aainstitutetext: Indian Institute of Science Education and Research, Pune,
Indiabbinstitutetext: SISSA International School for Advanced Studies, Via
Bonomea 265, 34136, Trieste, Italy

# Probing the initial state of inflation: analytical structure of cosmological
correlators

Diptimoy Ghosh a,b,2 Amartya Harsh Singh a,3 Farman Ullah
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the analytic structure of in-in correlation functions in a deSitter
background. The aim of this study is to probe the initial conditions for
inflation through the features of correlation functions of the field
fluctuations, and understand precisely how an in-in correlator responds to
particles in the initial state. We emphasize that the choice of vacuua and the
corresponding particle interpretation for these fluctuations is flexible, and
we clarify the role of this choice at the level of calculations and their
diagrammatic interpretation. We consider several possibilities aside from the
standard Bunch Davies (BD) vacuum prescription for the initial state, and
trace the change in pole structure as one begins adding excitations; starting
from just a single particle, to highly excited states and special cases such
as a coherent state. We also go beyond the pole structure of the bispectrum,
and calculate the 4 point classical and quantum correlators. We illustrate -
with the example of coherent states - the subtleties in concluding a Bunch
Davies initial state from the absence of physical poles in the bispectrum,
which is interesting in light of some recent literature. Initial states with a
finite number of excitations are plagued with disconnected diagrams isolated
in phase space, and we highlight their implications on the observation of
these signals, and how the situation changes as one begins to excite more and
more particles. We also comment about the implications of various initial
conditions on the squeezed limit of the bispectrum. These new pole structures
are a direct consequence of mixing of positive and negative frequency modes
which is a characteristic of curved spacetimes; in particular, we see in
detail how particles in an initial state replicate mode mixing structures.
This study aims to clarify the missing details that link quantum and classical
initial conditions, and sharpen our understanding of in-in correlators in
inflation.

##  1 Introduction

It is now widely believed that the structure in our universe was seeded by
quantum mechanical fluctuations generated during an epoch of exponential
expansion known as InflationGuth:1980zm ; Achucarro:2022qrl . During this
period, these quantum fluctuations were stretched to super-horizon distances
with unchanging amplitudesMukhanov:1981xt which later gave rise to the rich
structure that we observe around us. The predictions of inflationary models
are consistent with observations but there remain some unanswered questions,
such as what were the initial conditions for inflation? In most models, the
perturbations start their life in a special kind of vacuum state known as the
Bunch-Davies vacuum but this assumption is not free of
contention(Agullo:2010ws ; Ganc:2011dy ; Holman:2007na ; Easther:2002xe ;
Brandenberger:2002hs ; Meerburg:2009ys ). The primary motivation for this is
that we know nothing about the (possibly Transplanckian) physics that preceded
inflation, and assuming any boundary condition in the far past is an
extrapolation of intuition to scales we don’t understand. It turns out that
the most general vacuum state possible is the general Bogolyubov transformed
Bunch-Davies state (Holman:2007na ; Ganc:2011dy ) and the most general vacuum
state invariant under de Sitter isometries is the so called $\alpha$
vacuumShukla:2016bnu ; PhysRevD.32.3136 . The power spectrum alone cannot
distinguish between these different choices of vacua and initial states.
Therefore, we turn to higher point functions e.g., BispectrumAcquaviva_2003 ;
Maldacena_2003 ; Weinberg_2005 ; Bartolo_2004 and their analytic structures to
look for distinguishing features. It is believed that perturbations that had a
classical origin can give rise to present observations at the level of the
power spectrum, without appealing to any quantum origin (Green:2020whw ;
Green:2022fwg ); see Agullo:2022ttg for interesting recent developments. The
definition of what pole structure one calls classical, however, is unclear and
it is among the purposes of this paper to clarify this point. The crucial
insight is to study the analytic structure of the correlation functions as a
function of the initial conditions, and we find that a simple definition of
classical - such as a state with a large number of particles - is not enough
to indicate the presence of a different pole structure than that of a true
quantum vacuum. Coherent states for a Bunch Davies observer which are clearly
excited - under certain assumptions - behave exactly like a Bunch Davies
vacuum at the level of correlators. Moreover, any other choice of initial
state aside from the Bunch Davies vacuum and it’s coherent states displays a
recurring presence of physical poles. Therefore, in a sense, the analytic
structure arising from a Bunch Davies vacuum or coherent state is very unique;
with everything else showing a common additional structure in the form of
physical poles. It is the purpose of this work to trace precisely the origin
of this difference, and study in detail the new features that in-in
correlators start displaying as one changes the nature of the initial state.

The different pole structures can ultimately be tied down to the presence or
absence of mixing of positive and negative frequency modes-in curved space,
these can naturally arise depending on the initial condition. When we think of
inflation as being driven by some Effective field theory, we need to
understand what features of the physics before inflation are important to
understand. New physics can manifest itself as either irrelevant
operators(consistent with slow-roll conditions) or as initial state
modifications (backreaction effects must be consistently included/dropped -
this is a subtle issue that we don’t explore here). At the level of the power
spectrum, given a cutoff $\Lambda$; initial state effects can add corrections
of the order $\frac{H}{\Lambda}$, much larger than the
$\frac{H^{2}}{\Lambda^{2}}$ effects coming from irrelevant operators
Kaloper:2002cs , and therefore initial state effects cannot be disregarded,
despite their somewhat more obscure origin(see Cabass:2022avo for a recent
review of the EFT of cosmology). The observed power spectrum then allows us to
infer bounds on the scale of new physics, $\Lambda$. More generally, they
constrain the number density of particles in the initial state by constraining
the Bogolyubov parameters, that, in this case, modify also the normalization
of the power spectrum. However, higher point correlation functions are more
sensitive probes(Holman:2007na ) for the simple reason that they are more
sensitive to interactions of particles in the initial state, and have always
been an object of interest either from a bootstrapping perspective or as a
tool to systematically trace signatures of new physics in these correlators
https://doi.org/10.48550/arxiv.2205.10340 ; Pajer:2020wxk ; Baumann:2015nta ;
Arkani-Hamed:2018kmz ; Arkani-Hamed:2015bza ; Pimentel:2022fsc ; Chen:2022vzh
; Baumann:2022jpr ; Flauger:2022hie ; see Benincasa:2022gtd for a recent view.
For this matter, we turn our attention to the bispectrum and the 4-point
function Seery_2007 ; Chen_2009 ; Senatore_2011 ; Chen:2006nt ;
Bonifacio:2021azc with various initial conditions, and observe how their
structure changes. We don’t make any pretence about the kind of physics before
inflation that leads to the choice of initial state. In particular, we
emphasize that the universe has no obligation to exist in the vacuum state of
the perturbations.

Correlation functions of the inflaton (more precisely, the scalar mode of the
ADM metric after removing gauge freedom) are an obvious tool to study the
physics of inflation, encoding both the nature of the interactions in the EFT
of inflation, as well as the initial conditions for inflation itself. These
correlators, on the other hand, must not be sensitive to whatever fundamental
physics drove inflation.We have to be careful about the nature of
singularities that are generated by different interactions, as well as the
need for an IR regulator. For instance, the difference between a $\zeta^{3}$
and a $\zeta^{\prime 3}$ interaction in an in-in calculation reflects in the
singularities they generate in a correlation function. The former generates
logarithmic singularities, whereas the latter generates simple poles. The log
divergences are contact terms in Fourier space, and their origin can be traced
to the fact that conformal symmetry constrains position space correlators such
that they are singular at coincident points. This relationship deserves a
detailed analysis of it’s own(Kehagias:2012pd ; Antoniadis:2011ib ), and we
don’t dwell on this subject in this work. We’ll instead work with a derivative
interaction where the poles are simple poles in momenta and no regulator is
required, and we see how the correlation functions with this specific
interaction respond to changes in initial conditions. Three point functions,
in particular, are especially interesting because they are a measure of the
‘non-gaussianity’, which is a direct measure of the existence of interactions
in the inflationary action. The important thing to notice is the word
’interaction’, and this is where the nature of deSitter geometry and
corresponding vacuua becomes important. In flat space, one typically has the
convenience to turn off all interactions in the far past and future, so that
the vacuum state at the asymptotic boundaries is free of excitations. In
curved backgrounds, in contrast, one doesn’t have this luxury, and in fact,
the notion of particles and excitations is not well defined. This has
consequences on the nature of correlation functions evaluated with different
initial conditions. For example, if the initial state is excited relative to,
say, the Bunch Davies observer, then it is possible in some cases to find an
observer which sees no particles. These cases are precisely those which are
related via a Bugolyubov transform. The interpretation of correlators as
Feynman diagrams is therefore very different - the diagrams for a Bunch Davies
observer contains disconnected pieces because there are more particles in the
initial state than those that can interact at a vertex-these will then have
support only at localised points in phase space. The diagrams for an alpha-
observer however are fully connected as only the field insertions interact at
the vertex.

In either case, we see the presence of physical poles where the singularities
can be reached in the physical region. It is tempting to relate this to an
interpretation in terms of particle scattering and decays in the initial
state, but as we remarked, this picture is observer dependent. Clearly, the
situation is more complicated than a relative particle density between the
vacuum state of the modes and the initial state for inflation. An initial
state which is an alpha vacuum at the start of inflation produce a similar
structure to the one mentioned above, but here it is clear that there is no
particle self interaction interpretation that can be ascribed. If one insists
to work with an observer who sees particles, then it is true that an analogy
can be drawn with flat space scattering processes, but one must then deal with
the fact that the phase space where these poles have support is restricted.
Therefore, it isn’t straightforward to trace various pole structures to the
absence/presence of particles in the initial state. One has the same problem
in flat space-LSZ only gives us information about fully connected diagrams
that have the right pole structure, but the S-matrix contains contributions
from both connected and disconnected diagrams in a highly excited state.

