aainstitutetext: Indian Institute of Science Education and Research Pune,
India

# Mixed Graviton and Scalar Bispectra in the EFT of Inflation: Soft Limits and
Boostless Bootstrap

Diptimoy Ghosh a,2 Kushan Panchal a,3 Farman Ullah
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Boostless Bootstrap techniques have been applied by many in the literature to
compute pure scalar and graviton correlators. In this paper, we focus
primarily on mixed graviton and scalar correlators. We start by developing an
EFT of Inflation (EFToI) with some general assumptions, clarifying various
subtleties related to power counting. We verify explicitly the soft limits for
mixed correlators, showing how they are satisfied for higher derivative
operators beyond the Maldacena action. We clarify some confusion in the
literature related to the soft limits for operators that modify the power
spectra of gravitons or scalars. We then proceed to apply the boostless
bootstrap rules to operators that do not modify the power spectra. Towards the
end, we give a prescription that gives correlators for $\alpha$ vacua directly
once we have the correlator for the Bunch-Davies vacuum. This enables us to
bypass complicated in-in calculations for $\alpha$ vacua.

##  1 Introduction

It is believed that the structure in our universe was seeded by quantum
mechanical fluctuations generated during an epoch of exponential expansion
known as inflation STAROBINSKY198099 ; Guth:1980zm ; LINDE1982389 . During
this period, these quantum fluctuations were stretched to super-horizon
distances with amplitudes freezing post horizon exit starobinsky:1979ty ;
Mukhanov:1981xt . Inflation generates correlations between these fluctuations
which seed the late-time cosmological observables such as temperature
correlations on the CMB. Although current observational reach is limited to
deducing the scalar power spectrum and its tilt Hazra_2014 ; Huang_2015 ;
Planck:2018vyg , one can expect that in the future, higher point scalar
correlators, as well as correlators involving spinning fields such as the
graviton, will also be measured.

There are a wide variety of models for Inflation Tong:2004dx ;
Arkani_Hamed_2004  and one can compute observables for each of them. However,
one can adopt a more general (model-independent) approach by constructing an
Effective Field Theory of Inflation (EFToI) Cheung_2008 ; Weinberg_2008 ;
Piazza_2013  consistent with symmetries. This allows us to go beyond the
minimally coupled canonical single field slow-roll inflation Maldacena_2003 .
In this paper, we consider higher derivative operators which contribute to the
mixed three-point correlation functions (i.e.
$\langle\zeta\zeta\gamma\rangle$, $\langle\gamma\gamma\zeta\rangle$, which we
compute using the in-in formalism). It is important that the EFT respects soft
limits Maldacena_2003 ; Creminelli_2012 ; Bordin_2020  provided that we do not
violate any of the assumptions implicit in their derivations. We clearly state
under what assumptions these theorems are expected to hold and explicitly
check (for mixed correlators) that they are satisfied for operators that
modify both scalar as well as tensor power spectra. This provides important
consistency checks for our calculations.

It is also interesting to bootstrap these correlators from a pure boundary
perspective without referring to the bulk evolution. This approach has a lot
of advantages, for instance, it was shown in Creminelli_2014 ; Bordin_2017
that using certain field redefinitions (that vanish at the boundary) some
operators can be removed from the EFT without affecting the late-time
correlators. This redundancy, by construction, is not present once we have a
purely boundary perspective and therefore can potentially lead to a lot of
simplifications. There is a large amount of literature on both the
conformal/pure de-sitter bootstrap Mata_2013 ; Kundu_2016 ;
https://doi.org/10.48550/arxiv.1811.00024 ;
https://doi.org/10.48550/arxiv.2203.08121  as well as the “boostless
bootstrap” approach, the latter being more recent Pajer_2021 ; Cabass:2021fnw
; Goodhew_2021 . The boostless bootstrap program focuses on properties such as
the analytical structure of the correlators, soft limits etc. The analytical
structure of correlators on its own gives a lot of information such as the
initial state Green_2020 ; Green_2022 ;
https://doi.org/10.48550/arxiv.2207.06430  and the flat space amplitude
Bordin_2020 ; https://doi.org/10.48550/arxiv.1811.00024  corresponding to the
interaction. This technique has had considerable success for pure graviton
correlators (sometimes in conjunction with tools like spinor helicity
formalism and results related to parity and the cosmological optical theorem
Maldacena_2011 ; https://doi.org/10.48550/arxiv.2210.02907 ; Cabass:2021fnw ).
In this paper, we follow the rules entailed in Pajer_2021  to bootstrap three-
point mixed correlators arising from operators in our EFToI that do not modify
power spectra, and check to what extent the Boostless Bootstrap works. In
doing so, we separately consider local and non-local interactions.

Finally, we give a prescription for obtaining the correlators for $\alpha$\-
vacua (which is the most general family of vacuum states consistent with de
Sitter isometries Allen:1985ux ) once we have the answers for the Bunch Davies
(BD) vacuum. This offers significant computational benefit since one does not
have to repeat the cumbersome in-in calculation for $\alpha$\- vacua.

##  2 Notations and Conventions

For all the calculations that follow, the form of the metric (in unitary
gauge) is given by the standard ADM formulation Arnowitt:1962hi :

$\displaystyle ds^{2}$
$\displaystyle=-N^{2}dt^{2}+g_{ij}(dx^{i}+\widetilde{N^{i}}dt)(dx^{j}+\widetilde{N^{j}}dt)$
(1)

where we see that:

$\displaystyle g_{00}=-N^{2}+$ $\displaystyle
g_{ij}\widetilde{N^{i}}\widetilde{N^{j}}$ (2)

and we define:

$\displaystyle g^{0i}=N^{i}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
g_{ij}=a^{2}e^{2\zeta}e^{\gamma_{ij}}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\partial_{i}\gamma_{ij}=0\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \gamma_{ii}=0$
(3)

where $a$ is the scale factor and $\frac{\dot{a}}{a}=H$ is the Hubble
constant. $\gamma_{ij}$ is the gauge-invariant tensor perturbation and
$\zeta=\psi+H\frac{\delta\phi}{\bar{\dot{\phi}}}$ is the gauge-invariant
scalar perturbation. It is important to note that the quantity that actually
appears in $g_{ij}$ is $\psi$ but since in unitary gauge $\delta\phi=0$, we
have $\psi|_{\delta\phi=0}=\zeta$. For $\gamma_{ij}$, we denote the two
helicities by $h$ and the corresponding polarization tensors by $e^{h}_{ij}$.
We follow the normalization convention
$e^{h_{1}}_{ij}e^{h_{2}}_{ij}=2\delta_{h_{1}h_{2}}$ . In terms of $H$ and the
background inflaton field $\phi(t)$,the slow roll parameters are defined as:

$\displaystyle\epsilon=-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}^{2}}{2M_{pl}^{2}H^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\eta=-\frac{\ddot{\phi}}{\dot{\phi}H}+\frac{\dot{\phi}^{2}}{2M_{pl}^{2}H^{2}}$
(4)

##  3 EFToI: Explicit check of soft theorems

In this section, we will explicitly check the validity of soft theorems for
some of our EFT operators but first, we quickly review the key assumptions
implicit in their derivations. It was shown in Maldacena_2003 ;
Creminelli:2004yq ; Cheung:2007sv ; Creminelli_2012  that every single field
inflation model must satisfy consistency conditions relating three-point
functions (in the squeezed limit) to two-point functions.

$\displaystyle\lim_{\bm{k_{1}}\rightarrow
0}\langle\zeta_{\bm{k_{1}}}\zeta_{\bm{k_{2}}}\zeta_{\bm{k_{3}}}\rangle=-(n_{s}-1)\langle\zeta_{\bm{k_{1}}}\zeta_{-\bm{k_{1}}}\rangle\langle\zeta_{\bm{k_{2}}}\zeta_{-\bm{k_{2}}}\rangle\left(1{+\mathcal{O}(k^{2}_{1}/k^{2}_{2})}\right)$
(5) $\displaystyle\lim_{\bm{k_{1}}\rightarrow
0}\langle\zeta_{\bm{k_{1}}}\gamma^{h}_{\bm{k_{2}}}\gamma^{h^{\prime}}_{\bm{k_{3}}}\rangle=-n_{t}\langle\zeta_{\bm{k_{1}}}\zeta_{-\bm{k_{1}}}\rangle\langle\gamma^{h}_{\bm{k_{2}}}\gamma^{h^{\prime}}_{-\bm{k_{2}}}\rangle\left(1{+\mathcal{O}(k^{2}_{1}/k^{2}_{2})}\right)$
(6) $\displaystyle\lim_{\bm{k_{1}}\rightarrow
0}\langle\gamma^{h}_{\bm{k_{1}}}\zeta_{\bm{k_{2}}}\zeta_{\bm{k_{3}}}\rangle=-\langle\gamma^{h}_{\bm{k_{1}}}\gamma^{h}_{-\bm{k_{1}}}\rangle\epsilon^{h}_{ij}k_{2}^{i}k_{2}^{j}\frac{\partial}{\partial
k_{2}^{2}}\langle\zeta_{\bm{k_{2}}}\zeta_{-\bm{k_{2}}}\rangle\left(1{+\mathcal{O}(k^{2}_{1}/k^{2}_{2})}\right)$
(7)

where, $\zeta$ and $\gamma$ are defined in the Appendix A, and $n_{s}$,
$n_{t}$ are the scalar and tensor spectral tilts defined as

$\displaystyle
n_{s}-1=\frac{d\ln\left(k^{3}\langle\zeta_{-\vec{k}}\zeta_{\vec{k}}\rangle\right)}{d\ln
k}$ (8) $\displaystyle
n_{t}=\frac{d\ln\left(k^{3}\langle\gamma_{-\vec{k}}\gamma_{\vec{k}}\rangle\right)}{d\ln
k}$ (9)

For the leading order (two derivatives) minimally coupled canonical action, we
have $n_{s}-1=2(\eta-3\epsilon)$ and $n_{t}=-2\epsilon$ Maldacena_2003 .
Besides requiring only a single degree of freedom during inflation and the
condition that the curvature perturbation does not grow outside the horizon
(not true for non-attractor models Kinney:2005vj ; Huang:2013oya ;
Chen:2013eea ; Martin:2012pe ; Namjoo:2012aa ), these relations also rely on
the fact that there is a negligible contribution to the three-point function
when all the modes are within the horizon. In fact, these theorems assume that
the contribution to the correlators only starts becoming sizable when the long
mode has left the horizon and has already frozen acting as a classical
background. This is always the case when the initial state is the BD vacuum.
The BD three-point correlators only have a total energy pole which comes from
integrals of the form

$\langle\zeta_{\vec{k_{1}}}\zeta_{\vec{k_{2}}}\zeta_{\vec{k_{3}}}\rangle\sim\int_{-\infty}^{0}\tau^{m}e^{iK\tau}d\tau$
(10)

where, $K=k_{1}+k_{2}+k_{3}$. To compute such an integral, a regularisation
scheme is required. This integral can be regularized by taking
$\tau\rightarrow\tau(1-i\epsilon)$, and therefore, damping the contribution in
the far past. It only starts giving appreciable contribution once
$K\tau\sim-1$ which in the equilateral case ($k_{1}=k_{2}=k_{3}$) corresponds
to the epoch of horizon crossing for all the modes and in the squeezed case
($k_{1}\ll k_{2}\approx k_{3}$, relevant for soft theorems) corresponds to
epoch of horizon crossing for the short modes since $K\sim k_{s}$. Therefore,
in the squeezed limit the in-in contribution remains negligible till the epoch
of the horizon crossing for the short modes. By this time the long mode has
already frozen and therefore, the soft theorems must hold. There exist cases
where the soft theorems are violated even in single field models Namjoo:2012aa
; Chen:2013aj ; Cai:2018dkf ; https://doi.org/10.48550/arxiv.2207.06430 , for
instance, if one starts in an excited initial state, for e.g., $\alpha$-vacuum
Allen:1985ux  then apart from total energy pole, one also has a
$\left(k_{1}+k_{2}-k_{3}\right)$ kind of pole structure arising from mode-
mixing. Therefore the integral contains terms,

$\langle\zeta_{\vec{k_{1}}}\zeta_{\vec{k_{2}}}\zeta_{\vec{k_{3}}}\rangle\sim\int_{-\infty}^{0}\tau^{m}e^{i\left(k_{1}+k_{2}-k_{3}\right)\tau}d\tau$
(11)

which, in the squeezed limit ($k_{1}\ll k_{2}\approx k_{3}$) starts giving
appreciable contribution when $k_{1}\tau\sim-1$ i.e when the long mode is near
horizon exit. Therefore, the soft theorem proof does not hold in this case.

