},k_{3})-48\hat{e}_{2}^{2}k_{3}^{2}(\hat{e}_{1}^{2}-8\hat{e}_{2}+6\hat{e}_{1}k_{3}+k_{3}^{2}),$
(5.54)

using the symmetric polynomials in two variables: $\hat{e}_{1}=k_{1}+k_{2}$,
$\hat{e}_{2}=k_{1}k_{2}$.

Perhaps surprisingly we see that for these four examples the bispectra for
both helicity configurations can be written as a linear sum of the GR
bispectra and those that we will introduce in the following section, namely,
the Type-II bispectra. Indeed, once we subtract the GR contributions from
these Type-I bispectra, we are left with contributions that contain an overall
$e_{3}^{2}$ which is the defining property of Type-II bispectra as we will
show. This does not mean that the two building blocks operators are completely
degenerate with the three building block ones since observables could still
differ at higher order in perturbations e.g. the corresponding trispectra
could be different. However, this observation certainly motivates us to
construct these bispectra directly using consistency relations which may shed
light on why this apparent degeneracy arises. Our result here suggests that if
there is no correction to the power spectrum, then to satisfy the consistency
relations we require an overall $e_{3}^{2}$ and it would be nice to prove this
in generality. We will come back to this in the future.

###  5.4 Checking the consistency relations

Given that we are working within the EFToI, our bispectra should satisfy the
consistency relations of single-clock cosmologies which relate the soft limit
of $n$-point functions to lower point ones [15, 16, 17]. They arise from the
unbroken spatial diffeomorphisms and offer a good consistency check of our
results. The leading order graviton soft theorem is

$\langle\gamma^{h_{1}}_{\mathbf{q}}\gamma^{h_{2}}_{\textbf{k}-\mathbf{q}/2}\gamma^{h_{3}}_{-\textbf{k}-\mathbf{q}/2}\rangle^{\prime}\sim\frac{3}{2}(\epsilon_{ij}^{h_{1}}(\mathbf{q})\hat{\textbf{k}}_{i}\hat{\textbf{k}}_{j})P^{h_{1}h_{1}}_{\gamma}(q)P^{h_{2}h_{3}}_{\gamma}(k)\quad\text{as}\quad\frac{q}{k}\to
0,$ (5.55)

where we have introduced the notation $\langle\ldots\rangle^{\prime}$ to
denote a correlator with the momentum-conserving delta function stripped off.

We will focus on the consistency relation for the $+++$ configuration which is
enough to verify that our results are correct. We first need to compute the
tensor structures that appear on both sides of the soft theorem. In both cases
we can write these solely in terms of the three external energies by picking a
basis for the vectors and polarisation tensors. Using momentum conservation
and $SO(3)$ invariance, we can make each of the three external vectors lie in
the $(x,y)$ plane and we choose $\textbf{k}_{1}=k_{1}(1,0,0)$ without loss of
generality. We can then write the corresponding polarisation tensor as

$\displaystyle e^{\pm}(\textbf{k}_{1})$ $\displaystyle=\begin{pmatrix}0&0&0\\\
0&1&\pm i\\\ 0&\pm i&-1\end{pmatrix}\,,$ (5.56)

which is traceless, transverse to the momentum and has the correct
normalisation. The other vectors and polarisation tensors can easily be
extracted from these by performing rotations and using momentum conservation
(see e.g. [12]). It is then straightforward to see that222222Despite the
overall factor of $e_{3}^{-2}$, this object is not singular in the soft limit
thanks to the tunings between terms in the numerator.

$\displaystyle\text{SH}_{+++}=-8e^{h_{1}}_{ij}e^{h_{2}}_{jk}e^{h_{3}}_{ki}=-\frac{k_{T}^{3}}{e_{3}^{2}}\left(8e_{3}-4k_{T}e_{2}+k_{T}^{3}\right)\,,$
(5.57)

with a similar expression for $\text{SH}_{++-}$ [12].232323We note that the
polarisation tensors are matrices of rank $1$ which ensures that we can use
the spinor helicity formalism without loss of generality. We can therefore use
this expression for $\text{SH}_{+++}$ to easily write all tensor structures in
terms of the energies. Similarly, we have

$\displaystyle
e_{ij}^{h_{1}}\hat{k}_{i}^{2}\hat{k}_{j}^{3}=-\frac{k_{2}}{k_{3}}\left(1-\frac{(k_{1}^{2}+k_{2}^{2}-k_{3}^{2})^{2}}{4k_{1}^{2}k_{2}^{2}}\right).$
(5.58)