For example, if we had a cubic vertex $\phi^{3}$ and we computed the leading
order contribution to

$\int d^{4}x\langle n_{out}|\phi^{3}(x)|n_{in}\rangle\supset\int d^{4}x\langle
n_{k_{1}}+1,n_{k_{2}}+1,n_{k_{3}}-1,..|n_{in}\rangle\sim\delta(k_{1}-k_{2}-k_{3})A_{1\to
2}$ (1)

where $n\equiv\otimes_{k}(a^{\dagger}_{k})^{n}|0\rangle/\sqrt{n!}$. Here,
$A_{1,2}$ is now a disconnected amplitude with $k_{1},k_{2},k_{3}$ interacting
at a vertex, while the rest of the $n_{k}$’s spectate. LSZ kills this diagram,
and therefore it doesn’t contribute to the fully connected part of the
S-matrix; it exists nonetheless and since with inflation we don’t have the
luxury of an S-matrix formalism, we must make our peace with such processes.

We are typically interested only in the fully connected part of the S-matrix,
but such a requirement cannot be imposed when the initial states are highly
occupied. Going to higher orders in perturbation theory partially remedies
this situation. For instance, consider a $3\to 2$ process in flat space with a
cubic vertex. The Feynman diagrams for the first two orders in perturbation
theory are shown in figure 1. We see that going to higher orders in
perturbation theory is one way to generate connected diagrams. However, this
problem only worsens as we include more and more particles in the external
states, because then the fully connected diagrams get harder and harder to
measure. In curved space in-in correlators, as we shall see, the recipe to
circumvent these is to switch to a frame where there are no particles and the
state is the vacuum with respect to the new modes. We will then be left with
only fully connected diagrams, whose pole structure can be analysed. Generic
excited states display physical poles, but their signatures are clouded by the
restricted points in phase space where they have support. We will see that
these physical poles occur when particles in the initial state interact at a
vertex along with some of the field insertions, while the remaining field
insertion is a spectator and therefore has support only when it’s momentum
coincides with one of the momenta of the spectator particles in the initial
state. If we have all possible momenta in such a state(what one would expect
in a ’classical’ state), then atleast one of these physical poles will be
observable, but at an isolated point in phase space for a given momentum.
Scanning all possible momenta for the field insertions will then generate a
measurable physical pole. In the case where the initial state is a Bugolyubov
transform of the Bunch Davies vacuum, the physical poles are seen
transparently.

Interestingly, there is a class of excited states over the Bunch Davies
vacuum-the so called coherent state $|C\rangle$ Kundu_2012

$a_{\vec{k}}|C\rangle=C(\vec{k})|C\rangle$ (2)

which, under certain assumptions, reproduces the pole structure for the vacuum
state i.e. it displays no physical poles despite being an excited state. This
is in direct contradiction with the supposed association of physical poles
with excited states. If instead we had a coherent state that was built over
not the Bunch Davies vacuum, but that of its Bugolyubov transform, then
physical poles are restored.

Figure 1: (Left) Disconnected diagram at order $\lambda$. (Right) Fully
connected at order $\lambda^{3}$

Let us elaborate on the role of disconnected pieces. Once the order in
perturbation theory is fixed, we must allow the possibility of other particles
to spectate the interaction and contribute as disconnected pieces. LSZ will
have nothing to say about this, and therefore the flat space intuition for
physical poles in inflation is premature. This manuscript is devoted to
studying in detail the origin of pole structures in in-in correlators and the
information it carries about the initial state of inflation, and probing the
relationship between a highly excited quantum state versus a ’classical’
state.

The broad outline of this manuscript is as follows. In section 3, we will
analyse in-in 3-point correlators for a variety of initial states, including
the anomalous case of coherent states. In addition, subsection 3.7 is devoted
to checking explicitly, by computing the 4-point classical and quantum in-in
correlation function, that going to higher orders in perturbation theory
doesn’t change the basic inferences we draw from the bispectrum about the
singularity structure, extending the result of Green:2020whw . In section 4.1,
we review the features of LSZ in flat space, and the corresponding intuition
to be drawn for in-in correlators. Next, in 5 we study the role of derivative
interactions that require no IR regulator in generating singularities that are
simple poles. In section 6 we discuss the squeezed limit of the bispectrum.
Finally, in section 7 we clarify the link between physical poles in classical
correlators versus those in quantum in-in correlators.

##  2 From flat space to curved space

###  2.1 Background

Provided with the field content, background geometry, and symmetries of the
system; one can write an action functional that describes the theory. The
equations of motion following from this action are solved in terms of mode
functions $u_{k}(t,\vec{x})$ that multiply some coefficients $a_{k}$, and
their hermitian conjugates. In time independent, flat backgrounds we have
plane wave solutions for the mode functions of free fields;
$u_{k}(t,\vec{x})=\frac{1}{\sqrt{2\omega_{k}}}e^{i\vec{k}{x}-\omega t}$. We
promote $a_{k},a_{k}^{\dagger}$ to operators that satisfy canonical
commutation rules that then get imparted to the fields themselves, and we say
we have canonically quantized the theory. Studying this theory then amounts to
studying the corresponding Hilbert space, and this discussion begins by first
identifying a vacuum state from which other states are realised, defined as
$a_{k}|0\rangle=0$. In flat space, it is uniquely fixed by demanding
invariance under the underlying Poincare group. This means that the mode
functions and the quantization prescription of the theory is unique. This is
traced to the fact that $u(k)$ is Lorentz invariant, and this is where the
situation drastically changes in more general backgrounds.

If we allow for diffeomorphism invariance-a much larger group-then the $x$ and
$t$ dependence of the mode functions can change since
($\vec{k}\cdot\vec{x}-\omega t$) need not be invariant anymore. In fact, plane
waves simply needn’t be solutions to equations of motion in general
backgrounds. Moreover, in time dependent backgrounds-as is the case with
inflation-$\omega$ has a time dependence, which reflects as an additional time
dependence in the mode functions. Therefore, mode functions are in general not
unique. Being solutions to second order equations of motion, they require two
additional inputs to be fixed. Consequently, different mode expansions are
distinguishable in the sense that they differ, for example, in their
measurement of particle density in a given state. Note that this difference
lies in their different basis of modes (and consequently creation/annihilation
operators), but the definition of a field itself ofcourse is invariant under
these transformations. Again, one has to keep in mind that this phenomenon
occurs simply by allowing accelerated coordinate frames, even in flat space,
and are therefore tied to the definition of an inertial observer. In special
relativity these are restricted to those obtained from Poincare
transformations on a Minkowsky background; in general relativity we allow for
the more general case of whatever isometry preserves the background geometry.

Unlike flat-space where demanding the vacuum to be invariant under the
isometries of the background metric uniquely fixes the vacuum state, in de
Sitter the invariance under isometries does not completely fix the vacuum
statePhysRevD.32.3136 . Therefore, there is a class of vacua known as alpha
vacua which are invariant under de Sitter isometries and parametrized by
$\alpha$PhysRevD.32.3136 ; Shukla:2016bnu . The mode function corresonding to
$\alpha$ vacua are given by

$\displaystyle
u_{k}(\tau)=\frac{\Delta_{\zeta}}{\sqrt{k^{3}}}\left(\alpha(1-ik\tau)e^{ik\tau}+\beta(1+ik\tau)e^{-ik\tau}\right)$
(3)

where,

$|\alpha|^{2}-|\beta|^{2}=1$ (4)

(3) is the solution of Mukhanov-Sassaki equation

$\zeta_{k}^{\prime\prime}-\frac{2}{\tau}\zeta_{k}^{\prime}+k^{2}\zeta_{k}=0$
(5)

If we call the $\alpha$ modes $v_{k}$ and the Bunch Davies modes by
$u_{k}=(1-ik\tau)e^{ik\tau}$, then we can compare the associated initial
conditions through

$\displaystyle\dot{v_{k}}(\tau_{0})=-i\omega^{\prime}_{k}v_{k}(\tau),\dot{u}_{k}(\tau_{0})=-i\omega_{k}u_{k}(\tau)\implies\frac{\alpha_{k}}{\beta_{k}}=\frac{\omega^{\prime}_{k}-\omega_{k}}{\omega^{\prime}_{k}+\omega_{k}}$
(6)

The Bunch Davies vacuum ($\alpha=1$, $\beta=0$) corresponds to one particular
choice of $\alpha$ vacua. This choice of vacuum is usually considered the most
natural one since the modes that we are interested in were far inside the
horizon at the beginning of inflation ($\tau\rightarrow-\infty$ limit )
therefore these modes do not "feel" any de-Sitter curvature and can therefore
can be treated as Minkowski modes. But as mentioned above this is not a
settled issue. It is important to note that no co-moving/free falling particle
detector will register any particles in a particular $\alpha$ vacuum since all
these observers are related by de Sitter isometries and these vacua are
invariant under them.

###  2.2 What we learn

The purpose and findings of this present work can be summarized as follows-

  * •

Is there a unique pole structure for all in-in correlators? What distinguishes
the physics leading to different singularities? No, the pole’s functional form
depends on the choice of interaction. The physical origin we aim to see lies
not in the functional form of the singularity but the momentum configuration
for which it occurs. See 5 for a discussion.

  * •

Are total energy poles a characteristic of only the Bunch Davies vacuum intial
state with no excitations? Do physical poles arise only if the initial state
was an excited state? What is true is that most excited states relative to a
Bunch Davies vacuum display physical poles. However, a class of coherent
states displays the same pole structure as a state with no Bunch Davies
particles, and this is an important anomaly that we wish to highlight. This
result is derived in 31.

  * •

Do initial states with a fixed number of Bunch Davies particles display
observable physical poles?Is it necessary for each mode to have a high
occupation number in the initial state to replicate the structure of a
classical correlator? With a finite number of particles, the physical poles
are isolated in phase space. One needs to excite all possible momenta to
guarantee an observable physical pole. The important ingredient is to not miss
out on regions in phase space, and to excite every mode atleast once. This
state however has a large number of particles anyway, as one would expect for
a classical state See 3.5 and eq. 3.5.2,38,39 to understand the support for
these new poles.