###  3.1 EFToI

The Effective field theory of Inflation is an attempt to unify all models of
inflation by constructing the most general low energy action consistent with
symmetries. The action is written in terms of fluctuations themselves. In
cosmological perturbation theory (See Mukhanov:1990me  for a nice review) one
always decomposes any relevant quantity (e.g., $\phi(x)$) in the following
manner

$\phi(\vec{x},t)=\bar{\phi}(t)+\delta\phi(\vec{x},t)$ (12)

where $\bar{\phi}(t)$ is the solution of the following background EOM for the
scalar field (FRW coordinates)

$\ddot{\phi}(t)+3H\dot{\phi}(t)+V^{\prime}=0$ (13)

No matter what coordinates one works with, the split between background and
perturbation will always have the same time-dependent (no spatial dependence)
background function. The reason for such a split is the following: our
universe on very large scales looks very homogeneous (no spatial dependence)
and isotropic to co-moving observers. This has been verified by the Large
scale structure data and the CMB temperature profile Planck:2018vyg . Of
course, this is only an approximate statement and there are small fluctuations
present on top of this background. Since we are interested in studying these
fluctuations we always split any relevant quantity in any coordinates in terms
of an FRW background solution (homogeneous and isotropic) and perturbations
about it. If one thinks in terms of a background spacetime then this means
that there is a preferred choice of the time coordinate (slicing) in which the
background metric has no spatial dependence. For the inflationary background,
this slicing corresponds to slices of constant inflaton which act as a
physical clock (with values of this field keeping time). This is the reason
why sometimes single-field inflation is also referred to as single-clock
inflation.
As mentioned above, we are finally interested in a theory of fluctuations
about this time-dependent background and therefore time diffeomorphism while
still being a symmetry is now non-linearly realised on the fluctuations i.e
they are spontaneously broken. One can then choose a gauge to write the EFT in
the so-called unitary gauge which has no inflaton fluctuations
$\delta\phi(\vec{x},t)=0$. In this gauge, all fluctuations go into the metric.
The transformation rule for $\delta\phi(\vec{x},t)$ under $x^{\mu}\rightarrow
x^{\mu}+\epsilon^{\mu}$,

$\delta\tilde{\phi}(\tilde{x})=\delta\phi(x)-\epsilon^{0}\dot{\bar{\phi}}(t)$
(14)

indicates that unitary gauge just fixes the time diffs and therefore, the EFT
will only involve terms which are invariant under spatial diffeomorphisms. We
are allowed to write full diff invariant terms,
e.g.,${}^{(4)}\\!R_{\mu\nu}^{(4)}\\!R^{\mu\nu}$, terms with free upper time
indices, e.g., $\delta g^{00}$ (terms with lower free time indices are not
invariant under spatial diffs) and terms describing the slicing. Hence we
have:

$S_{EFT}=\int{}\leavevmode\nobreak\
d^{4}x\sqrt{-g}\mathcal{L}(^{(4)}\\!R_{\mu\nu\sigma\rho},^{(3)}\\!R_{ijkl},\nabla_{\mu},\delta
K_{ij},g^{00},\partial_{t},...)$ (15)

So far we have identified the correct degrees of freedom for the action. Now,
like for any sensible EFT, one must identify the correct expansion
parameter(s) to organise the terms. The EFToI is an expansion in the number of
metric perturbations and the number of derivatives on the metric perturbations
which (prior to any canonical normalization) have dimension 0. We start by
splitting the action into three parts,

$S_{EFT}=S_{0}+S_{2}+S_{\geq 3}$ (16)

where,

$\displaystyle\begin{split}S_{0}=\int{}\leavevmode\nobreak\
d^{4}x\leavevmode\nobreak\
\sqrt{-g}M^{2}_{pl}\left({\frac{{}^{(4)}\\!R}{2}}-M^{2}_{pl}\lambda(t)-c(t)\delta
g^{00}+D\delta K\right)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \\\ S_{2}=\int{}\leavevmode\nobreak\
d^{4}x\leavevmode\nobreak\ \sqrt{-g}M^{2}_{pl}\left(m_{1}\left(\delta
K^{i}_{j}\delta K^{j}_{i}\right)+m_{2}(\delta
K)^{2}+m_{3}\left({}^{(3)}\\!R\delta g^{00}\right)+M_{2}^{2}(\delta
g^{00})^{2}+M_{3}\left(\delta g^{00}\right)\delta K\right.\\\
\left.+\frac{1}{M_{4}}{{}^{(3)}\\!R_{ij}}\delta
K^{ij}+\frac{1}{M_{5}}{{}^{(3)}\\!R}\delta
K+\frac{1}{M^{2}_{6}}{{}^{(3)}\\!R}^{2}+\frac{1}{M^{2}_{7}}{{}^{(3)}\\!R}^{2}_{ij}+\frac{1}{M^{2}_{8}}{{}^{(3)}\\!R}^{2}_{ijkl}\right)+....\\\
\end{split}$ (17)

where, $S_{n}$ has terms which start from $n^{th}$ order in perturbations.
Here, $S_{0}$ is Einstein-Hilbert action coupled with a scalar field with
$\lambda(t)=-3H^{2}+\dot{H}$ and $c(t)=-\dot{H}$ so that the
background/zeroth-order equations of motion are satisfied. The dots in the
above equations indicate the same terms but with more derivatives.
$\\{m_{i}\\}$ have mass dimension zero whereas $\\{M_{i}\\}$ have mass
dimension one. We have not made any assumption about how the EFT operators are
generated from the UV, therefore every operator comes with a different Mass
scale, $M_{i}$. It is important to keep in mind that finally, we want a theory
in terms of the scalar fluctuations $\zeta$ and the tensor fluctuation
$\gamma_{ij}$. The EFT derivative power counting is clear only in terms of
these variables and to illustrate this consider the operator
$\partial_{i}\delta g^{00}\partial^{i}\delta g^{00}$. This naively looks like
a leading two-derivative term and therefore, is not suppressed or enhanced by
any mass scale. However, in terms of $\zeta$, depending on the constraint
solution, it can actually generate higher derivative terms. Therefore, some
terms in the relevant part of Lagrangian might actually turn out to be
irrelevant. Since we are ignorant about the short distance physics i.e. $k\gg
H$ and are interested in the modes which have exited the horizon and have left
an imprint on the CMB, the typical length/energy scale is $H$. Hence an
expansion in $H/M$ is obtained for our EFToI. Assuming for concreteness that
all dimensional coefficients are order unity and $H\ll M_{4,5,6,7}$, one can
easily arrange the action as a perturbative series. For instance, the
quadratic action, $S_{2}$, is

$\displaystyle
S_{2}\sim\left(m_{1}+m_{2}+m_{3}+\frac{M_{2}^{2}}{H^{2}}+\frac{M_{3}}{H}+\frac{H}{M_{4}}+\frac{H^{2}}{M^{2}_{5}}+\frac{H^{2}}{M^{2}_{6}}+\frac{H^{2}}{M^{2}_{7}}\right)+....$
(18)

For some explicit calculations, we take the action $S_{0}+S_{2}$ and write
only the operators which have at most 2 derivatives on the metric :

$\displaystyle S_{0}+$ $\displaystyle S_{2}=\int{}\leavevmode\nobreak\
d^{4}x\leavevmode\nobreak\
\sqrt{-g}M^{2}_{pl}\left(\frac{{}^{(4)}\\!R}{2}+m_{1}\delta K^{i}_{j}\delta
K^{j}_{i}+m_{2}(\delta K)^{2}+m_{3}^{(3)}\\!R\delta g^{00}+D\delta
K-M^{2}_{pl}\lambda(t)-c(t)g^{00}\right.$
$\displaystyle\left.+M_{1}g^{ij}\partial_{i}\delta g^{00}\partial_{j}\delta
g^{00}+M^{2}_{2}(\delta g^{00})^{2}+M_{3}\delta g^{00}\delta K\right)$ (19)

The leading (two derivatives) quadratic (scalar and tensor) actions are given
by Maldacena_2003  :

$\displaystyle S_{\zeta\zeta}=$
$\displaystyle\frac{1}{2}\int{}d^{4}x\leavevmode\nobreak\
\epsilon\left(a^{3}\dot{\zeta}^{2}-a(\partial\zeta)^{2}\right)$ (20)
$\displaystyle S_{\gamma\gamma}=$
$\displaystyle\frac{1}{8}\int{}d^{4}x\leavevmode\nobreak\
\left(a^{3}\dot{\gamma_{ij}}^{2}-a(\partial\gamma_{ij})^{2}\right)$ (21)

i.e the action derived for a canonical scalar field minimally coupled to
gravity. This is an element of the class of actions defined by equation (3.1)
where the action only contains the first three terms of $S_{0}$. In passing we
point out that from (20) and (21) one can easily see that the existence of
$\zeta$ is tied to the fact that inflation is quasi de Sitter ($\epsilon\neq
0$), and such a variable does not exist in pure de Sitter ($\epsilon=0$). On
the other hand, the tensor perturbation $\gamma_{ij}$ is also well defined in
de Sitter. Coming back to (3.1), there is a large number of unknown parameters
but as usual, the EFT parameters have to be determined/constrained
experimentally. Unfortunately in cosmology, the number of observables is very
small. This is due to the fact that unlike in flat space EFTs, we do not have
experimental control. Therefore, as we will show below, one can constrain or
fix only a handful of parameters in the EFToI:

  * •

If a mass term is generated for the scalar perturbations in the EFT
($c_{1}\dot{\zeta}^{2}-c_{2}(\partial\zeta)^{2}-c_{3}\zeta^{2}$, where
$m^{2}=c_{3}/c_{1}$ ) then from the observed tilt of the scalar spectrum one
can show that $\frac{m}{H}\ll 1$, where $m$ is the mass of the fluctuations.
This can be easily inferred from the expression for power spectrum for a
massive scalar in de-Sitter.

$P_{\zeta}(k)=\frac{H^{2}}{2k^{3}}\left(\frac{k}{aH}\right)^{3-2\nu}$ (22)

where, $\nu=\sqrt{\frac{9}{4}-\frac{m^{2}}{H^{2}}}$. Now, since we know
(through observations) that the spectral tilt is very close to unity,
therefore $\nu\sim\frac{3}{2}\implies\frac{m^{2}}{H^{2}}\ll 1$. This allows us
to strongly constrain the coefficients of operators which generate a mass term
for $\zeta$.

  * •

For operators which contribute both to the sound speed and non-gaussianities,
we can use experimental bounds to deduce the bounds on $c_{s}$ or the operator
coefficients. Further constraints on $c_{s}$ or the coefficients can be found
by applying the partial wave unitarity bound by going to the flat gauge
Cheung_2008 ; Baumann_2016 .

To illustrate the points mentioned above, we take two examples of quadratic
actions of the type mentioned in Equation (3.1). We first solve for the ADM
constraint variables up to 1st order in $\zeta$ (See Appendix A). To simplify
calculations, we take $m_{2}=-m_{1}$. We then consider the cases:

  * •

$M_{3}=0,m_{1},m_{2}\neq 0$. The canonical 2 derivative terms in the action
are given by:

$S_{0}+S_{2}=\int{}M^{2}_{pl}\left(a^{3}\left(\epsilon+4\left(\frac{M_{2}}{H}\right)^{2}\right)\dot{\zeta}^{2}-a\epsilon(\partial\zeta)^{2}\right)$
(23)

i.e. the usual canonical action with a different sound speed. The partial wave
unitarity bound in the flat space limit is satisfied in the limit $c_{s}\to 1$
Cheung_2008  , which gives $M_{2}\ll\sqrt{\epsilon}H$. For this theory, which
has no mass term, we have the spectral tilt:

$n_{s}-1=\frac{1}{H}\frac{d}{dt_{*}}\langle\zeta(\bm{k})\zeta(\bm{-k})\rangle=\frac{1}{H}\frac{d}{dt_{*}}\frac{H^{2}(t_{*})}{4\epsilon(t_{*})M_{pl}^{2}c_{s}(t_{*})k^{3}}=2\eta-6\epsilon+4c_{s}^{2}\left(\frac{M_{2}}{H}\right)^{2}(\frac{\eta}{\epsilon}-1)$
(24)

where the asterisk denotes the time of horizon crossing. Because of the
$c_{s}$ constraint, the spectral tilt is already small. As was shown in
Cheung_2008 , a small speed of sound ($c_{s}\ll 1)$ also implies large
interactions (non-gaussianities). These are derivative interactions (e.g.,
$\dot{\pi}^{3}$) and they naturally produce equilateral non-gaussianity since,
due to derivatives their contribution in the squeezed limit is negligible.
There are experimental constraints on $c_{s}$ from bounds on equilateral non-
gaussianity, $f^{equil}_{NL}$,

$c_{s}\gg 0.028$ (25)

  * •

$\bm{M_{3}\neq 0,m_{1}=m_{2}=0,M_{2}=0}$. Since calculating the action for
arbitrarily large values of $M_{3}$ is difficult, we take the case where
$g=M_{3}/H\ll 1$. For this we get upto 1st order in $\epsilon$ and $g$,

$S_{0}+S_{2}=\int{}M^{2}_{pl}\left(a^{3}\left(\epsilon-6g\right)\dot{\zeta}^{2}-a\epsilon(\partial\zeta)^{2}\right)$
(26)

Again, taking into account that mass terms only start appearing at
$\mathcal{O}(g^{2})$, the spectral tilt is small and we won’t explicitly
calculate it here. Going to the Stueckelberg gauge Cheung_2008 , we can write
the lagrangian in the flat space limit, in terms of the Stueckelberg boson
$\pi$ as :

$\mathcal{L}_{\pi}=M^{2}_{pl}\epsilon
H^{2}\left(\dot{\pi}^{2}-(\partial\pi)^{2}\left(1-\frac{M_{3}}{H\epsilon}\right)\right)-2\frac{M_{3}M_{pl}^{2}c_{s}^{3}}{\left(\sqrt{M^{2}_{pl}\dot{H}}\right)^{3}}\partial^{2}\pi(\dot{\pi}^{2}+(\partial\pi)^{2})+\leavevmode\nobreak\
\leavevmode\nobreak\ (\text{$\epsilon$ -suppressed terms})$ (27)

This lagrangian tells us that we must have $M_{3}>0$ for $c_{s}<1$ and the
partial wave unitarity bound for the tree level $\pi\pi$ amplitude gives:

$\frac{M^{2}_{3}}{\epsilon^{3}M^{2}_{pl}c_{s}^{2}}\left(c_{s}^{4}+4(1+c_{s}^{2})^{2}\right)<\pi$
(28)

where, in the line above, the $\pi$ is the numerical constant.

While all this is for the scalar action, the graviton case is simpler to deal
with since the only two operators contributing to the two derivative quadratic
EFT are ${}^{(3)}\\!R$ and $\delta K_{ij}\delta K^{ij}$ which can be removed
by suitable field re-definitions of $g_{\mu\nu}$ Creminelli_2014 ;
Creminelli_2012 . We also note that one can write the EFT in some other gauge,
for instance, the flat gauge where $\delta\phi\neq 0$ Maldacena_2003 . Doing
calculations in flat gauge can lead to simplifications as it can sometimes
directly give us an EFT ordered in $\epsilon$ Maldacena_2003 . One has to be
careful while doing the calculations in this gauge and converting them to the
unitary gauge answers. This is pointed out using an example in Appendix E.