With these expressions, and the bispectrum and power spectrum of GR given in
(5.38) and (5.40) respectively, we can easily see that the soft theorem is
satisfied in pure gravity. Using the general consistency relation at zeroth
order in the new EFToI couplings, we can then obtain a simplified relation at
first order in terms of the new bispectra and change in the power spectrum
given by

$\delta B^{+++}(q,|\textbf{k}-\mathbf{q}/2|,|\textbf{k}+\mathbf{q}/2|)\sim
2B_{\text{GR}}^{+++}(q,|\textbf{k}-\mathbf{q}/2|,|\textbf{k}+\mathbf{q}/2|)\frac{\delta
P^{++}_{\gamma}(k)}{P^{++}_{\gamma,\text{GR}}(k)}.$ (5.59)

It is then simple to see that this relationship holds for the examples we
wrote above at leading order in the soft momentum i.e. at
$\mathcal{O}(1/q^{3})$. We have also checked that the $++-$ configuration
satisfies the appropriate soft limits. If we take $\vec{k}_{3}$ soft then the
bispectrum contributes to the left-hand side of the soft theorem and the
correction to the power spectrum ensures that it is satisfied, while if we
take either $\vec{k}_{1}$ or $\vec{k}_{2}$ soft then to leading order there is
no contribution to the left-hand side of the correlator which is comforting
since the right-hand side would only be non-zero at leading order if the $+$
and $-$ modes were correlated, which is not the case. In checking this we see
the welcome appearance of $I_{3}^{2}$ in the $++-$ bispectrum: at leading
order in the soft momentum ($\vec{k}_{1}$ or $\vec{k}_{2}$) this combination
of energies vanishes thereby ensuring that there is no contribution to the
left-hand side of the soft theorem.

##  6 Type-II bispectra

Let’s now turn our attention to Type-II bispectra that come from three
building block operators. In this case our Lagrangian contains all tensor
structures derived in [12] with the freedom to add additional derivatives. The
only constraint coming from the fact these are EFToI operators is that each
$\gamma_{ij}$ must come with at least one time derivative, as we showed in
Section 3. This makes sense from the point of view of symmetries: the theory
should be invariant under spatial diffeomorphisms and if we write these
symmetry transformations as a Taylor expansion in $x^{i}$, $\gamma_{ij}$ (and
$\zeta$) [15], then the operators with the fewest powers of $\gamma_{ij}$ must
be invariant under the field-independent part of the symmetry transformation
which would simply be a sum of polynomials in $x^{i}$ under which
$\gamma^{\prime}_{ij}$ is invariant. Here there are no corrections to the
power spectrum, and therefore the cubic terms are those with the fewest powers
of $\gamma_{ij}$.

Now as was explained in detail in [12], to construct bispectra we take one of
the allowed tensor structures and multiply it by a solution to the MLT before
summing over permutations. In terms of polarisation tensors and spatial
momenta, the five allowed tensor structures, up to permutations, are (these
also follow from the action in (3.21))

$\displaystyle\alpha=0:~{}~{}$ $\displaystyle
e_{ij}^{h_{1}}e^{h_{2}}_{jk}e^{h_{3}}_{ki},$ (6.1)
$\displaystyle\alpha=2:~{}~{}$ $\displaystyle
e_{lm}^{h_{1}}e_{lm}^{h_{2}}e_{ij}^{h_{3}}k_{1}^{i}k_{2}^{j}~{}~{}~{}\text{and}~{}~{}~{}e_{lm}^{h_{1}}e_{il}^{h_{2}}e_{jm}^{h_{3}}k_{1}^{i}k_{1}^{j},$
(6.2) $\displaystyle\alpha=4:~{}~{}$ $\displaystyle
e_{lk}^{h_{1}}e^{h_{2}}_{mk}e^{h_{3}}_{ij}k_{1}^{i}k_{2}^{j}k_{3}^{l}k_{3}^{m},$
(6.3) $\displaystyle\alpha=6:~{}~{}$ $\displaystyle
e_{il}^{h_{1}}e^{h_{2}}_{jm}e^{h_{3}}_{kn}k_{1}^{m}k_{1}^{k}k_{2}^{i}k_{2}^{n}k_{3}^{l}k_{3}^{j}.$
(6.4)

As we discussed in the previous section, wavefunction coefficients and