  * •

If the initial state is a vacuum state, then do we not see physical poles? The
notion of vacuum state is not unique, and for any other vacuum other than
Bunch Davies, one sees physical poles despite there being an observer who sees
no particles in that state. See 3.3for discussion.

  * •

What is the underlying reason for similar mathematical structures of classical
correlators and those from excited quantum states? The classical Green’s
function has both signs of modes. If an excited quantum state is the vacuum
for an observer related via a Bogolyubov transform to the Bunch Davies vacuum,
then in the picture where we work in the frame of this observer who sees a
vacuum, the associated Wightman functions have both signs of modes too. If one
instead continues to use the Bunch Davies picture, then the physical poles
arise because of initial state particles interacting at a vertex with the
field insertions produce modes with a relative sign between momenta. See 7 for
details.

##  3 In-In Correlation functions and associated analytic structures

In this section, we will report case by case the analytic structure for
correlators corresponding to various choices of initial states and mode
functions(i.e. the choice of vacuum). We use the In-In formalism (see Appendix
B) to compute quantum correlators. While our primary interest will be the 3
point function, we also write down the power spectrum for completeness. We
will also look at the structure obtained with a coherent initial state and
contrast it with a general excited initial state. Finally, we will look at the
4 point function calculated using both the in-in formalism and a classical
perturbation theory formalism, and check if our intuition after studying 3
point functions holds true there too.

The results in this section will be organized in the following manner. We will
first fix the mode functions i.e. the vacuum state of the perturbations to be
either BD or alpha-modes, and then compute the correlation functions with the
following choices of initial states-

  * •

The initial state coincides with the vacuum state

  * •

The initial state is a one particle state of definite momentum

  * •

The initial state is an excited state with all modes occupied

  * •

The initial state is a highly excited state which is a Bogolyuobov transform
of Bunch Davies.

  * •

The initial state is a coherent state for either a Bunch Davies or a
Bogolyubov transformed observer

The results, to be detailed in the following sections, are summarized in the
following table-

Initial state | Pole structure
---|---
BD vacuum |  $k_{t}$ pole
Excited over BD with finite modes present | disconnected physical poles, connected $k_{t}$ pole
Excited over BD with all modes present | connected $k_{t}$ pole, observable physical pole
Excited over BD but a BG | both physical and $k_{t}$ poles
Excited over BD but coherent |  $k_{t}$ pole
Excited over BD but coherent for some BG | both $k_{t}$ and physical poles
Classical | both $k_{t}$ and physical poles

###  3.1 Normalization of the expectation value

We must recognize that computing the expectation value for any state requires
that we divide also by the norm of that state, which needn’t always be one.
For example, with a $|p\rangle$ state the norm is proportional to
$\delta^{(3)}(\vec{p}-\vec{p})$ which is formally divergent. However, a
correlation function of the form $\langle
p|\zeta_{1}\zeta_{2}\zeta_{3}|p\rangle$ will also contain factors of $\langle
p|p\rangle$ when we contract the two initial states together(nothing prevents
us from doing it at the level of correlation functions, this is just a
disconnected diagram). Therefore, after dividing by the normalization, we will
see that this disconnected piece’s contribution vanishes and we’re left with a
finite term whose pole structure can be read off. All other terms will be
suppressed by this (infinite) normalization factor, except when they have
localised support in phase space through other delta function singularities,
in which case they can peak at those points in phase space, but those points
only. If we take these correlators to represent NG, then these isolated pole
structures(and therefore the corresponding initial states) are ruled out by
observation. It is important to note that $\delta^{(3)}(0)$, wherever it
occurs, will be regulated by a volume $V$, which we will see ultimately drops
out of the correlator. Also, the factor of $2E_{p}$ that relativistically
normalize a $|p\rangle$ drop out. In the below correlators, therefore, such a
normalization is always implicit, even if not explicitly written.

###  3.2 BD vacuum initial state

We review the results from the most common choice of initial state.

####  3.2.1 Two point function

At $O(\lambda^{0})$ the two-point function is given by

$\bra{0}\zeta_{k_{1}}\zeta_{k_{2}}\ket{0}=\frac{\Delta^{2}_{\zeta}}{k_{1}^{3}}\delta^{3}(\vec{k_{1}}+\vec{k_{2}})$
(7)

####  3.2.2 Three point function

To calculate

$\displaystyle\bra{0}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\ket{0}$ (8)

At $O(\lambda^{0})$ the three point function vanishes so we compute at
$O(\lambda)$Green:2020whw

$\displaystyle\bra{0}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\ket{0}=\frac{4H^{-1}\Delta^{6}_{\zeta}\lambda}{(k_{1}+k_{2}+k_{3})^{3}k_{1}k_{2}k_{3}}\delta^{3}(\sum\vec{k_{i}})$
(9)

###  3.3 Excited initial state from Bogolyubov transform:$\alpha$-vacuum

In this section, we consider initial states which are excited relative to a
Bunch Davies observer, but can also be reached by a Bogolyuobov transform of
the Bunch Davies vacuum. That is, we consider excitations that a different
observer would see when observing a Bunch Davies vacuum. What this means is
that there exists an observer for which such a state is actually the vacuum,
while it has particles for a Bunch Davies observer. From a calculation
standpoint, this is a massive simplification, because vacuum states are
simpler to deal with than excited states. For concreteness, let us study an
initial state that would correspond to an $\alpha$ vacuum, which is also de
Sitter invariant and can be reached from the Bunch Davies initial state by a
simple Bogolyuobov transform.

Given the Bunch Davies mode functions $u_{k}(\tau)$ and it’s set of creation
and annihilation operators $a_{k},a^{\dagger}_{k}$, the Bogolyuobov creation
annihilation operators are $b_{k}=\beta^{*}a_{k}-\alpha^{*}a^{\dagger}_{-k}$

where the $\alpha,\beta$ are parametrized by a single parameter(it’s part of
the 2 parameter family of dS invariant vacua, and we set one of the parameters
to zero) $\tilde{\alpha}$;
${\beta,\alpha}={\cosh(\tilde{\alpha}),-i\sinh(\tilde{\alpha})}$. The $\alpha$
vacuum can be expressed in terms of the Bunch Davies expression through
(Shukla:2016bnu )

$|\alpha\rangle=\frac{1}{N}\exp\bigg(\frac{\alpha^{*}}{2\beta^{*}}\int\frac{d^{3}k}{(2\pi)^{3}}a^{\dagger}_{k}a^{\dagger}_{-k}\bigg{missing})|0\rangle$
(10)

Where $N$ is an overall normalization constant. We see straightaway that it
contains Bunch Davies excitations, but they come in pairs of equal and
opposite momenta and therefore the state contains no net momentum. One would
therefore hope that any disconnected pieces in the three point function will
only have support at zero momentum for some of the external momenta(since
there is no nonzero spectator particle momentum that requires to be conserved
along with them)-an uninteresting case. Let us explicitly see how the
calculation of the three point function goes through. We are interested in the
object

$\langle\alpha|\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}|\alpha\rangle=\frac{1}{N^{2}}\bigg{(}\langle
0|\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}|0\rangle+\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\alpha^{*}}{2\beta^{*}}\langle
0|\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}a^{\dagger}_{p}a^{\dagger}_{-p}|0\rangle\\\
\\\ +\int\frac{d^{3}p}{(2\pi)^{3}}\frac{\alpha}{2\beta}\langle
0|a_{-p}a_{p}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}|0\rangle+\frac{|\alpha|^{2}}{4|\beta|^{2}}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{d^{3}q}{(2\pi)^{3}}\langle
0|a_{-p}a_{p}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}a^{\dagger}_{q}a^{\dagger}_{-q}|0\rangle+...\bigg{)}$
(11)

Let us name these terms $I,II,III,IV$ so that the correlator on the left is
their sum. The terms can be arranged in powers of $\frac{\alpha}{\beta}$, and
we therefore consider represenative examples from each power. To start with,
we see that at zeroth power i.e. the term $I$, the bispectrum is the same as
that for Bunch Davies vacuum, and we have a total $k$ pole only with no
disconnected subgraphs. Let us move on to terms linear in $\alpha/\beta$;
consider $III$ at first order in perturbation theory-

$\int\frac{d^{3}P}{(2\pi)^{3}}\langle
0|a_{-P}a_{P}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}|0\rangle=2\textrm{Im}\prod_{i=1}^{3}\int\frac{d^{3}p_{i}}{(2\pi)^{3}}e^{i\vec{p}_{i}\cdot\vec{x}}\int\frac{d\tau}{\tau}\int\frac{d^{3}P}{(2\pi)^{3}}\\\
\\\ \times\langle
0|a_{-P}a_{P}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\zeta^{\prime}_{p_{1}}(\tau)\zeta^{\prime}_{p_{2}}(\tau)\zeta^{\prime}_{p_{3}}(\tau)|0\rangle$
(12)

$=2\textrm{Im}\prod_{i=1}^{3}\int\frac{d^{3}p_{i}}{(2\pi)^{3}}e^{i\vec{p}_{i}\cdot\vec{x}}\int\frac{d\tau}{\tau}\int\frac{d^{3}P}{(2\pi)^{3}}\langle
a_{P}\zeta_{k_{1}}\rangle\langle
a_{-P}\zeta^{\prime}_{p_{1}}\rangle\langle\zeta_{k_{2}}\zeta^{\prime}_{p_{2}}\rangle\langle\zeta_{k_{3}}\zeta^{\prime}_{p_{3}}\rangle+\textrm{perm.}\\\
\\\
=\int\delta(\vec{P}+\vec{k_{1}})\delta(\vec{P}-\vec{p_{1}})\delta(\vec{k_{2}}+\vec{p_{2}})\delta(\vec{k_{3}}+\vec{p_{3}})(u^{*}_{k_{1}}(0)u^{\prime*}_{p_{1}}(\tau))(u_{k_{2}}(0)u^{\prime*}_{p_{2}}(\tau))(u_{k_{3}}(0)u^{*}_{p_{3}}(\tau))\\\
\\\
=\delta(\vec{k}_{t})2\textrm{Im}\int\frac{d\tau}{\tau}\prod_{i=1}^{3}\big{(}u_{k_{i}}(0)u^{\prime*}_{k_{i}}(\tau)\big{)}$
(13)

Thus we find, upto permutations(we have used that $u_{k_{1}}(0)$ is purely
real in the first paranthesis)

$III=\frac{H^{6}}{8k_{1}k_{2}k_{3}}2\textrm{Im}\int_{-\infty}^{0}d\tau\tau^{2}e^{i(k_{1}+k_{2}+k_{3})\tau}=\frac{H^{6}}{2k_{1}k_{2}k_{3}}\frac{1}{(k_{1}+k_{2}+k_{3})^{3}}$
(14)

We see that we only have a total energy pole. Figure 2 explains
diagrammatically the situation.