###  3.2 Explicit Checks

In this section, we explicitly calculate and verify the soft limits mentioned
above. We start with the scalar three-point function calculated for the action
in 3.1 with only $M_{2}\neq 0$ as a check for calculations with $c_{s}\neq 1$
while for the mixed correlators, we take the canonical minimally coupled
quadratic action for simplicity of calculation. Although the scalar soft
theorems have been explicitly checked Cheung:2007sv  the mixed correlator ones
have not been explicitly checked for higher derivative EFToI operators and
therefore, according to our knowledge, this is the first such an attempt.

####  3.2.1 Soft limit for $\langle\zeta\zeta\zeta\rangle$

The terms which contribute to the soft 3-point correlator
$\langle\zeta(k_{1})\zeta(k_{2})\zeta(k_{3})\rangle|{k_{1}\to 0}$ are given
by:

$S_{0}+S_{2}=\int{}M_{pl}^{2}\left[\left(\epsilon^{2}\zeta_{c}\dot{\zeta_{c}}^{2}a^{3}+\epsilon^{2}\zeta_{c}(\partial\zeta_{c})^{2}a\right)+4\left(3\epsilon-2\eta\right)\left(\frac{M_{2}}{H}\right)^{2}\zeta_{c}\dot{\zeta_{c}}^{2}a^{3}\right]$
(29)

where the first two terms are from the Maldacena cubic action. It is important
to note that the final Maldacena cubic action is derived after performing a
field redefinition Maldacena_2003

$\zeta=\zeta_{c}+\frac{1}{2}(2\epsilon-\eta)\zeta^{2}_{c}+\frac{\dot{\zeta_{c}}\zeta_{c}}{H}+....$

which removes terms proportional to the equation of motion. Since $M_{2}$ also
gives quadratic corrections to $\zeta$ action, the above field redefinition
generates additional cubic terms. Therefore, the last term in 29 is a
combination of the cubic part generated by $M_{2}$ and the terms generated by
the redefinition. The soft correlator is given by (where $n_{s}$ is given by
24),

$\begin{split}\langle\zeta(\bm{k_{1}})\zeta(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{\bm{k_{1}}\to
0}=&\left(\frac{H^{2}}{4\epsilon
c_{s}M_{pl}^{2}}\right)^{2}\left(\frac{11}{2}\epsilon-2\eta+\frac{\epsilon}{2}c_{s}^{2}+4\frac{M^{2}_{2}}{H^{2}}c_{s}^{2}\left(\frac{3}{2}-\frac{\eta}{\epsilon}\right)\right)\frac{1}{k_{1}^{3}}\frac{1}{k_{2}^{3}}\\\
=&-(n_{s}-1)\langle\zeta(\bm{k_{1}})\zeta(\bm{-k_{1}})\rangle\langle\zeta(\bm{k_{2}})\zeta(\bm{-k_{2}})\rangle\end{split}$
(30)

####  3.2.2 Soft limits for mixed correlators

We classify the operators in the EFT into two classes, Cubic and Purely Cubic
operators. Cubic operators are those which contribute to
$\langle\zeta\zeta\rangle$ or $\langle\gamma\gamma\rangle$, as well as the
mixed bispectra, while purely cubic operators contribute only to the latter.
We take two cubic operators from 17 as examples:

$(1)$$\bm{\int d^{4}{x}\sqrt{-g}\frac{M^{2}_{pl}}{M_{5}}{{}^{(3)}\\!R}\delta
K\mathrel{\mathop{\mathchar 58\relax}}}$
Since $\delta K$ contains both $\delta N$ and $N^{i}$, i.e. the ADM constraint
variables, they get modified and are now given by:

$\displaystyle\delta N=$
$\displaystyle\frac{\dot{\zeta}}{H}+\frac{2\partial^{2}\zeta a^{-2}}{M_{5}H}$
(31) $\displaystyle N^{i}=$
$\displaystyle\partial_{i}\left(-\frac{\zeta}{H}a^{-2}+\epsilon\partial^{-2}\dot{\zeta}+\frac{2\epsilon\zeta
a^{-2}}{M_{5}}\right)$ (32)

Note that the two equations above are valid only when $\frac{H}{M_{5}}\ll 1$
i.e. they’re 1st order expressions in $1/M_{5}$. Using this, the corrections
to the quadratic and cubic actions for $\gamma\gamma\zeta$ are given up to
first order in $1/M_{5}$ by:

$\displaystyle\mathcal{O}_{\zeta\zeta}=$
$\displaystyle\int{}4\frac{M^{2}_{pl}}{M_{5}}a\partial^{2}\zeta\left(\epsilon\dot{\zeta}-\frac{\partial^{2}\zeta}{H}a^{-2}\right)$
(33) $\displaystyle\mathcal{O_{\gamma\gamma\zeta}}=$
$\displaystyle\int{}\frac{M^{2}_{pl}}{4M_{5}}\left(a\frac{\dot{\gamma_{ij}}\dot{\gamma_{ij}}}{H}\partial^{2}\zeta-2a^{-1}\frac{\partial_{l}\gamma_{ij}\partial_{l}\gamma_{ij}\partial^{2}\zeta}{H}+\epsilon
a\partial_{l}\gamma_{ij}\partial_{l}\gamma_{ij}\dot{\zeta}-2\epsilon\dot{\gamma_{ij}\partial_{l}\gamma_{ij}\partial_{l}\zeta}\right)$
(34)

The correction to the scalar power spectrum is given by,

$\delta
P_{\zeta}=-\frac{M^{2}_{pl}}{M_{5}}\frac{H^{3}}{4\epsilon^{2}M^{4}_{pl}k^{3}}(5-3\epsilon)$
(36)

For perturbation theory to work here, the quadratic correction $\delta
P_{\zeta}$ must be small compared to $P_{\zeta}(k)=\frac{H^{2}}{4\epsilon
M_{pl}^{2}k^{3}}$. We henceforth assume that $\frac{H}{M_{5}\epsilon}\ll 1$ so
that the enhancement to the power spectrum remains small. This also ensures
that we can expand the action (as well as the ADM constraints) in powers of
$M$ and the analysis above holds. In the soft limit,
$\mathcal{O}_{\gamma\gamma\zeta_{k\to 0}}=0$ i.e its contribution to the mixed
correlator in the soft limit vanishes at leading order. However, the following
“exchange diagram” diagram gives a non-zero contribution:

Figure 1: The MT (Maldacena term) vertex is the usual cubic
$\gamma\gamma\zeta$ vertex of the Maldacena action

where the Maldacena term refers to the mixed cubic vertex
($\zeta\gamma\gamma$) computed in Maldacena_2003 . This 3-point correlator is
a sum of two terms,

$\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{\bm{k_{3}}\to
0}=2\left(\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{LR,{\bm{k_{3}}\to
0}}-\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{RR,{\bm{k_{3}}\to
0}}\right)$ (37)

where $R$ and $L$ indicate the time and anti-time orderings of the interaction
Hamiltonian respectively. For instance, RR means both vertices are time
ordered (See Weinberg_2005  for more details on these notations and
conventions). The $RR$ contribution is given by 111The results given from here
on do not include the divergent terms of the type: $\lim_{\eta\to
0}\frac{cos(k\eta)}{\eta}$. These are ignored while calculating contact
diagrams since they contribute only to the imaginary part of the $L$ and $R$
vertices and so, they get cancelled. Here, however, these terms come from both
the vertices of the “exchange” diagram and get multiplied, because of which
they contribute to the real part. However, one can easily check that the
contributions from $RR,RL,LR$ and $LL$ add up to $0$.

$\displaystyle\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{RR},_{\bm{k_{3}}\to
0}=\left(\frac{H^{6}}{M^{6}_{pl}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)\left(\frac{M^{2}_{pl}\Lambda}{8\epsilon
HM_{7}^{2}}\right)e^{h_{1}}_{ij}(k_{1})e^{h_{2}}_{ij}(k_{2})\left[k_{1}^{2}k_{2}^{2}\left[\frac{5-3\epsilon}{4k_{T}}+\frac{(3-\epsilon)k_{3}}{k_{T}^{2}}-\right.\right.$
$\displaystyle\left.\left.\frac{\epsilon
k_{3}^{2}}{k_{T}^{3}}-\frac{5k_{3}^{3}}{k_{T}^{4}}-\frac{8k_{3}^{4}}{k_{T}^{5}}\right]-(\bm{k_{1}}\cdot\bm{k_{2}})\left[\frac{5-3\epsilon}{4}k_{T}+\epsilon
k_{3}I-\frac{5-3\epsilon}{2}k_{3}+(3-\epsilon)\frac{k_{3}^{2}}{k_{T}}-k_{1}k_{2}\left(\frac{5-3\epsilon}{4k_{T}}+\frac{5-3\epsilon}{2k_{T}^{2}}+\right.\right.\right.$
$\displaystyle\left.\left.\left.\frac{k_{3}^{2}}{k_{T}^{3}}-2\frac{k_{3}^{3}}{k_{T}^{4}}\right)+(k_{2}k_{3}+k_{1}k_{3})\left(\frac{5-3\epsilon}{4k_{T}}+\frac{(1+\epsilon)k_{3}}{2k_{T}^{2}}+\frac{k_{3}^{2}}{k_{T}^{3}}+\frac{2k_{3}^{3}}{k_{T}^{4}}\right)+k_{1}k_{2}k_{3}\left(\frac{5-3\epsilon}{4k_{T}^{2}}+\frac{(1+\epsilon)k_{3}}{k_{T}^{3}}+\frac{3k_{3}^{2}}{k_{T}^{4}}\right.\right.\right.$
$\displaystyle\left.\left.\left.+\frac{8k_{3}^{3}}{k_{T}^{5}}\right)\right]-\left[k_{2}^{2}(\bm{k_{1}}\cdot\bm{k_{3}})\left(\frac{5-3\epsilon}{4k_{T}}+\frac{5-3\epsilon}{4k_{T}^{2}}(k_{1}+2k_{3})+\frac{k_{1}k_{2}(5-3\epsilon)+k_{3}^{2}}{k_{T}^{3}}+\frac{3k_{1}k_{3}^{2}-2k_{3}^{3}}{k_{T}^{4}}-\frac{8k_{3}^{3}k_{1}}{k_{T}^{5}}\right)\right.\right.$
$\displaystyle\left.\left.-(k_{1}\leftrightarrow k_{2})\right]\right]$ (38)

where $I=\gamma_{E}\leavevmode\nobreak\ +\leavevmode\nobreak\
\text{log}(-k_{T}\eta_{0})$ , $\eta_{0}\to 0$ and $k_{T}=\sum_{a=1}^{3}k_{a}$.
It is easy to show that the $LR$ contribution is simply given by,

$\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{LR}=\frac{B_{MT}\delta
P_{\zeta}(k_{3})}{4P_{\zeta}(k_{3})}$ (39)

Therefore, the final 3-point function is given by:

$\displaystyle\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{RR}=\frac{B_{MT}\delta
P_{\zeta}(k_{3})}{2P_{\zeta}(k_{3})}-2\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{RR}$
(40)

where $B_{MT}$ is the mixed Bispectrum of the Maldacena cubic action. One can
easily check that:

$\displaystyle\langle\gamma(\bm{k_{1}})\gamma(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{\bm{k_{3}}\to
0}=-\frac{n_{t}P_{\gamma}(k_{1})P_{\zeta}(k_{3})}{2P_{\zeta}(k_{3})}-\frac{1}{2}n_{t}P_{\gamma}(k_{1})\delta
P_{\zeta}(k_{3})=-n_{t}P_{\gamma}(k_{1})\delta P_{\zeta}(k_{3})$ (41)

i.e the soft limit 6 is satisfied.

$(2)$$\int
d^{4}x\sqrt{-g}\frac{M^{2}_{pl}}{M^{2}_{7}}{{}^{(3)}\\!R_{ij}}{{}^{(3)}\\!R_{ij}}$:
This operator gives the corrections:

$\displaystyle\mathcal{O}_{\gamma\gamma\zeta}=$
$\displaystyle\int{}\frac{M^{2}_{pl}}{M^{2}_{7}}a^{-1}\left[-\frac{1}{4}\zeta\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}+\frac{\dot{\zeta}}{4H}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}\right]$
(42) $\displaystyle O_{\zeta\zeta}$
$\displaystyle=\int{}6\frac{M^{2}_{pl}}{M^{2}_{7}}a^{-1}(\partial^{2}\zeta)^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
O_{\gamma\gamma}=\int{}\frac{M^{2}_{pl}}{4M^{2}_{7}}a^{-1}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}$
(43)

The corresponding corrections to the power spectra are,

$\displaystyle\delta
P_{\zeta}=15\frac{M^{2}_{pl}}{M^{2}_{7}}\frac{H^{4}}{8M^{4}_{pl}\epsilon^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle\delta
P_{\gamma}=5\frac{M^{2}_{pl}}{M^{2}_{7}}\frac{H^{4}}{2M^{4}_{pl}k^{3}}\delta_{h_{1}h_{2}}$
(44)

Again, as also noted in Cabass:2021fnw  one should be worried about the
$\frac{\Delta\langle\zeta\zeta\rangle}{\langle\zeta\zeta\rangle_{0}}\sim\frac{M^{2}_{pl}}{M^{2}_{7}}H^{2}/\epsilon
M^{2}_{pl}$ enhancement of the power spectrum and thus we can remove the
correction by taking an extra $(^{(3)}\\!R)^{2}$ term or (for the purposes of
this paper )assume that this ratio is small, which places a lower bound on
$M_{7}$. As pointed out in Cabass:2021fnw , after the field redefinition
$\gamma_{ij}\rightarrow\gamma_{ij}-\dot{\gamma_{ij}}\zeta/H+...$ which removes
the terms proportional to the equation of motion in the Maldacena mixed cubic
action, and after integration by parts, we get:

$\mathcal{O}=\int{}\sqrt{-g}\frac{M^{2}_{pl}}{M^{2}_{7}}{{}^{(3)}\\!R_{ij}}{{}^{(3)}\\!R_{ij}}+\frac{1}{2}\frac{M^{2}_{pl}}{M^{2}_{7}}\zeta\partial^{2}\frac{\dot{\gamma_{ij}}}{H}\partial^{2}\gamma_{ij}a^{-1}=-\int{}\frac{1}{4}\epsilon\zeta\frac{M^{2}_{pl}}{M^{2}_{7}}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}a^{-1}+....$
(45)

After taking the soft limit $\zeta_{\bm{k_{3}}\rightarrow 0}$, the terms
represented by dots go to 0 at leading order as mentioned in Cabass:2021fnw .
The term proportional to $\epsilon$ however, does contribute and we have:

$\Delta_{A}\langle\zeta(\bm{k_{3}})\gamma^{h_{1}}(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\rangle_{\bm{k_{3}}\to
0}=-\frac{5}{8}\frac{M^{2}_{pl}}{M^{2}_{7}}\left(\frac{H}{M_{pl}}\right)^{6}\frac{1}{k_{1}^{3}}\frac{1}{k_{3}^{3}}\delta_{h_{1}h_{2}}$
(46)

This term 45 is not mentioned in Cabass:2021fnw  and there, the 3-point
function in the soft limit is shown to vanish at
$\mathcal{O}((k_{1}/k_{3})^{0})$. The difference lies in the order of
calculations, i.e. whether you compute and simplify the operator first and
then do the in-in computation or just do the in-in computation for all the
terms and add the answers as done in Cabass:2021fnw . The difference in the
answers arises because $H$ is not a constant when we simplify the operators
but is taken to be time-independent while doing the in-in integrals. The
reason for this is that the in-in integral pick up most of the contribution
near horizon crossing and therefore, it is a good approximation to take
$H(t)=H_{*}$ i.e Hubble at horizon crossing.222for instance if the operator is
$\mathcal{O}=\int{}a^{3}M_{pl}({M_{2}}/{H})^{2}\dot{\zeta}^{3}$ and we want to
compute $\langle\zeta\zeta\zeta\rangle$ we’ll compute it as follows
$\langle\zeta\zeta\zeta\rangle\sim\left(\frac{M_{2}}{H_{*}}\right)^{2}\int_{-\infty}^{0}d\eta
k_{1}^{2}k_{2}^{2}k_{3}^{2}\eta^{4}e^{ik_{T}\eta}$ i.e. we keep $H_{*}$
outside the integral. There are also multiple factors of $H_{*}$ (and
$M_{pl}$), which come from $a=-1/H\eta$ and the mode functions that are also
taken outside. To get the correct soft limit, we should thus always try to
eliminate $H$ dependence in the action and get everything ordered in powers of
$\epsilon,\eta$, as in 45, as much as possible before calculating the
correlator. This prescription is also followed in standard Maldacena action
calculations Maldacena_2003 . Proceeding with the calculations, we again have
the “exchange diagrams” as before as shown below. The left one gives a
contribution:

$\displaystyle\Delta_{B}\langle\zeta(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle_{\bm{k_{1}}\to
0}=$ $\displaystyle 2\left(\frac{B_{MT}|_{k_{3}\to 0}\delta
P_{\gamma}(k_{1})}{2P_{\gamma}(k_{1})}\right)-2\langle\zeta(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle_{RR},_{\bm{k_{1}}\to
0}$ (47)
$\displaystyle=2\left(\frac{5}{8}\frac{M^{2}_{pl}}{M^{2}_{7}}\frac{H^{6}}{M^{6}_{pl}}\frac{1}{k_{1}^{3}k_{3}^{3}}\delta_{h_{1}h_{2}}\right)+\frac{15}{8}\frac{M^{2}_{pl}}{M^{2}_{7}}\frac{H^{6}}{M^{6}_{pl}}\frac{1}{k_{1}^{3}k_{3}^{3}}\delta_{h_{1}h_{2}}$
(48)

where the first term in the last line just follows from the Maldacena soft
limit and the second term is calculated in Appendix B. Calculating the
modified spectral tilt of $\langle\gamma\gamma\rangle$ yields:

$\displaystyle\widetilde{n_{t}}=\frac{1}{H}\partial_{t}\text{log}\langle\gamma\gamma\rangle=-2\epsilon-5\epsilon\frac{M^{2}_{pl}}{M^{2}_{7}}\frac{H^{2}}{M^{2}_{pl}}$
(49)

Adding both the contributions $\Delta_{A}$ and $\Delta_{B}$ gives:

$\displaystyle\langle\zeta(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle_{A+B,\leavevmode\nobreak\
\bm{k_{1}}\to 0}=-\widetilde{n_{t}}(P_{\gamma}(k_{1})+\delta
P_{\gamma}(k_{1}))P_{\zeta}(k_{3})$ (50)

The second diagram is just like the one we considered for ${}^{(3)}\\!R\delta
K$, and hence adding it to the previous answer gives:

$\displaystyle\langle\zeta(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle_{\bm{k_{1}}\to
0}=-\widetilde{n_{t}}(P_{\gamma}(k_{1})+\delta
P_{\gamma}(k_{1}))(P_{\zeta}(k_{3})+\delta P_{\zeta}(k_{3}))$ (51)

where the equality holds at $\mathcal{O}((H/M_{pl})^{6})$. These calculations
for the soft limits of $\langle\gamma\gamma\zeta\rangle$ also hold for
$\langle\gamma\zeta\zeta\rangle$ and one can verify it for ${}^{(3)}\\!R\delta
K$ (see Appendix C).

Figure 2: The MT (Maldacena term) vertex is the usual cubic vertex of the
Maldacena action and $\mathcal{O}_{\gamma\gamma},\mathcal{O}_{\zeta\zeta}$ the
respective quadratic correction operators from
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}$

Purely Cubic Operators: We have shown how the soft limits change at the
leading order in soft momenta for cubic operators. However, when we take
purely cubic operators, we see from the derivative structure of $\delta
K,\delta K_{ij},^{(3)}\\!R_{ij},^{(3)}\\!R$ (which are the building blocks for
purely cubic operators) that we just have 333Note that 53 doesn’t work for
operators involving terms with 4 indices like ${{}^{(3)}\\!R_{ijkl}}$. In that
case, the RHS of 53 is the same as 52. However, in this paper, we’re only
dealing with operators constructed from
${{}^{(3)}\\!R_{ij}},{{}^{(3)}\\!R},\delta K_{ij},\delta K,\delta g^{00}$ and
$\widetilde{N}_{i}$ for which there’s always a $\partial^{2}$ or
$\partial_{t}$ acting on $\gamma_{ij}$, due to which the given limit holds. :

$\displaystyle\mathcal{B}(\zeta_{q}\gamma\gamma)|_{\bm{q}\to 0}=$
$\displaystyle\mathcal{B}(\zeta_{q}\zeta\gamma)|_{\bm{q}\to 0}=0$ (52)
$\displaystyle\mathcal{B}(\gamma_{q}\gamma\zeta)|_{\bm{q}\to 0}=$
$\displaystyle\mathcal{B}(\gamma_{q}\zeta\zeta)|_{\bm{q}\to 0}=\widetilde{0}$
(53)

where $\widetilde{0}$ represents the fact that the correlator is 0 as a
function without taking $\bm{k_{1}}+\bm{k_{2}}+\bm{k_{3}}=0$. Hence these
operators only give $\mathcal{O}(q^{2})$ corrections on the RHS of the soft
limits 6 and 7.

We have thus shown explicitly how the soft limits are obeyed for various
higher derivative operators and models (i.e. $c_{s}=1$ or otherwise). These
provide a consistency check for models beyond the Maldacena action and as
we’ll see next, these have an important role in the boostless bootstrap of
correlators.

##  4 Boostless bootstrap

Cosmological bootstrap, like any other bootstrap program, tries to fix the
mathematical form of physical observables without doing explicit computations
and by using very general principles such as symmetries, analytical structure,
etc. As mentioned before, as of yet we have measured only the two-point scalar
correlation function and its tilt Hazra_2014 ; Huang_2015 . The 2-point
function can be bootstrapped just by using rotational, translational and scale
invariance. Note that the two-point function is not exactly scale-invariant
but since the deviation i.e the spectral tilt is very small, one can use scale
invariance to fix the leading contribution (i.e. the result which survives in
the pure dS limit). Since, the answer is completely fixed without using dS
boosts, from the bootstrap perspective it is not clear whether boosts are good
symmetries of cosmological correlators or not. In single-field slow-roll
models of inflation, both boost and scale invariance are broken by a slowly-
evolving background (governed by $\dot{\phi}(t)$) parametrised by the slow-
roll parameter $\epsilon$. This allows one to bootstrap these correlations up
to slow-roll corrections i.e up to $\epsilon$ corrections using the full de
Sitter isometries Pajer:2016ieg . In terms of $\zeta$, it is not very clear
that this should give the correct answer since it is a gauge mode in pure de
Sitter Maldacena_2003  but one can instead bootstrap $\delta\phi$ correlators
and use
$\zeta=\frac{\delta\phi}{\sqrt{2\epsilon}M_{pl}}-\frac{\eta\delta\phi^{2}}{8\epsilon
M^{2}_{pl}}$ to get the corresponding correlators for the curvature
perturbations. In a general theory of inflation, boost breaking can be very
large while still preserving approximate scale invariance. For e.g., as
already mentioned in the first example of the EFToI section, $M_{2}$ can
potentially give a speed of sound $c_{s}\ll 1$, thus breaking boosts by a
large amount. Therefore, one can develop either a boost-preserving i.e. a
conformal bootstrap https://doi.org/10.48550/arxiv.1811.00024 ; Shukla_2016 ;
Rychkov_2017  or a boost-breaking or boostless bootstrap Pajer_2021 ;
Cabass:2021fnw ; https://doi.org/10.48550/arxiv.2210.02907 .
The conformal bootstrap basically uses Ward Identities of Boost symmetries,
along with results related to the Operator product expansion(OPE) Rychkov_2017
and soft limits to solve for the relevant correlators. The boostless
bootstrap, however, focuses more on the analytical structure of the
correlators along with results motivated by locality, soft limits etc. In the
upcoming sections, we explore the latter and use the rules mentioned in
Pajer_2021  to construct mixed correlators from higher derivative (more than
two) operators in the EFToI.

###  4.1 Relevant terms for $\langle\gamma\gamma\zeta\rangle$

Table 1:  Quadratic and cubic operators and their contributions to $\langle\gamma\gamma\zeta\rangle$ . The 3rd column shows what powers of $\epsilon$ can be present in the terms generated by the operators, while the last column shows the coefficients of the operators in terms of some energy scale $\Lambda$. The term in red is redundant as it can be removed using $\int{}A(t)^{(3)}\\!R_{ij}K_{ij}=\int{}A(t)^{(3)}\\!RK+\dot{A(t)}{{}^{(3)}\\!R}/2N+\text{total derivative},$ while the term in blue is present in the Maldacena action. Here $\delta g^{00}=g^{00}+1$. Term | No. of Derivatives | $\mathcal{O}(\epsilon)$ |  $\mathcal{O}(\Lambda)$ or $\mathcal{O}(M_{pl})$
---|---|---|---
${}^{(3)}\\!R$ | 2 | 1 | $M_{pl}^{2}$
${}^{(3)}\\!R\delta g^{00}$ | 2 | 1 | $\Lambda^{2}$
$\delta K_{ij}\delta K_{ij}$ | 3 | $1,\epsilon$ | $M^{2}_{pl}$
$\delta K_{ij}\delta K_{ij}\delta g^{00}$ | 3 | $1$ | $\Lambda^{2}$
$\delta K_{ij}\delta K_{jk}\delta K_{ki}$ | 3 | $1,\epsilon$ | $\Lambda$
$\delta K_{ij}\delta K_{ji}\delta K$ | 3 | $1,\epsilon$ | $\Lambda$
${}^{(3)}\\!R_{ij}\delta K_{ij}$ | 3 | $1,\epsilon$ | $\Lambda$
${}^{(3)}\\!R\delta K$ | 3 | $1,\epsilon$ | $\Lambda$
${}^{(3)}\\!R_{ij}\delta K_{ij}\delta g^{00}$ | 3 | $1$ | $\Lambda$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}$ | 4 | 1 | 1
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}\delta g^{00}$ | 4 | 1 | 1
${}^{(3)}\\!R_{ij}\delta K_{jk}\delta K_{ki}$ | 4 | $1,\epsilon$ | 1
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{jk}\delta K_{ki}$ | 5 | $1,\epsilon$ | 1/$\Lambda$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}\delta K$ | 5 | $1,\epsilon$ | $1/\Lambda$
${}^{(3)}\\!R_{ij}^{(3)}\\!R\delta K_{ij}$ | 5 | $1,\epsilon$ | $1/\Lambda$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{jk}^{(3)}\\!R_{ki}$ | 6 | 1 | $1/\Lambda^{2}$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}^{(3)}\\!R$ | 6 | 1 | $1/\Lambda^{2}$

As mentioned before, While writing the operators in Table LABEL:tab:t-1, we
use covariant objects (i.e. which are covariant, at least w.r.t. the 3d
metric). We also take the operators defined in Bordin_2017  :

$\displaystyle V=$ $\displaystyle\frac{\dot{\delta N}-N^{i}\partial_{i}N}{N}$
(54) $\displaystyle A_{\mu}=$
$\displaystyle\frac{h^{\nu}_{\mu}\nabla_{\nu}N}{N}$ (55)

where $h_{\mu\nu}$ is the 3d spatial metric. Note that using $A_{0}$ and $V$
we can obtain $\dot{\delta N}$ and $\widetilde{N^{i}}\partial_{i}N$, so we’ll
use them instead. Also, using $\tilde{N_{i}}$ and $\partial_{i}N$ we can
obtain $A_{i}$’s contribution to various operators. and so, we will not use
$A_{i}$ explicitly. For $\langle\gamma\gamma\zeta\rangle$, only $\dot{\delta
N}$ is relevant, which is just a higher derivative operator derived from
$\delta N$ and hence it has not been included in Table LABEL:tab:t-1. All the
higher derivative operators will of course be suppressed by an energy or mass
scale $\Lambda$. Some of the operators give non-local terms which we shall
discuss below.