Figure 2: (Left) the initial state particles are spectators. (Right) the
initial state particles interact at a vertex. There is a sum over all possible
$p$ values.

Let’s check if this changes at the next order in $(\alpha/\beta)$. Take $IV$-

$\int\frac{d^{3}P}{(2\pi)^{3}}\frac{d^{3}Q}{(2\pi)^{3}}\langle
0|a_{-P}a_{P}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}a^{\dagger}_{Q}a^{\dagger}_{-Q}|0\rangle=2\textrm{Im}\prod_{i=1}^{3}\int\frac{d^{3}p_{i}}{(2\pi)^{3}}e^{i\vec{p}_{i}\cdot\vec{x}}\int\frac{d\tau}{\tau}\int\frac{d^{3}P}{(2\pi)^{3}}\\\
\\\ \times\langle
0|a_{-P}a_{P}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\zeta^{\prime}_{p_{1}}(\tau)\zeta^{\prime}_{p_{2}}(\tau)\zeta^{\prime}_{p_{3}}(\tau)a^{\dagger}_{Q}a^{\dagger}_{-Q}|0\rangle$
(15)

This will obviously include disconnected pieces because there are more terms
to contract than there are external points. For example, consider the
contraction

$IV\supset\int\langle a_{P}\zeta_{k_{1}}\rangle\langle
a_{-P}\zeta_{k_{2}}\rangle\langle\zeta^{\prime}_{p_{1}}a^{\dagger}_{Q}\rangle\langle\zeta^{\prime}_{p_{2}}a^{\dagger}_{-Q}\rangle\langle\zeta_{k_{3}}\zeta^{\prime}_{p_{3}}\rangle\propto\delta({\vec{k}_{3}})\delta(\vec{k}_{2}+\vec{k}_{1})$
(16)

which is disconnected and has support only when one of the momenta is zero.

Consider instead the contraction

$\langle
a_{P}\zeta_{k_{1}}\rangle\langle\zeta_{k_{2}}\zeta^{\prime}_{p_{2}}\rangle\langle\zeta_{k_{3}}\zeta^{\prime}_{p_{3}}\rangle\langle\zeta^{\prime}_{p_{1}}a^{\dagger}_{-Q}\rangle\langle
a_{-P}a^{\dagger}_{Q}\rangle$

This yields a piece proportional to the integral of
$\delta(\vec{k}_{t})e^{i(-k_{1}+k_{2}+k_{3})\tau}$, which is fully connected
but has a physical pole at $k_{1}=k_{2}+k_{3}$. On the other hand, consider
the contraction

$\langle\zeta_{k_{1}}a^{\dagger}_{-Q}\rangle\langle\zeta_{k_{2}}\zeta^{\prime}_{p_{2}}\rangle\langle\zeta_{k_{3}}\zeta^{\prime}_{p_{3}}\rangle\langle
a_{P}\zeta^{\prime}_{p_{1}}\rangle\langle a_{-P}a^{\dagger}_{Q}\rangle$

This piece yields again a connected result(proportional to
$\delta(\vec{k}_{t})$), but this time with a total energy pole at
$(k_{1}+k_{2}+k_{3})$.

Thus, with this way of calculating things, we see we that we obtain both total
and physical energy poles, but their physical meaning is obscured because of
the fact that some of these might be disconnected. This diagrammatic
interpretation in terms of connected/disconnected diagrams is obviously a
handicap because of the unclear particle interpretation in curved spaces-in
particular, we could use a different basis to expand the $\zeta$s and the
resulting calculations would be a mess. One would hope that all such
calculations are ultimately identical, but it is far from obvious at this
stage.

But it is precisely this freedom of choosing a particle interpretation that
rescues us. If the initial state is chosen to be among a family of Bogolyubov
transforms of the Bunch Davies states, then there is a natural choice of the
mode functions where the action of the fields on the state is simple. For
example, if the initial state is an $\alpha$ vacuum, it would be natural to
use the mode expansion with $\alpha$ modes. This approach should be free of
any artificial disconnected diagrams, because the initial state is now a
vacuum state for an observer using this basis. Note that the field itself is
fully invariant-observers are distinguished by their choice of mode functions
and accompanying creation operators. Using the freedom of being able to choose
any frame, we use the frame where the initial state is a vacuum for the choice
of observer, and then we are left with fully connected pieces. With this
approach, we can expand our field using the $\alpha$ mode functions
$v_{k}=\alpha u_{k}^{*}e^{ik\tau}+\beta u_{k}e^{-ik\tau}$ with corresponding
modifications to the creation operators via the Bugolyubov transform, and the
calculation is extremely simple because the action of these on the external
states is simple.

With this choice, we find the new mode functions, which we now call $u_{k}$

$u_{k}(\tau)=\alpha(1-ik\tau)e^{ik\tau}+\beta(1+ik\tau)e^{-ik\tau}$ (17)

where, $\alpha$, $\beta$ are complex constants. These vacua parametrized by
$\alpha$ and $\beta$ are invariant under de-Sitter isometriesPhysRevD.32.3136
. In other words, all co-moving/freefalling observers will agree on the choice
of $\alpha$, $\beta$.

The free two point ($O(\lambda^{0})$) function is given by

$\displaystyle\langle\zeta_{k_{1}}\zeta_{k_{2}}\rangle=A\frac{\Delta^{2}_{\zeta}}{k_{1}^{3}}\delta^{3}(\vec{k_{1}}+\vec{k_{2}})$
(18)

where, $A=|\alpha+\beta|^{2}$

We have at $O(\lambda)$ (all non-zero contractions are equal)

$\begin{split}=\textit{Re}\left(-2i\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})(\alpha+\beta)^{3}\frac{\Delta^{6}_{\zeta}H^{-1}}{k_{1}k_{2}k_{3}}\int_{-\infty}^{0}d\tau^{\prime}\tau^{\prime
2}\left(\alpha^{*}e^{-ik_{1}\tau^{\prime}}+\beta^{*}e^{ik_{1}\tau^{\prime}}\right)\right.\\\
\times\left(\alpha^{*}e^{-ik_{2}\tau^{\prime}}+\beta^{*}e^{ik_{2}\tau^{\prime}}\right)\left(\alpha^{*}e^{-ik_{3}\tau^{\prime}}+\beta^{*}e^{ik_{3}\tau^{\prime}}\right)\end{split}$
(19)
$\displaystyle\begin{split}=4\lambda\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})(\alpha+\beta)^{3}\frac{\Delta^{6}_{\zeta}H^{-1}}{2k_{1}k_{2}k_{3}}\left[\left(\beta^{*\,2}\alpha^{*}-\alpha^{*\,2}\beta^{*}\right)\left(\frac{1}{(k_{1}+k_{2}-k_{3})^{3}}+\frac{1}{(k_{1}-k_{2}-k_{3})^{3}}\right.\right.\\\
\left.\left.+\frac{1}{(-k_{1}+k_{2}-k_{3})^{3}}\right)+\left(\beta^{*3}-\alpha^{*\,3}\right)\frac{1}{(k_{1}+k_{2}+k_{3})^{3}}\right]+\textit{c.c}\end{split}$
(20)

In the limit where $\beta=0$ i.e we are back to Bunch-Davies vacuum, the
physical poles vanish therefore everything is consistent. Notice that there
are by definition no $\alpha$ particles in $\alpha$ vacuum but still the
three-point function exhibits physical poles. Therefore, the particle
interpretation for physical poles Green:2020whw does not hold in this case. It
seems that the particle interpretation is only clear when the observer is
using a Bunch-Davies basis.

###  3.4 Coherent States

There is yet another special class of excited states-the so called coherent
states which are eigenstates for annihilation operators. Since this is an
observer dependent statement in time dependent backgrounds, it is interesting
to ask what poles these will generate, and will the answer change if they are
defined with respect to different observers. In fact, as coherent states in
quantum mechanics are known to show classical behaviour( see appendix D for a
review), their pole structure will be extremely instructive to sharpen our
understanding of classical vs quantum initial states. It turns out that
coherent states for a Bunch Davies observer with certain assumptions can have
absence of physical poles Kundu_2012 (in contrast with Green:2020whw ). We do
the same calculation with a $\frac{\lambda}{3!}\dot{\zeta^{3}}$.This is an
outlier case which goes against currently accepted correspondence between
initial state and pole structure.
The coherent states are simply eigenstates of the lowering operator

$a_{\vec{k}}\ket{C}=C(\vec{k})\ket{C}$ (21)

The calculation of the Bispectrum is done again using a
$\frac{\lambda}{3!}\dot{\zeta}^{3}$ with one additional assumption i.e

$C(\vec{k})+C^{*}(-\vec{k})=0$ (22)

Which is equivalent to setting the one point function to zero at $\tau=0$,
that is $\langle C|\zeta(0)|C\rangle=0$. This can always be done at one point
in time, and in quantum mechanics it just reflects the choice of phase of the
eigenvalue of the coherent state. We need to compute at $O(\lambda)$

$\bra{C}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\ket{C}$ (23)

We can compute it in the following way. By wick’s theorem

$\zeta\zeta\zeta....\zeta=:\zeta\zeta\zeta....\zeta:+:\textit{all possible
contractions}:$ (24)