###  4.2 Purely Cubic Local Terms

Let us take a local term (which is not present in the Maldacena action) as an
example of bootstrapping. We shall primarily use 4 rules of bootstrapping,
developed in Pajer_2021 . They are:

  * •

Symmetry between identical bosons. For instance,
$\langle\zeta(\bm{k_{1}})\gamma(\bm{k_{2}})\gamma(\bm{k_{3}})\rangle$ should
be symmetric under $k_{2}\leftrightarrow k_{3}$.

  * •

Amplitude rule Goodhew_2021 :

$\lim_{k_{T}\to
0}\langle\zeta_{1}\zeta_{2}\zeta_{3}..\zeta_{n}\rangle=\frac{(-1)^{n}H^{p+n-1}(p-1)!}{2^{n-1}}\frac{\text{Re}(i^{n+p+1}A_{n})}{k_{T}^{p}\prod_{a=1}^{n}k_{a}}$
(56)

Here $p$ is found by the total number of space and time derivatives in the
interaction Hamiltonian.

  * •

Manifest Locality Test(MLT)Pajer_2021

$\frac{\partial\mathcal{B}}{\partial
k_{1}}\Bigg{|}_{k_{1}=0}=\frac{\partial\mathcal{B}}{\partial
k_{2}}\Bigg{|}_{k_{2}=0}=\frac{\partial\mathcal{B}}{\partial
k_{3}}\Bigg{|}_{k_{3}=0}=0$ (57)

where $\mathcal{B}(k_{1},k_{2},k_{3})\sim\leavevmode\nobreak\
\langle\zeta_{1}\zeta_{2}\zeta_{3}\rangle_{tr}k_{1}^{3}k_{2}^{3}k_{3}^{3}$,
“tr” signifying that the tensor contractions have been trimmed. This equation
is valid only for local operators 444This can be seen easily by taking the
most general local operator
$\mathcal{O}\sim\zeta\cdot\zeta^{(m)}\cdot\zeta^{(n)}$ where $n,m$ denote the
no. of derivatives on $\zeta$. Suppose $\bm{q}$ is the soft mode. Note that if
all three $\zeta$’s had derivatives then the MLT would be trivially satisfied
due to powers of $q$ coming from the derivatives. Hence the non-trivial
contribution comes when we have $\bm{q}$ associated with the $\zeta$ with no
derivatives. In this case, the main term to focus on is:
$\langle\zeta_{\bm{q}\to 0}\zeta\zeta\rangle_{tr}\sim\frac{\partial}{\partial
q}\int_{-\infty}^{0}d\eta(1-iq\eta)e^{ik_{T}\eta}|_{\bm{q}\to
0}+\frac{\partial}{\partial
q}\int_{-\infty}^{0}d\eta(1+iq\eta)e^{-ik_{T}\eta}|_{\bm{q}\to 0}=0$ where we
have omitted the mode functions involving the other 2 momenta. The analysis is
easily extended to correlators with $\gamma$. .

  * •

Soft limits for purely cubic operators i.e. Equations (52) and (53). We shall
not bootstrap correlators arising from cubic operators since the procedure
would require knowing what diagrams we need, and in this case, an explicit in-
in computation might be much more simpler and economical.

We take the purely cubic vertex

$R^{(3)}_{ij}\delta K_{ij}\delta
g^{00}=\int{}a\frac{\Lambda}{H}\partial^{2}\gamma_{ij}\dot{\gamma_{ij}}\dot{\zeta}$
(58)

Direct in-in calculation gives:

$\langle\zeta(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle=\frac{\Lambda}{H}e^{h_{2}}_{ij}(\bm{k_{2}})e^{h_{3}}_{ij}(\bm{k_{3}})k^{2}_{1}k_{2}^{2}k_{3}^{2}\frac{H^{6}}{\epsilon
M^{6}_{pl}k^{3}_{1}k^{3}_{2}k^{3}_{3}}\left[\frac{2}{(k_{1}+k_{2}+k_{3})^{3}}+3\frac{k_{2}+k_{3}}{(k_{1}+k_{2}+k_{3})^{4}}\right]$
(59)

we have $p=4$ for this vertex and 555note that there’s no way to bootstrap the
amplitude since any function of the form $[23]^{4}f(k_{1},k_{2},k_{3})$, where
f is a degree 4 polynomial, is a valid $p=4$ amplitude. Here [ ] is the
relevant helicity bracket.

$A[1^{0},2^{h_{2}},3^{h_{3}}]\sim
e^{h_{2}}_{ij}(\bm{k_{2}})e^{h_{3}}_{ij}(\bm{k_{3}})k_{1}k_{2}k_{3}(k_{2}+k_{3})$
(60)

where we have kept the normalization arbitrary (which contains information
about things like $\Lambda$ and $\epsilon$ dependence). From this we write an
ansatz using the first rule(where $k_{T}=k_{1}+k_{2}+k_{3}$ ,
$e_{1}=k_{2}+k_{3}$, $e_{2}=k_{2}k_{3}$ and $e_{3}=k_{1}k_{2}k_{3}$):

$\begin{split}\langle\zeta(\bm{k_{1}})\gamma&(\bm{k_{2}})\gamma(\bm{k_{3}})\rangle\sim\frac{e^{h_{2}}_{ij}(\bm{k_{2}})e^{h_{3}}_{ij}(\bm{k_{3}})}{k^{4}_{T}e^{3}_{3}}\left[e_{3}k_{1}k_{2}k_{3}(k_{2}+k_{3})+k_{T}(A_{1}e_{2}^{3}+A_{2}e_{2}^{2}e_{1}^{2}+A_{3}e_{2}e_{1}^{4}+A_{4}e_{1}^{6})+\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
\left.k_{T}^{2}(A_{5}e_{2}^{2}e_{1}+A_{6}e_{2}e_{1}^{3}+A_{7}e_{1}^{5})+k_{T}^{3}(A_{8}e_{2}^{2}+A_{9}e_{2}e_{1}^{2}+A_{10}e_{1}^{4})+k_{T}^{4}(A_{11}e_{2}e_{1}+A_{12}e_{1}^{3})\right.\\\
&\left.+k_{T}^{5}(A_{13}e_{2}+A_{14}e_{1}^{2})+A_{15}k_{T}^{6}e_{1}+A_{16}k_{T}^{7}\right]\\\
\end{split}$ (61)

We get the following set of equations after applying MLT and soft limits for
various momenta:

$\displaystyle\text{Soft limit for $k_{2},k_{3}$:}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
A_{4}=A_{7}=A_{10}=A_{12}=A_{14}=A_{15}=A_{16}=0$ (62) $\displaystyle\text{MLT
for $k_{2},k_{3}$:}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ A_{3}=A_{6}=A_{9}=A_{11}=A_{13}=0$ (63)
$\displaystyle\text{Soft limit for $k_{1}$:}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ A_{2}+A_{5}+A_{8}=0$ (64)
$\displaystyle\text{MLT for $k_{1}$:}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ A_{1}=0\leavevmode\nobreak\ \leavevmode\nobreak\
3A_{2}+2A_{5}+A_{8}=0$ (65)

which fixes our correlator to be:

$\langle\zeta(\bm{k_{1}})\gamma(\bm{k_{2}})\gamma(\bm{k_{3}})\rangle\sim\frac{e^{h_{2}}_{ij}(\bm{k_{2}})e^{h_{3}}_{ij}(\bm{k_{3}})}{k^{4}_{T}e^{3}_{3}}e_{3}^{2}\left[A_{2}k_{T}+e_{1}\right]$
(66)

which means we’re able to fix the bispectra up to an overall factor and
another arbitrary constant. This expression agrees with the explicit
calculation 59 with $A_{2}=2/3$.

###  4.3 Non-local terms

We have the following non-local terms (see Appendix D) at various orders in
$\epsilon$ and energy scales:

Operator | $\mathcal{O}(\epsilon)$ |  $\mathcal{O}(H/M_{pl})$ or $\mathcal{O}(H/\Lambda)$
---|---|---
$\delta K_{ij}\delta K_{ij}$ | $\epsilon$ | $H/M_{pl}$
$\delta K_{ij}\delta K_{jk}\delta K_{ki}$ | $\epsilon$ | $(H/M_{pl})^{2}(\Lambda/M_{pl})$
$\delta K_{ij}\delta K_{jk}{{}^{(3)}\\!R}_{ki}$ | $\epsilon$ | $(H/M_{pl})^{3}$
$\delta K_{ij}{{}^{(3)}\\!R}_{jk}{{}^{(3)}\\!R}_{ki}$ | $\epsilon$ | $(H/M_{pl})^{3}(H/\Lambda)$

(For the sake of brevity we have not included the odd parity terms but their
contributions are of the same order as the last 3 terms). The second term in
the table gives:

$\int\sqrt{-g}\leavevmode\nobreak\ \Lambda\delta K_{ij}\delta K_{jk}\delta
K_{ki}=\int-\frac{3}{4}a^{3}\Lambda\epsilon\dot{\gamma_{ij}}\dot{\gamma_{jk}}\partial_{i}\partial_{k}\partial^{-2}\dot{\zeta}$
(67)

for which the explicit in-in calculation yields:

$\langle\zeta(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle=-\frac{3}{2}\frac{H^{6}}{M_{pl}^{6}}\left(\frac{\Lambda}{H}\right)e^{h_{2}}_{ij}(\bm{k_{2}})e^{h_{3}}_{jm}(\bm{k_{3}})k_{1i}k_{1m}\frac{k_{2}^{2}k_{3}^{2}}{k_{T}^{3}e_{3}^{3}}$
(68)

The non-local term in 67 naively doesn’t seem to have a proper flat space
counterpart, but as pointed out in Pajer_2021 , it can be considered to come
from a toy model:

$S_{flat}=\int{}d^{4}x\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon(\partial_{\mu}\zeta)^{2}-\frac{1}{2}(\partial_{i}X)^{2}+\frac{M_{pl}^{2}}{8}(\partial_{\mu}\gamma_{ij})^{2}-\frac{3\Lambda\epsilon\dot{\zeta_{0}}}{8}X+\frac{3\Lambda\epsilon}{8\dot{\zeta_{0}}}X\dot{\zeta}^{2}+\Lambda\dot{\gamma_{ij}}\dot{\gamma_{jk}}\partial_{i}\partial_{k}X$
(69)

Here $\zeta_{0}(t)=-\sqrt{2\epsilon}\bar{\phi}(t)$ is the background value of
the scalar field. Integrating out the field $X$ above gives us an EFT with the
desired non-local term, which gives an amplitude (with no of derivatives,
$p=3$):

$A[1^{0},2^{h_{2}},3^{h_{3}}]=e^{h_{2}}_{ij}(\bm{k_{2}})e^{h_{3}}_{jm}(\bm{k_{3}})k_{1i}k_{1m}\frac{k_{2}k_{3}}{k_{1}}$
(70)

Taking the ansatz as before(the soft limit for $k_{2},k_{3}$ has already been
taken):

$\langle\zeta(\bm{k_{1}})\gamma(k\bm{k_{2}})\gamma(\bm{k_{3}})\rangle\sim\frac{e^{h_{2}}_{ij}(\bm{k_{2}})e^{h_{3}}_{jm}(\bm{k_{3}})k_{1i}k_{1m}}{k_{T}^{3}e_{3}^{3}}\left[\frac{k_{2}k_{3}}{k_{1}}e_{3}+Ak_{T}e_{1}e_{2}+Bk_{T}^{2}e_{2}\right]$
(71)

applying MLT with respect to $k_{2}$ and $k_{3}$ fixes $A=B=0$ and hence, the
correlator up to an overall factor, matching with 68. Note that as the value
of $p$ increases, we’ll get more and more unknown parameters. Specifically for
the non-local terms, we cannot use the MLT w.r.t the non-local momenta, which
removes one condition and increases the no. of arbitrary constants. From the
conditions we have used, one can simply find that:

  * •

For local terms and odd $p$ with $p\geq 3$, one has either $(p+1)(p-3)/4+1$ or
$(p-3)^{2}/4$ parameters (which correspond to either the polarization tensors
contracted with each other or with external momenta) that can’t be fixed
(excluding the overall factor). For even $p$, this number is either
$(p^{2}-2p-4)/4$ or $(p-4)(p-2)/4$.

  * •

For non-local terms and odd $p$ with $p\geq 3$, one has either $(p^{2}-5)/4$
or $(p-3)(p-1)/4$ parameters that can’t be fixed (excluding the overall
factor). For even $p$, this number is either $(p^{2}-4)/4$ or $(p-2)^{2}/4$.

We want to point out that the source of these non-localities is rooted in the
fact that not all metric components are dynamical variables. Since, we have
constraint equations for these non-dynamical variables, one plugs in their
formal solution in the action which can potentially involve inverse
differential operators since the constraint equations are differential
equations.

###  4.4 Extending results to $\langle\gamma\zeta\zeta\rangle$

All operators in this case have enough derivatives on $\gamma$ and $\zeta$ so
that the soft limits 52, 53 are still valid. A similar bootstrap analysis can
be carried out for non-local and local terms separately. One again finds that
the non-local terms start appearing at $\mathcal{O(\epsilon)}$. The operators
are summarized in Table LABEL:table2 below

Table 2: Quadratic and cubic operators and their contributions to $\langle\gamma\zeta\zeta\rangle$. Again, the blue terms are Maldacena terms, and the red term is removable by the identity mentioned below Table LABEL:tab:t-1. Term | No. of Derivatives | $\mathcal{O}(\epsilon)$ |  $\mathcal{O}(\Lambda)$ or $\mathcal{O}(M_{pl})$
---|---|---|---
$c(t)\delta g^{00}$ | 0 | $\epsilon$ | $M^{2}_{pl}$
$\delta K_{ij}\delta K_{ij}$ | 2 | $1,\epsilon$ | $M^{2}_{pl}$
$\delta K_{ij}\widetilde{N_{i}}\partial_{j}\delta g^{00}$ | 2 | $1,\epsilon$ | $\Lambda^{2}$
${}^{(3)}\\!R_{ij}\delta K_{ij}$ | 3 | $1,\epsilon$ | $\Lambda$
$\delta K_{ij}\delta K_{jk}\delta K_{ki}$ | 3 | $1,\epsilon,\epsilon^{2}$ | $\Lambda$
$\delta K_{ij}\delta K_{ji}\delta K$ | 3 | $1,\epsilon,\epsilon^{2}$ | $\Lambda$
${}^{(3)}\\!R_{ij}\delta K_{ij}\delta g^{00}$ | 3 | $1,\epsilon$ | $\Lambda$
$\delta K_{ij}\partial_{i}\delta g^{00}\partial_{j}\delta g^{00}$ | 3 | 1 | $\Lambda$
${}^{(3)}\\!R_{ij}\widetilde{N_{i}}\partial_{j}\delta g^{00}$ | 3 | $1,\epsilon$ | $\Lambda$
${}^{(3)}\\!R_{ij}\delta K_{ij}\delta K$ | 4 | $1,\epsilon,\epsilon^{2}$ | 1
${}^{(3)}\\!R_{ij}\delta K_{jk}\delta K_{ki}$ | 4 | $1,\epsilon,\epsilon^{2}$ | 1
${}^{(3)}\\!R_{ij}\partial_{i}\delta g^{00}\partial_{j}g^{00}$ | 4 | 1 | 1
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}\delta g^{00}$ | 4 | 1 | $1/\Lambda$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{jk}\delta K_{ki}$ | 5 | $1,\epsilon$ | $1/\Lambda$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}\delta K$ | 5 | $1,\epsilon$ | $1/\Lambda$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{jk}^{(3)}\\!R_{ki}$ | 6 | 1 | $1/\Lambda^{2}$
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}^{(3)}\\!R$ | 6 | 1 | $1/\Lambda^{2}$

We take the term (which is the operator $\mathcal{O}_{3}$ in Appendix D) :

$\mathcal{O}=\delta K_{ij}\partial_{i}\delta g^{00}\partial_{j}\delta
g^{00}\sim\int
a\frac{1}{\Lambda}\dot{\gamma_{ij}}\partial_{i}\dot{\zeta}\partial_{j}\dot{\zeta}$
(72)

The explicit in-in calculation gives:

$\langle\gamma^{h}(\bm{k_{1}})\zeta(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle=6\frac{H^{7}}{\epsilon^{2}M_{pl}^{6}\Lambda}\frac{e_{ij}^{h}(\bm{k_{1}})k_{2i}k_{3j}}{e_{3}k_{T}^{5}}$
(73)

We have the corresponding flat space amplitude:

$A[1^{h},2^{0},3^{0}]\sim e_{3}e^{h}_{ij}(\bm{k_{1}})k_{2i}k_{3j}$ (74)

which gives us the ansatz for the correlator to be:

$\displaystyle\langle\gamma^{h}(\bm{k_{1}})\zeta(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle\sim\frac{e^{h}_{ij}(\bm{k_{1}})k_{2i}k_{3j}}{k_{T}^{5}e_{3}^{3}}\left[e_{3}^{2}+k_{T}(A_{1}e_{2}^{2}e_{1}+A_{2}e_{2}e_{1}^{3}+A_{3}e_{1}^{5})+k_{T}^{2}(A_{4}e_{2}^{2}+A_{5}e_{2}e_{1}^{2}+A_{6}e_{1}^{4})\right.$
$\displaystyle\left.k_{T}^{3}(A_{7}e_{2}e_{1}+A_{8}e_{1}^{3})+k_{T}^{4}(A_{9}e_{2}+A_{10}e_{1}^{2})+A_{11}k_{T}^{5}e_{1}+A_{12}k_{T}^{6}\right]$
(75)

Soft limits and MLTs give the following equations:

Soft limit $\bm{k_{1}}\to 0$ $\displaystyle\begin{cases}A_{1}+A_{4}=0\\\
A_{2}+A_{5}+A_{7}+A_{9}=0\\\
A_{3}+A_{6}+A_{8}+A_{10}+A_{11}+A_{12}=0\end{cases}$ (76)

MLT for $k_{1}$ $\displaystyle\begin{cases}4A_{1}+3A_{4}=0\\\
4A_{2}+3A_{5}+2A_{7}+A_{9}=0\\\
4A_{3}+3A_{6}+2A_{8}+A_{10}-A_{12}=0\end{cases}$ (77) Soft limit
$\bm{k_{2}}\to 0$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
A_{3}+2A_{6}+4A_{8}+8A_{10}+16A_{11}+32A_{12}=0$ (78) MLT for $k_{2}$
$\displaystyle\begin{cases}5A_{3}+A_{2}+2A_{6}=0\\\ A_{3}=0\\\
A_{5}+4A_{6}+3A_{8}=0\\\ A_{7}+3A_{8}+4A_{10}=0\\\ A_{9}+2A_{10}+5A_{11}=0\\\
A_{11}+6A_{12}=0\end{cases}$ (79)

which leads to $A_{i}=0$ for all $i$, fixing correlator upto an arbitrary
factor:

$\langle\gamma^{h}(\bm{k_{1}})\zeta(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle\sim\frac{e^{h}_{ij}(\bm{k_{1}})k_{2i}k_{3j}}{k_{T}^{5}e_{3}^{3}}e_{3}^{2}$
(80)

##  5 Going to $\alpha$ vacua

In our calculations, we are not compelled to fix the initial condition (and
the subsequent evolution) by taking the Bunch-Davies (BD) vacuum. One can take
the well-known family of $\alpha$ vacua Allen:1985ux ; Shukla_2016  as well
since they respect the symmetries of the quasi dS background. Bootstrapping
$\alpha$-vacua answers directly using the boostless bootstrap is difficult
since we don’t have the soft limit conditions such as 53, because of the pole
structures mentioned in Section 3. However, once we have bootstrapped
$\mathcal{B}(k_{1},k_{2},k_{3})$(as defined above) for BD, we can extend the
result to $\alpha$ vacua easily. Just by noting the form of the mode
functions, which are given by :

$u_{k}(\eta)=\text{cosh}\alpha(1-ik\eta)e^{ik\eta}-i\text{sinh}\alpha(1+ik\eta)e^{-ik\eta}$
(81)

we can give an ansatz for the $\alpha$\- vacua bispectra as follows:

$\begin{split}\mathcal{B}_{\alpha}(k_{1},k_{2},k_{3})=\text{Re}&\left[\left(\text{cosh}\alpha-i\text{sinh}\alpha\right)^{3}\left(\mathcal{B}_{BD}(k_{1},k_{2},k_{3})\text{cosh}^{3}\alpha+\sum_{\text{cyclic}}\mathcal{B}_{BD}(-k_{1},k_{2},k_{3})i\text{cosh}^{2}\alpha\leavevmode\nobreak\
\text{sinh}\alpha\right.\right.\\\
&\left.\left.-\sum_{\text{cyclic}}\mathcal{B}_{BD}(-k_{1},-k_{2},k_{3})\text{cosh}\alpha\leavevmode\nobreak\
\text{sinh}^{2}\alpha-\mathcal{B}_{BD}(-k_{1},-k_{2},-k_{3})i\text{sinh}^{3}\alpha\right)\right]\end{split}$
(82)

Using this equation to bootstrap $\langle\gamma\gamma\gamma\rangle$ for the
Maldacena action, we take the well-known result for BD which was bootstrapped
in Pajer_2021  (also explicitly calculated in Maldacena_2003 ), and get :

$\begin{split}\langle\gamma^{h_{1}}(\bm{k_{1}})&\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle_{\alpha}=-\frac{2H^{4}}{M_{pl}^{4}}\frac{1}{(\prod^{3}_{a=1}k_{a}^{3})}e^{h_{1}}_{ii^{\prime}}e^{h_{2}}_{jj^{\prime}}e^{h_{3}}_{kk^{\prime}}t_{ijk}t_{i^{\prime}j^{\prime}k^{\prime}}\left[\left(-k_{T}+\frac{\sum
k_{i}k_{j}}{k_{T}}+\frac{k_{1}k_{2}k_{3}}{k_{T}^{2}}\right)+\right.\\\
&\left.\text{sinh}^{2}2\alpha\left(-(-k_{1}+k_{2}+k_{3})+\frac{k_{2}k_{3}-k_{1}k_{2}-k_{1}k_{3}}{(-k_{1}+k_{2}+k_{3})}-\frac{k_{1}k_{2}k_{3}}{(-k_{1}+k_{2}+k_{3})^{2}}\right)+\text{cyclic}\right]\\\
\\\ &\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
t_{ijk}=k_{2i}\delta_{jl}+k_{3j}\delta_{li}+k_{1k}\delta_{ij}\end{split}$ (83)

which agrees with the explicit in-in result in Kanno_2022 . We also get the
following expression for the pure scalar correlator:

$\begin{split}&\langle\zeta(\bm{k_{1}})\zeta(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{\alpha}=\frac{H^{4}}{32M_{pl}^{4}\epsilon^{2}}\frac{1}{(\prod_{a=1}^{3}k_{a}^{3})}\left[2(\epsilon-\eta)\sum_{a}k^{3}_{a}+\epsilon\left(\sum_{a}k^{3}_{a}+\sum_{a\neq
b}k^{2}_{a}k_{b}+\sum_{a>b}\frac{k_{a}^{2}k_{b}^{2}}{k_{T}}\right)+\right.\\\
&\left.\text{sinh}^{2}2\alpha\left(2(\epsilon-\eta)\sum_{a}k_{a}^{3}+\epsilon\left(\sum_{a}k_{a}^{3}+\sum_{a\neq
b}k^{2}_{a}k_{b}+\sum_{a>b}\frac{k_{a}^{2}k_{b}^{2}}{k_{2}+k_{3}-k_{1}}+\frac{k_{a}^{2}k_{b}^{2}}{k_{3}+k_{1}-k_{2}}+\frac{k_{a}^{2}k_{b}^{2}}{k_{1}+k_{2}-k_{3}}\right)\right)\right]\end{split}$
(84)

which agrees with the calculation done in Shukla_2016 . This demonstrates that
the precription given above indeed works.

Similarly, using the BD results, we get the following for mixed correlators
for Maldacena action in $\alpha$ vacua:

$\begin{split}\langle\gamma^{h}(\bm{k_{1}})&\zeta(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle_{\alpha}=\frac{H^{4}}{4M^{4}_{pl}\epsilon}\frac{1}{(\prod_{a=1}^{3}k_{a}^{3})}e^{h}_{ij}(k_{1})k_{2i}k_{3j}\left[\left(-k_{T}+\frac{\sum
k_{i}k_{j}}{k_{T}}+\frac{k_{1}k_{2}k_{3}}{k_{T}^{2}}\right)+\right.\\\
&\left.\text{sinh}^{2}2\alpha\left(-(-k_{1}+k_{2}+k_{3})+\frac{k_{2}k_{3}-k_{1}k_{2}-k_{1}k_{3}}{(-k_{1}+k_{2}+k_{3})}-\frac{k_{1}k_{2}k_{3}}{(-k_{1}+k_{2}+k_{3})^{2}}\right)+\text{cyclic}\right]\end{split}$
(85)

$\begin{split}\langle&\zeta(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\gamma^{h_{3}}(\bm{k_{3}})\rangle_{\alpha}=\frac{H^{4}}{8M^{4}_{pl}}\frac{1}{(\prod_{a=1}^{3}k_{a}^{3})}e^{h_{2}}_{ij}(k_{2})e^{h_{3}}_{ij}(k_{3})\left[\left(-\frac{1}{4}k_{1}^{3}+\frac{1}{2}k_{1}(k_{2}^{2}+k_{3}^{2})+4\frac{k^{2}_{2}k^{2}_{3}}{k_{T}}\right)\right.\\\
&\left.+\text{sinh}^{2}2\alpha\left(-\frac{1}{4}k_{1}^{3}+\frac{1}{2}k_{1}(k_{2}^{2}+k_{3}^{2})+4\frac{k^{2}_{2}k^{2}_{3}}{(k_{2}+k_{3}-k_{1})}+4\frac{k^{2}_{2}k^{2}_{3}}{(k_{3}+k_{1}-k_{2})}+4\frac{k^{2}_{2}k^{2}_{3}}{(k_{1}+k_{2}-k_{3})}\right)\right]\end{split}$
(86)

We see that the prescription saves us the effort of doing the cumbersome in-in
calculation which involves simplifying a lot of integrals.

##  6 Conclusions

In this article, we aimed to understand the mixed graviton and scalar
bispectra in the EFT of inflation. A summary of the main results of the paper
is as follows:

  * •

Following the methods prescribed in Cheung_2008 , we wrote a general EFToI and
attempted to organize terms in the order of the number of derivatives on them
w.r.t the metric perturbations. We also clarified the energy scale counting in
terms of $H/M_{i}$ where $M_{i}$’s are the UV cutoffs/high mass scales
appearing in our EFTs.