Since, now we are not computing in the vacuum state therefore all terms on the
RHS of (24) contribute but the condition (22) makes everything vanish except
the term where all fields are contracted and we get the usual BD answer. A
simple proof is as follows. In every term that is not fully contracted, there
exists, say, an uncontracted $a^{\dagger}$. There must also be a term which is
different from this only by the replacement $a^{\dagger}\to a$. Adding these
terms gives a result proportional to $C(\vec{k})+C^{*}(-\vec{k})$, which
vanishes by assumption. To illustrate it further, consider a term with just
two contractions

$\displaystyle\wick{\sim\lambda\bra{C}:\chi 1\zeta_{k_{1}}\chi
2\zeta_{k_{2}}\zeta_{k_{3}}\chi 1\zeta^{\prime}_{q_{1}}(\tau)\chi
2\zeta^{\prime}_{q_{2}}(\tau)\zeta^{\prime}_{q_{3}}(\tau):\ket{C}}$ (25)

where, we have omitted the overall integral and the overall delta function.
This simplifies to

$\displaystyle\begin{split}\wick{\sim\lambda\chi 1\zeta_{k_{1}}\chi
1\zeta^{\prime}_{q_{1}}(\tau)\chi 2\zeta_{k_{2}}\chi
2\zeta^{\prime}_{q_{2}}(\tau)}\bra{C}a_{k_{3}}^{\dagger}a_{q_{3}}^{\dagger}u_{k_{3}}u^{\prime}_{q_{3}}(\tau)+a_{k_{3}}^{\dagger}a_{-q_{3}}u_{k_{3}}u^{\prime*}_{q_{3}}(\tau)+a_{q_{3}}^{\dagger}a_{-k_{3}}u^{*}_{k_{3}}u^{\prime}_{q_{3}}(\tau)\\\
+a_{-k_{3}}a_{-q_{3}}u^{*}_{k_{3}}u^{\prime*}_{q_{3}}(\tau)\ket{C}\end{split}$
(26)

where, $u_{k}$’s are the de Sitter mode functions at $\tau=0$. Using the fact
that $a_{k}\ket{C}=C(k)\ket{C}$, we get

$\displaystyle\begin{split}\wick{\sim\lambda\chi 1\zeta_{k_{1}}\chi
1\zeta^{\prime}_{q_{1}}(\tau)\chi 2\zeta_{k_{2}}\chi
2\zeta^{\prime}_{q_{2}}(\tau)}\bra{C}u^{\prime}_{q_{3}}\left(C^{*}(\vec{k_{3}})+C(-\vec{k_{3}})\right)C^{*}(\vec{q_{3}})+u^{*}_{q_{3}}\left(C^{*}(\vec{k_{3}})\right.\\\
\left.+C(-\vec{k_{3}})\right)C(-\vec{q_{3}})\ket{C}\end{split}$ (27)

Finally, using $C^{*}(-\vec{k})+C(\vec{k})=0$ one can easily check that the
expression vanishes. In this way all partially contracted terms can be shown
to vanish. Schematically, representing uncontracted operators in red, terms
can be pairwise combined and cancelled as follows

$\langle
C|a^{\dagger}a^{\dagger}..{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a^{\dagger}_{K}}..aaa|C\rangle+\langle
C|a^{\dagger}a^{\dagger}..{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{K}}..aaa|C\rangle\propto(C(K)+C^{*}(-K))=0$
(28)

thereby leaving behind only the fully contracted piece,

$\displaystyle\begin{split}\text{Re}-2i\int\left(\prod_{i}d^{3}\vec{q}_{i}\right)\delta^{3}(\vec{k_{1}}+\vec{q_{1}})\delta^{3}(\vec{k_{2}}+\vec{q_{2}})\delta^{3}(\vec{k_{3}}+\vec{q_{3}})\delta^{3}(\vec{q_{1}}+\vec{q_{2}}+\vec{q_{3}})\\\
\frac{q^{2}_{1}q^{2}_{2}q^{2}_{3}\Delta^{6}_{\zeta}H^{-1}\lambda}{\sqrt{(k_{1}k_{2}k_{3}q_{1}q_{2}q_{3})^{3}}}\int_{-\infty}^{0}d\tau^{\prime}\tau^{\prime
2}e^{i(q_{1}+q_{2}+q_{3})\tau^{\prime}}\end{split}$ (29)
$=\text{Re}-2i\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})\frac{\Delta^{6}_{\zeta}H^{-1}\lambda}{k_{1}k_{2}k_{3}}\int_{-\infty}^{0}d\tau^{\prime}\tau^{\prime
2}e^{i(k_{1}+k_{2}+k_{3})\tau^{\prime}}$ (30)

which then yields a result identical to the Bunch Davies case upto the
normalization $\langle C|\rangle$, which drops out of the overall expectation
value. Thus,

$\bra{C}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\ket{C}=\frac{4H^{-1}\Delta^{6}_{\zeta}\lambda}{(k_{1}+k_{2}+k_{3})^{3}k_{1}k_{2}k_{3}}\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})$
(31)

This results seems to contradict the conclusion of Green:2020whw that a total
energy pole will uniquely single out BD Vacuum as the initial state. The above
class of coherent states, as seen above, also shares the same feature. One can
think of the physical poles vanishing precisely because of the normal ordered
term vanishing. As we saw in earlier examples, physical poles occur when one
two the field insertions interact with an external state at a vertex, and the
third spectates. There are 2 ways to contract with the external state, either
with the left or with the right. With a coherent state, each is accompanied
with a factor of either $C(k)$ or $C^{*}(-k)$, and their sum vanishes because
of our assumption. In fact, note that the vacuum state is merely a trivial
special case of a coherent state with $C(\vec{k})=0$. Therefore, one can
invert the train of thought and say that the vacuum state must have the same
pole structure as that of a coherent state (with vanishing one point
function), which implies a total energy pole for Bunch-Davies vacuum.

Since the coherent state’s definition is observer dependent-more precisely,
dependent on the basis of annihilation/ creation operators used; one can ask
what happens if the initial state is coherent not for a Bunch Davies observer
but for an observer that is related by a Bogolyubov transform, say the
$\alpha$-vacuum. We therefore now consider

$b_{\vec{k}}|C\rangle_{\alpha}=C(\vec{k})|C\rangle_{\alpha}$ (32)

This is not coherent for a Bunch Davies observer because $b\sim a-a^{\dagger}$
and the second piece renders it to be not an eigenstate of both $b$ and $a$.
Therefore, it is a generic excited state for a Bunch Davies observer, and one
would expect to see physical poles. Indeed, by expanding the field insertions
with an $\alpha$ observer’s mode functions, one finds

${}_{\alpha}\langle
C|\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}|C\rangle_{\alpha}=4\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})(\alpha+\beta)^{3}\frac{\Delta^{6}_{\zeta}H^{-1}\lambda}{2k_{1}k_{2}k_{3}}\bigg{[}(\beta^{*2}\alpha^{*}-\alpha^{*2}\beta^{*})\\\
\left(\frac{1}{(k_{1}+k_{2}-k_{3})^{3}}+\frac{1}{(k_{1}-k_{2}-k_{3})^{3}}\right.\\\
\left.+\frac{1}{(-k_{1}+k_{2}-k_{3})^{3}}\right)+(\beta^{*3}-\alpha^{*3})\frac{1}{(k_{1}+k_{2}+k_{3})^{3}}\bigg{]}+\textit{c.c}$
(33)

The normal ordered product of the $b$’s vanishes for the same reason as
before. We therefore see plainly physical poles, and this is in tune with our
intuition that a state that is coherent for an $\alpha$ observer is just some
generic excited state for a BD observer and therefore must display physical
poles.

###  3.5 BD modes, one particle state

The purpose of this section is to see the precise origin of physical poles
starting from a single excitation, and their support in phase space

####  3.5.1 Two point function

Performing the contractions one gets at $O(\lambda^{0})$

$=\frac{\Delta^{2}_{\zeta}}{k^{3}}\delta^{3}(\vec{k_{1}}+\vec{k_{2}})\left(1+\frac{1}{V}\left[\delta^{3}(\vec{k_{1}}+\vec{p})+\delta^{3}(\vec{k_{1}}-\vec{p})\right]\right)$
(34)

The $k$ dependence does not change.

####  3.5.2 Three point function

At $O(\lambda^{0})$ the correlator vanishes. Therefore, we look at
$O(\lambda)$ contribution, where a typical contraction gives terms like

$\displaystyle\begin{split}-2i\int\left(\prod_{i}d^{3}\vec{q}_{i}\right)\delta^{3}(\vec{k_{1}}-\vec{p})\delta^{3}(\vec{q_{3}}-\vec{p})\delta^{3}(\vec{k_{2}}+\vec{q_{1}})\delta^{3}(\vec{k_{3}}+\vec{q_{2}})\delta^{3}(\vec{q_{1}}+\vec{q_{2}}+\vec{q_{3}})\\\
\frac{q^{2}_{1}q^{2}_{2}q^{2}_{3}\Delta^{6}_{\zeta}H^{-1}\lambda}{\sqrt{(k_{1}k_{2}k_{3}q_{1}q_{2}q_{3})^{3}}}\int_{-\infty}^{0}d\tau^{\prime}\tau^{\prime
2}e^{-i(q_{1}+q_{2}+q_{3})\tau^{\prime}}\end{split}$ (35)
$=-2i\delta^{3}(\vec{k_{1}}-\vec{p})\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})\frac{\Delta^{6}_{\zeta}H^{-1}\lambda}{k_{1}k_{2}k_{3}}\int_{-\infty}^{0}d\tau^{\prime}\tau^{\prime
2}e^{-i(k_{1}+k_{2}+k_{3})\tau^{\prime}}$ (36)

Adding contributions from all contractions we get

$\displaystyle=\frac{-4\lambda\Delta^{6}_{\zeta}H^{-1}}{k_{1}k_{2}k_{3}}\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})\left[\left(\frac{1}{k_{1}+k_{2}+k_{3}}\right)\right.$
$\displaystyle\left.+\frac{1}{V}\left(\frac{1}{(k_{1}+k_{2}+k_{3})}\delta^{3}(\vec{k_{1}}-\vec{p})+\frac{1}{(-k_{1}+k_{2}+k_{3})}\delta^{3}(\vec{k_{1}}+\vec{p})+k_{1}\leftrightarrow
k_{2}+k_{1}\leftrightarrow k_{3}\right)\right]$ (37)

Since this is real therefore the real part is the above expression itself.
Apart from a total momentum conserving delta function in (6.2) and (6.7) we
also have additional delta functions involving $\vec{p}$. Therefore, this is a
disconnected diagram which contributes only at isolated points in phase space,
where there is an additional $V$ from the numerator that removes the delta
function singularity in the denominator. The total energy pole always
survives, and reflects the fact that the vacuum fluctuations can always
annihilate at a vertex among themselves without any reference to initial state
particles.