  * •

We gave some general constraints on the EFT parameters, namely the bounds due
to small spectral tilt, unitarity bound in flat space limit and experimental
bounds of non-gaussianities Cheung_2008 . We gave 2 simple examples where
these bounds constrained some of the arbitrary EFT coefficients.

  * •

We explicitly checked the soft limits 5, 6 and 7 for EFT operators, which
change both the quadratic and cubic action for $\zeta$ or $\gamma$. These
limits as we checked, are obeyed for cubic operators at leading order in soft
momenta and leading order in the “couplings” (i.e. $\epsilon,\eta$ etc.) and
$H/M_{pl}$. We clarified some confusion in the literature related to what
diagrams to take and how to organise terms in order to get the correct soft
limits. Hence, as expected from the general derivation of the soft limits
Creminelli_2014  (also see Section 3), they can be extended to models beyond
the standard Maldacena action Maldacena_2003 .

  * •

We attempted to bootstrap the three-point correlators from purely cubic
operators, i.e. operators that do not change the quadratic action, by noticing
that they don’t contribute to the soft limit at leading order in soft momenta
(see Equations 52, 53). Using the bootstrap prescription in Pajer_2021 , we
found (as expected) that the number of undetermined parameters increases with
the number of derivatives. Furthermore, this bootstrap method heavily relies
on knowing the interaction hamiltonian since we use the amplitude of the
interaction to determine the residue of the total energy pole Goodhew_2021 .
Hence, the bootstrap method is more of an alternative to doing the in-in
integrals than an ideal “boundary perspective” bootstrap.

  * •

Since the background symmetry allows us to take $\alpha$ vacua, we have shown
that the results of BD can be readily extended to give the $\alpha$ vacua
3-point correlators. This helps us in bypassing a lot of in-in integrals.

It will be interesting to explore mixed quartic operators for and beyond the
Maldacena action and check the soft limits for these since some exchange
diagrams also come into the picture here as they’re of a similar order in
“couplings” and $H/M_{pl}$ as contact diagrams. The right-hand side of the
soft limits might be tricky to evaluate due to the momentum dependence of the
polarization tensors. We would also like to extend our $\alpha$ vacua results
to four-point correlators and explore the implications of the new kinds of
pole structures we get.

## Acknowledgments

DG acknowledges support through the Ramanujan Fellowship and MATRICS Grant of
the Department of Science and Technology, Government of India. KP thanks
Enrico Pajer for clarifications related to soft limits via email. We also
thank Sachin Jain and Muhammad Ali for useful discussions.

##  Appendix A Calculating the action in Unitary Gauge

From the definitions of the ADM metric variables, we have
$\widetilde{N^{i}}=-g_{00}N^{i}=N^{2}N^{i}$. We further write $N=1+\delta N$
and then consider the following quadratic action:

$\displaystyle\mathcal{L}_{2}$
$\displaystyle=\sqrt{-g}M^{2}_{pl}\left(\frac{1}{2}R^{(4)}+m_{1}\delta
K^{i}_{j}\delta K^{j}_{i}+m_{2}(\delta K)^{2}+D\delta
K-M^{2}_{pl}\lambda(t)-c(t)g^{00}\right.$
$\displaystyle\left.+M_{1}g^{ij}\partial_{i}(g^{00}+1)\partial_{j}(g^{00}+1)+M^{2}_{2}(g^{00}+1)^{2}+M_{3}(g^{00}+1)\delta
K+m_{3}^{(3)}\\!R(g^{00}+1)\right)$ (87)

where $c(t)$,$\lambda(t)$ are as defined before. Taking
$\widetilde{N^{i}}=\partial_{i}\chi$, the equations of motion for $N$ and
$\widetilde{N^{i}}$ gives

$\displaystyle\delta N$
$\displaystyle\left((m_{1}+3m_{2}-1)H-M_{3}\right)+(m_{1}+m_{2})\partial^{2}\chi=\dot{\zeta}(m_{1}+3m_{2}-1)$
(88) $\displaystyle\partial^{2}\chi$
$\displaystyle=\frac{-1}{(m_{1}+3m_{2}-1)H-M_{3}}\left(-\partial^{2}\zeta
a^{-2}+c(t)\delta N+4M^{2}_{2}\delta N-4a^{-2}M_{1}\partial^{2}\delta
N\right)-3\dot{\zeta}$ (89)
$\displaystyle+\frac{3\dot{\zeta}H(m_{1}+3m_{2}-1)}{(m_{1}+3m_{2}-1)H-M_{3}}-\frac{3\dot{\zeta}H(m_{1}+3m_{2}-1)M_{3}}{((m_{1}+3m_{2}-1)H-M_{3})^{2}}$
(90)

To solve these equations, one will have to take $m_{1}+m_{2}=0$ so that the
equations separate.

##  Appendix B Calculating the RR vertex for
${}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}$

The expression for the diagram where both the vertices are time ordered, i.e.
RR vertices Weinberg_2005  after taking the soft limit is given by:

$\displaystyle\langle$
$\displaystyle\gamma^{h_{1}}(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\zeta(\bm{k_{3}})|_{\bm{k_{3}}\to
0}\rangle|_{RR}=\frac{H^{6}}{M^{6}_{pl}k_{1}^{9}4\epsilon
k_{3}^{3}}\left(\frac{M_{pl}^{2}\epsilon}{M_{7}^{2}(4\cdot 8)}2\cdot 2\cdot
2\cdot 2\leavevmode\nobreak\
e^{h_{1}}_{ij}(\bm{k_{1}})e^{h_{2}}_{ij}(\bm{k_{2}})\right)$
$\displaystyle\left[\int_{-\infty}^{0}\left(k_{1}^{2}k_{2}^{2}e^{2ik_{1}\eta}-(k_{1}\cdot
k_{2})\frac{(1-ik_{1}\eta)(1-ik_{2}\eta)}{\eta^{2}}e^{2ik_{1}\eta}\right)\left(\int_{\eta}^{0}(1+ik_{2}\eta^{\prime})k_{2}^{4}(1-ik_{2}\eta^{\prime})\right)d\eta
d\eta^{\prime}+\right.$
$\displaystyle\left.\int_{-\infty}^{0}\left(k_{1}^{2}k_{2}^{2}-(k_{1}\cdot
k_{2})\frac{(1-ik_{1}\eta)(1+ik_{2}\eta)}{\eta^{2}}\right)\left(\int_{-\infty}^{\eta}(1-ik_{2}\eta^{\prime})^{2}k_{2}^{4}e^{2ik_{2}\eta}\right)d\eta
d\eta^{\prime}\right]$ (91)

where we have taken $\bm{k_{3}}\to 0$ but have not yet put
$\bm{k_{1}}=\bm{-k_{2}}$ for clarity. All the combinatorial and
numerical/coupling factors are in the bracket in the first line. Note that for
the Maldacena vertex, we have not taken the non-local term Maldacena_2003 , as
that term is 0 (or rather subleading) in the soft limit. After putting
$\bm{k_{1}}=\bm{-k_{2}}$ and simplifying we get:

$\langle\gamma^{h_{1}}(\bm{k_{1}})\gamma^{h_{2}}(\bm{k_{2}})\zeta(\bm{k_{3}})|_{k_{3}\to
0}\rangle|_{RR}=-\frac{15c_{1}H^{6}}{16M^{6}_{pl}k_{1}^{3}k_{3}^{3}}$ (92)

##  Appendix C Soft limit of $\langle\gamma\zeta\zeta\rangle$ for
${}^{(3)}\\!R\delta K$

The “exchange diagra” is the same as the right diagram in Figure 2 and we
have, to the 1st order in $c_{1}$:

$\displaystyle O_{2\zeta}$ $\displaystyle=\int
4a\frac{M^{2}_{pl}}{M_{5}}\partial^{2}\zeta\left(\epsilon\dot{\zeta}-\frac{\partial^{2}\zeta}{H}a^{-2}\right)$
(93) $\displaystyle O_{\gamma\zeta\zeta}$ $\displaystyle=\text{Maldacena
terms}\leavevmode\nobreak\ \leavevmode\nobreak\
+4\frac{M^{2}_{pl}}{M_{5}}\left(2\frac{\gamma_{ij}a^{-1}}{H}\partial_{i}\partial_{j}\zeta\partial^{2}\zeta-a\gamma_{ij}\partial_{i}\partial_{j}\zeta\dot{\zeta}\right)$
(94)

This gives a correction to the 3-point function, in the limit $\bm{k_{1}}\to
0$

$\displaystyle\langle\gamma^{h}(\bm{k_{1}})\zeta(\bm{k_{2}})\zeta(\bm{k_{3}})\rangle=B_{LL}+B_{RR}-B_{RL}-B_{LR}+B_{\text{contact}}$
(95)

where the contact vertex is the one from $\mathcal{O}_{\gamma\zeta\zeta}$. One
finds that $B_{RR}+B_{LL}=B_{\text{contact}}$ and we can easily see that

$\begin{split}B_{LR}(k_{1}\to 0)=B_{RL}(k_{1}\to
0)=\frac{1}{2}\frac{B_{MT}(k_{1}\to 0,k_{2}=k_{3})\delta
P_{\zeta}(k_{3})}{P_{\zeta}(k_{3})}=\frac{3}{4}e^{h}_{ij}(\bm{k_{1}})\frac{1}{k_{2}^{2}}k_{2i}k_{2j}P_{\gamma}(k_{1})\delta
P_{\zeta}(k_{2})\end{split}$ (96)

and hence the soft limit (Equation 7) is satisfied.

##  Appendix D Purely Cubic Operators

Here, we give explicit expressions for operators that contribute to the mixed
correlators. Note that for these calculations, we have taken 20, 21 as the
quadratic actions, i.e. the Maldacena quadratic action. $\Lambda_{i}$’s are
the UV cutoffs for each term while $c_{i}^{\prime}s$ are dimensionless
quantities.

Purely Cubic Operators for $\langle\gamma\gamma\zeta\rangle$

$\displaystyle\mathcal{O}_{1}$
$\displaystyle=\int{}{}\sqrt{-g}\Lambda_{1}^{2}\delta K_{ij}\delta
K_{ij}\delta
g^{00}=\int{}\Lambda_{1}^{2}a^{3}\frac{1}{2H}\dot{\gamma_{ij}}\dot{\gamma_{ij}}\dot{\zeta}$
(97) $\displaystyle\mathcal{O}_{2}$
$\displaystyle=\int{}{}\sqrt{-g}\Lambda_{2}^{(3)}\\!R_{ij}\delta K_{ij}\delta
g^{00}=\int{}\Lambda_{2}a\frac{1}{2H}\partial^{2}\gamma_{ij}\dot{\gamma_{ij}}\dot{\zeta}$
(98) $\displaystyle\mathcal{O}_{3}$
$\displaystyle=\int{}\sqrt{-g}c_{3}^{(3)}\\!R_{ij}^{(3)}\\!R_{ij}\delta
g^{00}=\int{}c_{3}a^{-1}\frac{1}{2H}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}\dot{\zeta}$
(99) $\displaystyle\mathcal{O}_{4}$
$\displaystyle=\int{}{}\sqrt{-g}c_{4}^{(3)}\\!R_{ij}\delta K_{jk}\delta
K_{ki}=\int{}c_{4}a\left(\frac{1}{2}\partial^{2}\gamma_{ij}\dot{\gamma_{ij}}\dot{\zeta}-\frac{1}{2H}a^{-2}\partial^{2}\gamma_{ij}\dot{\gamma_{jk}}\partial_{i}\partial_{k}\zeta\right.$
(100) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left.+\frac{1}{2}\epsilon\partial^{2}\gamma_{ij}\dot{\gamma_{jk}}\partial_{i}\partial_{k}\partial^{-2}\dot{\zeta}-\frac{1}{4}\dot{\gamma_{ij}}{\dot{\gamma_{jk}}}\partial_{i}\partial_{k}\zeta-\frac{1}{4}\dot{\gamma_{ij}}\dot{\gamma_{ij}}\partial^{2}\zeta\right)$
(101) $\displaystyle\mathcal{O}_{5}$
$\displaystyle=\int{}{}\sqrt{-g}\frac{1}{\Lambda_{5}}{{}^{(3)}\\!R}_{ij}^{(3)}\\!R_{jk}\delta
K_{ki}=\int{}\frac{1}{\Lambda_{5}}\left(\frac{1}{4H}a^{-3}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{jk}\partial_{k}\partial_{i}\zeta\right.$
(102) $\displaystyle\left.-\frac{1}{4}\epsilon
a^{-1}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{jk}\partial_{k}\partial_{i}\partial^{-2}\dot{\zeta}\right)+\frac{1}{2}a^{-1}\partial^{2}\gamma_{ij}\dot{\gamma_{ik}}\partial_{j}\partial_{k}\zeta+\frac{1}{2}a^{-1}\partial^{2}\gamma_{ij}\dot{\gamma_{ij}}\partial^{2}\zeta$
(103) $\displaystyle\mathcal{O}_{6}$
$\displaystyle=\int{}{}\sqrt{-g}\frac{1}{\Lambda_{6}}{{}^{(3)}\\!R}_{ij}^{(3)}\\!R_{ij}\delta
K=\int{}\frac{1}{\Lambda_{6}}\left(\frac{1}{4H}a^{-3}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}\partial^{2}\zeta-\frac{1}{4}\epsilon
a^{-1}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}\dot{\zeta}\right)$ (104)
$\displaystyle\mathcal{O}_{7}$
$\displaystyle=\int{}\sqrt{-g}\frac{1}{\Lambda_{7}}{{}^{(3)}\\!R}_{ij}\delta
K_{ij}{{}^{(3)}\\!R}=\int{}\frac{1}{\Lambda_{7}}a^{-1}\partial^{2}\gamma_{ij}\dot{\gamma_{ij}}\partial^{2}\zeta$
(105) $\displaystyle\mathcal{O}_{8}$
$\displaystyle=\int{}\sqrt{-g}\frac{1}{\Lambda_{8}^{2}}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}_{jk}{{}^{(3)}\\!R}_{ki}=\int{}\frac{1}{\Lambda_{8}^{2}}\left(-\frac{3}{4}a^{-3}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{jk}\partial_{k}\partial_{i}\zeta-\frac{3}{4}a^{-3}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}\partial^{2}\zeta\right)$
(106) $\displaystyle\mathcal{O}_{9}$
$\displaystyle=\int{}\sqrt{-g}\frac{1}{\Lambda_{9}^{2}}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}=\int{}-\frac{1}{\Lambda_{9}^{2}}a^{-3}\partial^{2}\gamma_{ij}\partial^{2}\gamma_{ij}\partial^{2}\zeta$
(107) $\displaystyle\mathcal{O}_{10}$
$\displaystyle=\int{}\sqrt{-g}\Lambda_{10}\delta K_{ij}\delta K_{jk}\delta
K_{ki}=\int{}\leavevmode\nobreak\
\frac{3}{4}a^{3}\Lambda_{10}\dot{\gamma_{ij}}\dot{\gamma_{jk}}\left(\partial_{k}\partial_{i}\frac{\zeta}{H}a^{-2}-\epsilon\partial_{k}\partial_{i}\partial^{-2}\dot{\zeta}\right)$
(108) $\displaystyle\mathcal{O}_{11}$
$\displaystyle=\int{}\sqrt{-g}\Lambda_{11}\delta K_{ij}\delta K_{ji}\delta
K=\int{}\leavevmode\nobreak\
\frac{3}{4}a^{3}\Lambda_{11}\dot{\gamma_{ij}}\dot{\gamma_{ji}}\left(\partial^{2}\frac{\zeta}{H}a^{-2}-\epsilon\dot{\zeta}\right)$
(109)