Let us carefully analyse this expression in light of the discussion in
literature Green:2020whw ; Green:2022fwg where the presence of physical poles
was ascribed to physical scatterings and decays in the initial- possibly
’classical’-state. Note that there are indeed physical poles that correspond
to scattering-like processes in the initial state(wherein the support of the
disconnected pieces depends on the initial state momentum, as required by
momentum conservation in any process), as well as a total energy pole that
corresponds to vacuum fluctuations. Note that now if we divide by the norm of
the state, the only unsuppressed contribution is the last term, which has the
same structure as a Bunch Davies vacuum 3 point function. It is true that
formally there are physical poles that when extrapolated to flat space would
indicate interactions between the initial state particle and the field
insertions-what this means is that there exist diagrams with interaction
vertices which can connect the external $p$ state with two of the $k_{i}$
modes. The remaining $p$ state is then contracted with the remaining field
insertion, and momentum conservation produces the offending delta function
that makes the support localized. Dividing by the norm then forces this term
to always vanish, except when the support is reached i.e. the unique point in
phase space where one of the field insertions has the same momentum as the
initial state. It is natural to ask what happens when the initial state
contains all possible momenta instead of a single particle with a fixed
momentum $p$. The physical pole should now be more likely to be observable,
because of the larger support. In fact, it was shown in Martin:1999fa that an
initial state with a finite number of quanta is already at odds with
data(which is largely compatible with Bunch Davies initial conditions), and
therefore it is natural to ask how this changes when there is a very large
number of quanta in the initial state-atleast one for every momentum. In this
case, the isolated support for the physical poles is expected to smear out,
because there always be a momentum in the initial state that has support at
the same momentum as one of the $k_{i}$. Then, while the physical poles will
still formally have support at isolated momenta, the fact that the phase space
of support is continuous will ensure that atleast one of the physical poles is
always reached. Let us see how this happens explicitly. Consider, an initial
state where multiple momenta are occupied; $\sqrt{2p_{1}\cdot 2p_{2}\cdot
2p_{3}\cdot\cdot\cdot}|p_{1},p_{2},p_{3},....\rangle$. This state will be
normalized with respect to $\prod\langle p_{i}|p_{i}\rangle$; all other
contractions are zero by orthogonality. Then, the 3 point function will be
schematically-

$\frac{\langle
p_{1},p_{2},p_{3},...|\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}|p_{1},p_{2},p_{3},..\rangle}{\prod\langle
p_{i}|p_{i}\rangle}\propto\delta(\vec{k_{t}})\bigg{[}\frac{1}{(k_{1}+k_{2}+k_{3})^{3}}\\\
\\\
+\frac{1}{(k_{1}+k_{2}+k_{3})^{3}}\sum_{i}\frac{\delta(\vec{k}_{1}-\vec{p}_{i})}{\langle
p_{i}|p_{i}\rangle}+\frac{1}{(k_{2}+k_{3}-k_{1})^{3}}\sum_{i}\frac{\delta(\vec{k}_{1}+\vec{p}_{i})}{\langle
p_{i}|p_{i}\rangle}+\textrm{perm.}\bigg{]}$ (38)

All other contractions vanish because they involve spectator terms like
$\delta(\vec{p}_{i}-\vec{p}_{j})$ which for unequal momenta vanish. Note now
that in comparison to the 1 particle state, the physical poles have more
support - at each of the $p_{i}$s now, and the observational imprint of these
states. If the momenta were continuously occupied-that is to say that $p_{i}$s
were a continuous variable, then we would expect a continuous distribution.
Therefore, taking the continuum limit and making the replacement

$\sum_{i}\frac{\delta(\vec{k}_{1}\pm\vec{p}_{i})}{\langle
p_{i}|p_{i}\rangle}\to\sum_{\vec{p}\in\mathbb{R}^{3}}\frac{\delta(\vec{k}_{1}\pm\vec{p})}{\langle
p|p\rangle}=1$ (39)

because the sum now has all possible momenta, one of which must be $k_{1}$ and
therefore this term always has a support. The numerator and denominator are
then both $\delta^{(3)}(0)$ which can be regulated in the usual way
Schwartz:2014sze to be taken to $V$, and therefore the limit $V\to\infty$ of
this expression is just $1$.This is similar to how the infinities coming from
phase space volumes drop out in the final formulae for cross sections. One can
think of this infinite sum schematically as an integral $\int
d^{3}k\delta(\vec{k}_{1}\pm\vec{p})$=1, where it is understood that in the
denominator the limit $d^{3}p\to 0,V\to\infty$ is taken such that the
$d^{3}pV\to\textrm{finite}$.

We therefore see that in the case where all momenta states are occupied, the
physical pole is always reached. While it is true that it comes from a
disconnected diagram, the point now is that now there always exists a
disconnected diagram that has the required support to be observable. We can
therefore conclude that

$\textrm{initial
state}=\prod_{k\in\mathbb{R}^{3}}|\vec{k}\rangle\implies\textrm{sum of
disconnected pieces always contains observable physical pole}$ (40)

It turns out that if one wants a connected contribution from $\ket{p}$ state
correlation function then one needs to go to higher order in perturbation
theory. e.g., for our case, we need to compute contribution at
$O(\lambda^{3})$. The in-in formula at $O(\lambda^{3})$ gives

$-i\int_{-\infty}^{0}dt_{1}\int_{-\infty}^{t_{1}}dt_{2}\int_{-\infty}^{t_{2}}dt_{3}[H(t_{3}),[H(t_{2}),[H(t_{1}),Q(0)]]]$
(41)

Clearly, there will be many terms but it serves our purpose to just pick one
of these terms and one of the contractions. We pick

$\displaystyle\begin{split}-2\textit{Re}\int_{-\infty}^{0}\frac{d\tau_{1}}{H\tau_{1}}\int_{-\infty}^{\tau_{1}}\frac{d\tau_{2}}{H\tau_{2}}\int_{-\infty}^{\tau_{2}}\frac{d\tau_{3}}{H\tau_{3}}i\langle
Q(0)H(t_{1})H(t_{2})H(t_{3})\rangle\\\
=-2\textit{Re}\int_{-\infty}^{0}dt_{1}\int_{-\infty}^{t_{1}}dt_{2}\int_{-\infty}^{t_{2}}dt_{3}i\prod_{i=1}^{9}\int
d^{3}\vec{q_{i}}\delta^{3}(\vec{q_{1}}+\vec{q_{2}}+\vec{q_{3}})\delta^{3}(\vec{q_{4}}+\vec{q_{5}}+\vec{q_{6}})\\\
\times\delta^{3}(\vec{q_{7}}+\vec{q_{8}}+\vec{q_{9}})\bra{p}\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\zeta^{\prime}_{q_{1}}(\tau_{1})\zeta^{\prime}_{q_{2}}(\tau_{1})\zeta^{\prime}_{q_{3}}(\tau_{1})\zeta^{\prime}_{q_{4}}(\tau_{2})\\\
\times\zeta^{\prime}_{q_{5}}(\tau_{2})\zeta^{\prime}_{q_{6}}(\tau_{2})\zeta^{\prime}_{q_{7}}(\tau_{3})\zeta^{\prime}_{q_{8}}(\tau_{3})\zeta^{\prime}_{q_{9}}(\tau_{3})\ket{p}\end{split}$
(42)

It can be easily checked that if one performs the following contraction

$\langle
a_{p}\zeta_{q_{1}}(\tau_{1})\rangle\langle\zeta_{q_{2}}(\tau_{1})\zeta_{q_{4}}(\tau_{2})\rangle\langle\zeta_{q_{5}}(\tau_{2})\zeta_{q_{7}}(\tau_{3})\rangle\langle\zeta_{k_{1}}\zeta_{q_{3}}(\tau_{1})\rangle\langle\zeta_{k_{2}}\zeta_{q_{6}}(\tau_{2})\rangle\langle\zeta_{k_{3}}\zeta_{q_{9}}(\tau_{3})\rangle\langle\zeta_{q_{8}}(\tau_{3})a_{p}^{\dagger}\rangle$
(43)

then after performing all momenta integrals one is left with only one delta
function i.e $\delta^{3}(\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}})$ which means we
have a connected diagram. In the next section, we will see in detail the
nature of contractions for a highly occupied initial state, by which we mean
not only that all momenta are excited, but also each is excited by a large
number. In Appendix C, we show that if instead we have a 1-particle state for
an $\alpha$ observer, we see physical poles without isolated support.

###  3.6 BD modes, large occupation number initial state

FollowingGreen:2022fwg and the preceding discussion, we compute the
correlation functions for excited high occupation number states above the
Bunch-Davies vacuum. All momentum modes for this $\ket{n}$ state are highly
occupied. To be more precise we have

$\ket{n}\equiv\bigotimes_{\vec{k_{i}}}\ket{n_{\vec{k}}}$ (44)

where,

$\ket{n_{\vec{k}}}=\frac{1}{\sqrt{n!}}\left(a_{\vec{k}^{\dagger}}\right)^{n}\ket{0}$
(45)

where $n$ is very large.

####  3.6.1 Two-Point Function

The two point function at $O(\lambda^{0})$ is given by

$\displaystyle\bra{n}\zeta_{k_{1}}\zeta_{k_{2}}\ket{n}=(2\pi)^{3}\delta^{3}(\vec{k_{1}}+\vec{k_{2}})\left((n_{k_{1}}+1)u^{*}_{k_{2}}u_{k_{1}}+n_{k_{1}}u_{k_{2}}u^{*}_{k_{1}}\right)$
(46)
$\displaystyle=(2pi)^{3}\delta^{3}(\vec{k_{1}}+\vec{k_{2}})\frac{\Delta^{2}_{\zeta}}{k_{1}^{3}}(2n_{k}+1)$
(47)

where, the action of field $\zeta_{k}$ on the state is given as follows

$\zeta_{k}\ket{n}=\sqrt{n_{-k}+1}u_{k_{1}}\ket{n_{-k_{1}}+1}\ket{\hat{n};-\vec{k_{1}}}+\sqrt{n_{k}}u^{*}_{k}\ket{n_{k}-1}\ket{\hat{n};\vec{k}}$
(48)

####  3.6.2 Three-Point Function

At $O(\lambda)$ the three point function is the following

$\displaystyle\textit{Re}\bra{0}\prod_{i}(a_{p_{i}})^{n}\left(-2i\zeta^{I}_{k_{1}}\zeta^{I}_{k_{2}}\zeta^{I}_{k_{3}}\int\left(\prod_{i}d^{3}\vec{q}_{i}\right)\delta^{3}(\sum\vec{q_{i}})\int_{-\infty}^{0}\frac{d\tau^{\prime}}{H\tau^{\prime}}\left(\zeta^{\prime
I}_{q_{1}}\zeta^{\prime I}_{q_{2}}\zeta^{\prime
I}_{q_{3}}\right)\right)\prod_{i}(a^{\dagger}_{p_{i}})^{n}\ket{0}$ (49)

It is clear from the above expression and previous computations that there are
contraction where $\zeta^{\prime}_{q}$s on the right can be contracted with
lowering operators on the left and the $\zeta_{k}$s on the left with raising
operators on the right. These types of contractions will give rise to physical
poles. One such contraction is shown below

$\wick{\prod_{i}\chi 1(a_{p_{i}})^{n}\chi 2a_{p_{j}}(-2i\chi
3\zeta^{I}_{k_{1}}\chi 4\zeta^{I}_{k_{2}}\chi
5\zeta^{I}_{k_{3}}\int\left(\prod_{i}d^{3}\vec{q}_{i}\right)\delta^{3}(\sum\vec{q_{i}})\int_{-\infty}^{0}\frac{d\tau^{\prime}}{H\tau^{\prime}}\left(\chi
5\zeta^{\prime I}_{q_{1}}\chi 4\zeta^{\prime I}_{q_{2}}\chi 2\zeta^{\prime
I}_{q_{3}})\right)\chi 3a^{\dagger}_{p_{j}}\prod_{i}\chi
1(a^{\dagger}_{p_{i}})^{n}}$ (50)

Such a contraction will produce an integral of the form

$\displaystyle\sim\int_{-\infty}^{0}\tau^{\prime
2}e^{-i\left(k_{2}+k_{3}-k_{1}\right)\tau}d\tau^{\prime}$ (51)

giving rise to physical poles but again there will be disconnected diagrams.

###  3.7 Four point function

All these calculations assume that there is no nongaussianity in the initial
state. Then the Nongaussianity results only from the interaction Hamiltonian
time evolving the initial state to the present state. Moreover, one can
explicitly write the BD interacting vacuum in terms of the free theory
eigenstate using standard perturbation theory, and Trivedi shows that the
limit $\tau_{initial}\to-\infty$ reduces the interacting theory vacuum to free
vacuum only for cubic self coupling. Our analysis has more possibilities,
where the initial state needn’t match the vacuum of the mode functions, and
therefore really has no access to convenient methods like the WKB
approximation to interpret results. We aim to only study the analytic
structures allowed, and in this section we extend the results of Green:2020whw
to four-point functionsGoodhew_2021 ; Arroja_2009 ; Chen_2009 ;
https://doi.org/10.48550/arxiv.2106.05294 . We calculate both classical and
quantum four-point functions and comment about their pole structures.

####  3.7.1 Classical correlator

Consider a massless field in de Sitter space with the following action

$S=\int{d\tau d^{3}x\left(\frac{1}{2}\left(a^{2}\zeta^{\prime
2}-a^{2}\partial^{2}\zeta\right)+a\frac{\lambda}{6}\zeta^{\prime 3}\right)}$
(52)

where $a$ is the scale factor and prime denotes a derivative w r t to
conformal time $\tau$. The equation of motion is given by

$\frac{1}{a^{2}}(\zeta^{\prime\prime}+2\frac{a^{\prime}}{a}\zeta^{\prime}-\partial^{2}\zeta)=-\frac{\lambda}{a^{4}}(a\zeta^{\prime}\zeta^{\prime\prime}+\frac{a^{\prime}\zeta^{\prime
2}}{2})$ (53)

The Greens function corresponding to the operator on the left acting on
$\zeta$ and its derivative are given byGreen:2020whw ; van_der_Meulen_2007

$\displaystyle
G_{\vec{k}}(\tau,\tau^{\prime})=\theta(\tau-\tau^{\prime})\frac{2\Delta_{\zeta}^{2}}{k^{3}}\left((1+k^{2}\tau\tau^{\prime})\sin{k(\tau-\tau^{\prime})}-k(\tau-\tau^{\prime})\cos{k(\tau-\tau^{\prime})}\right)$
(54)
$\displaystyle\partial_{\tau}G_{\vec{k}}(\tau,\tau^{\prime})=2\Delta_{\zeta}^{2}[k^{-1}\tau^{\prime}\sin{k(\tau-\tau^{\prime})}-\tau\tau^{\prime}\cos{k(\tau-\tau^{\prime})}]$
(55)

We can solve Eq(5.2) perturbatively by expanding $\zeta$ as follows

$\zeta=\zeta^{(1)}+\lambda\zeta^{(2)}+\lambda^{2}\zeta^{(3)}+O(\lambda^{3})$
(56)

Plugging this into (5.2) we get the following equations

$\displaystyle\frac{1}{a^{2}}({\zeta^{\prime\prime}}^{(1)}+2\frac{a^{\prime}}{a}{\zeta^{\prime}}^{(1)}-\partial^{2}\zeta^{(1)})=0$
(57)
$\displaystyle\frac{1}{a^{2}}({\zeta^{\prime\prime}}^{(2)}+2\frac{a^{\prime}}{a}{\zeta^{\prime}}^{(2)}-\partial^{2}\zeta^{(2)})+\frac{1}{a^{4}}\left(a{\zeta^{\prime}}^{(1)}{\zeta^{\prime\prime}}^{(1)}+\frac{a^{\prime}{\zeta^{\prime}}^{(1)2}}{2}\right)=0$
(58)
$\displaystyle\frac{1}{a^{2}}({\zeta^{\prime\prime}}^{(3)}+2\frac{a^{\prime}}{a}{\zeta^{\prime}}^{(3)}-\partial^{2}\zeta^{(3)})+\frac{1}{a^{4}}\left(a{\zeta^{\prime}}^{(1)}{\zeta^{\prime\prime}}^{(2)}+a{\zeta^{\prime}}^{(2)}{\zeta^{\prime\prime}}^{(1)}+a^{\prime}{\zeta^{\prime}}^{(1)}{\zeta^{\prime}}^{(2)}\right)=0$
(59)

Therefore the solution to (5.7) and (5.8) can be written in terms of Greens
function given above.

$\displaystyle\zeta^{(2)}(\tau,\vec{x})=\int
d^{3}x^{\prime}\int{d\tau^{\prime}\frac{a}{2}\partial_{\tau^{\prime}}G(x,x^{\prime})(\zeta^{\prime(1)2})}$
(60) $\displaystyle\zeta^{(3)}(\tau,\vec{x})=\int
d^{3}x\int{d\tau^{\prime}a\left(\partial_{\tau^{\prime}}G(x,x^{\prime})\zeta^{\prime(1)}\zeta^{\prime(2)}\right)}$
(61)

where we have neglected a total derivative since $G(\tau,\tau+0^{+})$ is zero
(retarded GF) and $G(\tau,-\infty)$ is also set to zero with an appropriate
$i\epsilon$ prescription. These equations in fourier space read

$\displaystyle\zeta^{(2)}_{\vec{k}}(\tau)=\int{d\tau^{\prime}\frac{d^{3}p}{2(2\pi)^{3}}a\left(\partial_{\tau^{\prime}}G_{\vec{k}}(\tau,\tau^{\prime}){\zeta^{\prime}}^{(1)}_{\vec{p}}{\zeta^{\prime}}^{(1)}_{\vec{k}-\vec{p}}\right)}$
(62)
$\displaystyle\zeta^{(3)}_{\vec{k}}(\tau)=\int{d\tau^{\prime}\frac{d^{3}p}{(2\pi)^{3}}a\left(\partial_{\tau^{\prime}}G_{\vec{k}}(\tau,\tau^{\prime}){\zeta^{\prime}}^{(2)}_{\vec{p}}{\zeta^{\prime}}^{(1)}_{\vec{k}-\vec{p}}\right)}$
(63)

The free field solution is given by

$\zeta^{(1)}_{\vec{k}(\tau)}=\frac{\Delta_{\zeta}}{\sqrt{k^{3}}}\left(a_{\vec{k}}^{\dagger}u_{k}(\tau)+a_{-\vec{k}}u_{k}^{*}(\tau)\right)$
(64)

defined with the following statistics

$\displaystyle<a_{\vec{k}}a_{\vec{k^{\prime}}}^{\dagger}>=<a_{\vec{k}}^{\dagger}a_{\vec{k^{\prime}}}>=\frac{1}{2}\delta^{3}(\vec{k}-\vec{k^{\prime}})$
(65)

where, $u_{\vec{k}}(\tau)=(1-ik\tau)e^{ik\tau}$
This produces the correct two point function which is the only observable we
have measured so far. Note that this statistical nature is due to our lack of
knowledge and not due to some inherent uncertainty as is the case for quantum
mechanical operators.