Purely Cubic operators for $\langle\gamma\zeta\zeta\rangle$

$\displaystyle\mathcal{O}_{1}$
$\displaystyle=\int\sqrt{-g}\Lambda_{1}^{2}\delta
K_{ij}\widetilde{N_{i}}\partial_{j}\delta
g^{00}=\int{}\Lambda_{1}^{2}\left(-a\frac{\dot{\gamma_{ij}}}{H^{2}}\partial_{i}\zeta\partial_{j}\dot{\zeta}+a^{3}\frac{\epsilon}{H}\dot{\gamma_{ij}}\partial_{i}\dot{\zeta}\partial_{j}\partial^{-2}\dot{\zeta}\right)$
(110) $\displaystyle\mathcal{O}_{2}$
$\displaystyle=\int\sqrt{-g}\Lambda_{2}{{}^{(3)}\\!R}_{ij}\delta K_{ij}\delta
g^{00}=\int{}\Lambda_{2}\left(-a^{-1}\frac{\partial^{2}\gamma_{ij}}{H^{2}}\partial_{i}\partial_{j}\zeta\dot{\zeta}+\epsilon
a\frac{\partial^{2}\gamma_{ij}}{H^{2}}\partial_{i}\partial_{j}\dot{\zeta}\dot{\zeta}\right)$
(111) $\displaystyle\mathcal{O}_{3}$
$\displaystyle=\int{}\sqrt{-g}\Lambda_{3}\delta K_{ij}\partial_{i}\delta
g^{00}\partial_{j}\delta g^{00}=\int
2\Lambda_{3}a\dot{\gamma_{ij}}\partial_{i}\dot{\zeta}\partial_{j}\dot{\zeta}$
(112) $\displaystyle\mathcal{O}_{4}$
$\displaystyle=\int{}\sqrt{-g}\Lambda_{4}{{}^{(3)}\\!R}_{ij}\widetilde{N_{i}}\partial_{j}\delta
g^{00}=\int{}\Lambda_{4}\left(a^{-1}\frac{\partial^{2}\gamma_{ij}}{H^{2}}\partial_{i}\zeta\partial_{j}\dot{\zeta}-\epsilon
a\frac{\partial^{2}\gamma_{ij}}{H}\partial_{i}\dot{\zeta}\partial_{j}\partial^{-2}\dot{\zeta}\right)$
(113) $\displaystyle\mathcal{O}_{5}$
$\displaystyle=\int{}\sqrt{-g}c_{5}{{}^{(3)}\\!R}_{ij}\delta K_{ij}\delta
K=\int{}c_{5}\left(-a^{-3}\frac{\partial^{2}\gamma_{ij}}{2H^{2}}\partial_{i}\partial_{j}\zeta\partial^{2}\zeta+\epsilon
a^{-1}\frac{\partial^{2}\gamma_{ij}}{2H}\partial^{2}\zeta\partial_{i}\partial_{j}\partial^{-2}\dot{\zeta}\right.$
(114) $\displaystyle\left.+\leavevmode\nobreak\ \epsilon
a^{-1}\frac{\partial^{2}\gamma_{ij}}{2H}\dot{\zeta}\partial_{i}\partial_{j}\zeta-\epsilon^{2}a\frac{\partial^{2}\gamma_{ij}}{2}\dot{\zeta}\partial_{i}\partial_{j}\partial^{-2}\dot{\zeta}\right)$
(115) $\displaystyle\mathcal{O}_{6}$
$\displaystyle=\int{}\sqrt{-g}c_{6}{{}^{(3)}\\!R}_{ij}\delta K_{jk}\delta
K_{ki}=\int{}\leavevmode\nobreak\
-\frac{1}{2}c_{6}a\partial^{2}\gamma_{ij}\partial_{j}\partial_{k}\left(-\frac{\zeta}{H}a^{-2}+\epsilon\partial^{-2}\dot{\zeta}\right)\partial_{i}\partial_{k}\left(-\frac{\zeta}{H}a^{-2}+\epsilon\partial^{-2}\dot{\zeta}\right)$
$\displaystyle-
c_{6}a\dot{\gamma_{ij}}\partial_{j}\partial_{k}\zeta\partial_{k}\partial_{i}\left(-a^{-2}\frac{\zeta}{H}+\epsilon\partial^{-2}\dot{\zeta}\right)-c_{6}a\dot{\gamma_{ij}}\partial^{2}\zeta\partial_{j}\partial_{i}\left(-a^{-2}\frac{\zeta}{H}+\epsilon\partial^{-2}\dot{\zeta}\right)$
(116) $\displaystyle\mathcal{O}_{7}$
$\displaystyle=\int{}\sqrt{-g}c_{7}{{}^{(3)}\\!R}_{ij}\partial_{i}\delta
g^{00}\partial_{j}\delta
g^{00}=\int{}-2c_{7}a^{-1}\frac{\partial^{2}\gamma_{ij}}{H^{2}}\partial_{i}\dot{\zeta}\partial_{j}\dot{\zeta}$
(117) $\displaystyle\mathcal{O}_{8}$
$\displaystyle=\int{}\sqrt{-g}c_{8}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}_{ij}\delta
g^{00}=\int{}\leavevmode\nobreak\
2c_{8}a^{-1}\partial^{2}\gamma_{ij}\partial_{i}\partial_{j}\zeta\dot{\zeta}$
(118) $\displaystyle\mathcal{O}_{9}$
$\displaystyle=\int{}\sqrt{-g}\frac{1}{\Lambda_{9}}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}_{jk}\delta
K_{ki}=\int{}\frac{1}{\Lambda_{9}}\left(a^{-3}\frac{\partial^{2}\gamma_{ij}}{H}\partial_{i}\partial_{k}\zeta\partial_{j}\partial_{k}\zeta+a^{-3}\partial^{2}\frac{\gamma_{ij}}{H}\partial_{i}\partial_{j}\partial^{2}\zeta\right.$
(119)
$\displaystyle\left.+\frac{1}{2}a^{-1}\dot{\gamma_{ij}}\partial_{i}\partial_{k}\zeta\partial_{j}\partial_{k}\zeta+a^{-1}\dot{\gamma_{ij}}\partial_{i}\partial_{j}\zeta\partial^{2}\zeta-\epsilon
a^{-1}\partial^{2}\gamma_{ij}\partial_{j}\partial_{k}\zeta\partial_{k}\partial_{i}\partial^{-2}\dot{\zeta}-\epsilon
a^{-1}\partial^{2}\gamma_{ij}\partial^{2}\zeta\partial_{i}\partial_{j}\partial^{-2}\dot{\zeta}\right)$
(120) $\displaystyle\mathcal{O}_{10}$
$\displaystyle=\int{}\sqrt{-g}\frac{1}{\Lambda_{10}}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}_{ij}\delta
K=\int{}\frac{1}{\Lambda_{10}}\left(a^{-3}\frac{\partial^{2}\gamma_{ij}}{H}\partial_{i}\partial_{j}\zeta\partial^{2}\zeta-\epsilon
a^{-1}\partial^{2}\gamma_{ij}\partial_{i}\partial_{j}\zeta\dot{\zeta}\right)$
(121) $\displaystyle\mathcal{O}_{11}$
$\displaystyle=\int{}\frac{1}{\Lambda_{11}^{2}}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}_{jk}{{}^{(3)}\\!R}_{ki}=\int{}-\frac{3}{2}\frac{1}{\Lambda_{11}^{2}}a^{-3}\partial^{2}\gamma_{ij}\partial_{i}\partial_{k}\zeta\partial_{j}\partial_{k}\zeta-3\frac{c_{11}}{\Lambda^{2}}a^{-3}\partial^{2}\gamma_{ij}\partial_{i}\partial_{j}\zeta\partial^{2}\zeta$
(122) $\displaystyle\mathcal{O}_{12}$
$\displaystyle=\int{}\sqrt{-g}\frac{1}{\Lambda_{12}^{2}}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}_{ij}{{}^{(3)}\\!R}=\int{}-4\frac{1}{\Lambda_{12}^{2}}a^{-3}\partial^{2}\gamma_{ij}\partial_{i}\partial_{j}\zeta\partial^{2}\zeta$
(123) $\displaystyle\mathcal{O}_{13}$
$\displaystyle=\int{}\sqrt{-g}\Lambda_{13}\delta K_{ij}\delta K_{jk}\delta
K_{ki}=\frac{3}{2}a^{3}\Lambda_{13}\dot{\gamma_{ij}}\partial_{j}\partial_{k}\left(\frac{\zeta}{H}a^{-2}-\epsilon\partial^{-2}\dot{\zeta}\right)\partial_{k}\partial_{i}\left(\frac{\zeta}{H}a^{-2}-\epsilon\partial^{-2}\dot{\zeta}\right)$
(124) $\displaystyle\mathcal{O}_{14}$
$\displaystyle=\int{}\sqrt{-g}\Lambda_{14}\delta K_{ij}\delta K_{ji}\delta
K=a^{3}\Lambda_{14}\dot{\gamma_{ij}}\partial_{j}\partial_{i}\left(\frac{\zeta}{H}a^{-2}-\epsilon\partial^{-2}\dot{\zeta}\right)\left(\partial^{2}\frac{\zeta}{H}a^{-2}-\epsilon\dot{\zeta}\right)$
(125)

##  Appendix E Issue with Gauge

The EFT of inflation is written after fixing the time diffs, so naturally, the
operators in the EFT do not respect time diffs. Therefore, they are NOT gauge-
invariant. Consider the following operator

$\displaystyle\sqrt{-g}\left(\delta g^{00}(t)\right)^{2}{{}^{(3)}\\!R(t)}$
(126)

Naively if we just calculate the cubic $\zeta$ interactions coming from this
operator, one can easily see that we get a non-zero answer in the unitary
gauge, but zero in the flat gauge. Now, one can make this operator completely
gauge-invariant by introducing the Stueckelberg field, $\pi$. The gauge
invariant operator reads

$\displaystyle\sqrt{-g}\left(\frac{\partial(t+\pi)}{\partial
x^{\mu}}\frac{\partial(t+\pi)}{\partial
x^{\nu}}g^{\mu\nu}+1\right)^{2}{{}^{(3)}\\!R(t+\pi)}=\sqrt{-g}\left[(1+\dot{\pi})^{2}g^{00}+\partial_{i}\pi\partial_{j}\pi
g^{ij}\right.$ (127)
$\displaystyle\left.+2g^{0i}\partial_{i}\pi(1+\dot{\pi})+1\right]^{2}{{}^{(3)}\\!R(t+\pi)}$
(128)

Since we are interested in the cubic vertex, we need the expression for
${}^{(3)}\\!R$ only up to first order.

${}^{(3)}\\!R=\partial_{k}\Gamma_{ii}^{k}-\partial_{i}\Gamma_{ik}^{k}$ (129)
$\displaystyle=-a^{-2}\partial^{2}g_{ii}$ (130)

The operator becomes

$\displaystyle=-\left[(1+\dot{\pi})^{2}g^{00}+\partial_{i}\pi\partial_{j}\pi
g^{ij}+2g^{0i}\partial_{i}\pi(1+\dot{\pi})+1\right]^{2}e^{-2\rho}\partial^{2}g_{ii}(t+\pi)$
(131)

Let us evaluate this operator in the two gauges Maldacena_2003
Flat gauge: $\delta\phi\neq 0$

$\displaystyle=-6(\delta N|_{\psi=0}-\dot{\pi})^{2}H\partial^{2}\pi$ (132)
$\displaystyle=-6\left(\frac{\dot{\zeta}}{H}\right)^{2}\partial^{2}\zeta$
(133)

Co-moving gauge: $\delta\phi=0$

$\displaystyle=-6\frac{\dot{\zeta}^{2}}{H^{2}}\partial^{2}\zeta$ (134)

The vertex for $\zeta$ matches in the two gauges as expected. Since, this
operator just contains derivatives, at leading order, this gives a vanishing
contribution to the local bispectrum.

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