The two point function is given by

$<\zeta^{(1)}_{\vec{k}}(\tau)\zeta^{(1)}_{\vec{k^{\prime}}}(\tau^{\prime})>=\frac{\Delta_{\zeta}^{2}}{k^{3}}\delta^{3}(\vec{k}+\vec{k^{\prime}})Re[u_{k}(\tau)u_{k}^{*}(\tau^{\prime})]$
(66)

Let us now calculate the classical 4 point function upto $O(\lambda^{2})$

$\displaystyle\begin{split}<\zeta\zeta\zeta\zeta>=<(\zeta^{(1)}_{k_{1}}+\lambda\zeta^{(2)}_{k_{1}}+\lambda^{2}\zeta^{(3)}_{k_{1}})(\zeta^{(1)}_{k_{2}}+\lambda\zeta^{(2)}_{k_{2}}+\lambda^{2}\zeta^{(3)}_{k_{2}})(\zeta^{(1)}_{k_{3}}+\lambda\zeta^{(2)}_{k_{3}}+\lambda^{2}\zeta^{(3)}_{k_{3}})\times\\\
(\zeta^{(1)}_{k_{4}}+\lambda\zeta^{(2)}_{k_{4}}+\lambda^{2}\zeta^{(3)}_{k_{4}})>\end{split}$
(67)

Terms of $O(\lambda)$ vanish (odd number of terms) and the non-trivial
contribution to the correlator comes from two types of terms:

I:
$<\zeta^{(1)}_{k_{1}}\zeta^{(1)}_{k_{2}}\zeta^{(2)}_{k_{3}}\zeta^{(2)}_{k_{4}}>$

II:
$<\zeta^{(1)}_{k_{1}}\zeta^{(1)}_{k_{2}}\zeta^{(1)}_{k_{3}}\zeta^{(3)}_{k_{4}}>$

Rest of the terms are just permutations of these two terms.

Let us first evaluate I:

$\displaystyle\begin{split}\textbf{I}=\int\frac{d\tau^{\prime}d\tau^{\prime\prime}d^{3}pd^{3}q}{(2\pi)^{6}}(-H\tau^{\prime})^{-1}(-H\tau^{\prime\prime})^{-1}\partial_{\tau^{\prime}}G_{k_{3}}(\tau,\tau^{\prime})\partial_{\tau^{\prime}}G_{k_{3}}(\tau,\tau^{\prime\prime})\times\\\
<\zeta_{k_{1}}^{(1)}\zeta_{k_{2}}^{(1)}{\zeta^{\prime}}_{\vec{k_{3}}-\vec{p}}^{(1)}(\tau^{\prime}){\zeta^{\prime}}_{p}^{(1)}(\tau^{\prime}){\zeta^{\prime}}_{\vec{k_{4}}-\vec{q}}^{(1)}(\tau^{\prime\prime}){\zeta^{\prime}}_{q}^{(1)}(\tau^{\prime\prime})>\end{split}$
(68)

Using wick’s theorem for these gaussian random fields, we get

$\displaystyle\begin{split}=\frac{12\Delta^{10}_{\zeta}H^{-2}k_{13}}{6k_{1}k_{2}k_{3}k_{4}}\left[\int
d\tau^{\prime}\tau^{\prime
2}\cos{k_{1}\tau^{\prime}}\cos{k_{13}\tau^{\prime}}\sin{k_{3}\tau^{\prime}}\int
d\tau^{\prime}\tau^{\prime
2}\cos{k_{2}\tau^{\prime\prime}}\cos{k_{13}\tau^{\prime\prime}}\sin{k_{4}\tau^{\prime\prime}}\right]\\\
+k_{1}\leftrightarrow k_{2}\end{split}$ (69)

where, $k_{13}=|\vec{k_{1}}+\vec{k_{3}}|$. After some more algebra this
simplifies to

$\frac{4\Delta^{10}_{\zeta}H^{-2}}{k_{1}k_{2}k_{3}k_{4}}\left(k_{13}\times\alpha\times\beta+k_{1}\leftrightarrow
k_{2}\right)$ (70)

where,

$\displaystyle\alpha=\frac{1}{(k_{1}+k_{3}+k_{13})^{3}}+\frac{1}{(k_{1}+k_{3}-k_{13})^{3}}+\frac{1}{(-k_{1}+k_{3}+k_{13})^{3}}-\frac{1}{(k_{1}-k_{3}+k_{13})^{3}}$
(71)
$\displaystyle\beta=\frac{1}{(k_{2}+k_{4}+k_{13})^{3}}+\frac{1}{(k_{2}+k_{4}-k_{13})^{3}}+\frac{1}{(-k_{2}+k_{4}+k_{13})^{3}}-\frac{1}{(k_{2}-k_{4}+k_{13})^{3}}.$
(72)

Now, term II gives

$\displaystyle\begin{split}=\int\frac{d\tau^{\prime}d^{3}p}{(2\pi)^{3}}(-H\tau^{\prime})^{-1}\partial_{\tau^{\prime}}G_{k_{4}}(\tau,\tau^{\prime})<\zeta_{k_{1}}^{(1)}\zeta_{k_{2}}^{(1)}\zeta_{k_{3}}^{(1)}{\zeta^{\prime}}_{\vec{k_{4}}-\vec{p}}^{(1)}(\tau^{\prime})\times\\\
\partial_{\tau^{\prime}}\left(\int\frac{d\tau^{\prime\prime}d^{3}q}{(2\pi)^{3}}(-H\tau^{\prime\prime})^{-1}\partial_{\tau^{\prime\prime}}G_{p}(\tau^{\prime},\tau^{\prime\prime}){\zeta^{\prime}}_{q}^{(1)}(\tau^{\prime\prime}){\zeta^{\prime}}_{\vec{p}-\vec{q}}(\tau^{\prime\prime})\right)>\end{split}$
(73)

Using Leibniz rule to simply this further

$\displaystyle\begin{split}\int\frac{d\tau^{\prime}d^{3}pd^{3}qd\tau^{\prime\prime}}{(2\pi)^{6}}(-H\tau^{\prime})^{-1}(-H\tau^{\prime\prime})^{-1}\partial_{\tau^{\prime}}G_{k_{4}}(\tau,\tau^{\prime})\partial_{\tau^{\prime}}\partial_{\tau^{\prime\prime}}G_{p}(\tau^{\prime},\tau^{\prime\prime})\\\
<\zeta_{k_{1}}^{(1)}\zeta_{k_{2}}^{(1)}\zeta_{k_{3}}^{(1)}{\zeta^{\prime}}_{\vec{k_{4}}-\vec{p}}^{(1)}(\tau^{\prime}){\zeta^{\prime}}_{q}^{(1)}(\tau^{\prime\prime}){\zeta^{\prime}}^{(1)}_{\vec{p}-\vec{q}}(\tau^{\prime\prime})>\end{split}$
(74)

We proceed as before and after some lengthy algebra one gets

$\frac{4H^{-2}\Delta^{10}_{\zeta}k_{14}}{16k_{1}k_{2}k_{3}k_{4}}\bigg{[}\frac{4}{u_{x1}^{3}}\bigg{(}\frac{1}{x^{3}}+\frac{6}{xu^{2}_{x1}}\bigg{)}+\frac{4}{u_{x2}^{3}}\bigg{(}\frac{1}{x^{3}}+\frac{6}{xu^{2}_{x2}}\bigg{)}+\frac{4}{u_{x3}^{3}}\bigg{(}\frac{1}{{x}^{3}}+\frac{6}{xu^{2}_{x3}}\bigg{)}\\\
+\frac{4}{u_{x4}^{3}}\bigg{(}\frac{1}{{x}^{3}}+\frac{6}{xu^{2}_{x4}}\bigg{)}+\frac{12}{x^{2}}\bigg{(}-\frac{1}{u^{4}_{x1}}+\frac{1}{u^{4}_{x2}}-\frac{1}{u^{4}_{x3}}+\frac{1}{u^{4}_{x4}}\bigg{)}+(u_{y},y)+(u_{z},z)+(u_{w},w)\\\
+k_{1}\leftrightarrow k_{2}(\textrm{of previous term})+k_{2}\leftrightarrow
k_{3}(\textrm{of previous term})\bigg{]}$ (75)

where,

$\displaystyle(u_{m},m)=\frac{4}{u_{m1}^{3}}\left(\frac{1}{m^{3}}+\frac{6}{mu^{2}_{m1}}\right)+\frac{4}{u_{m2}^{3}}\left(\frac{1}{m^{3}}+\frac{6}{mu^{2}_{m2}}\right)+\frac{4}{u_{m3}^{3}}\left(\frac{1}{{m}^{3}}+\frac{6}{mu^{2}_{m3}}\right)$
(76)
$\displaystyle+\frac{4}{u_{m4}^{3}}\left(\frac{1}{{m}^{3}}+\frac{6}{mu^{2}_{m4}}\right)+\frac{12}{m^{2}}\left(-\frac{1}{u^{4}_{m1}}+\frac{1}{u^{4}_{m2}}-\frac{1}{u^{4}_{m3}}+\frac{1}{u^{4}_{m4}}\right)$
(77) $\displaystyle x=k_{14}+k_{2}+k_{3}$ (78) $\displaystyle
y=k_{14}+k_{2}-k_{3}$ (79) $\displaystyle z=k_{14}-k_{2}+k_{3}$ (80)
$\displaystyle w=k_{14}-k_{2}-k_{3}$ (81) $\displaystyle
u_{x1}=k_{1}+k_{4}+k_{14}-x$ (82) $\displaystyle u_{x2}=k_{1}+k_{4}-k_{14}-x$
(83) $\displaystyle u_{x3}=-k_{1}+k_{4}-k_{14}-x$ (84) $\displaystyle
u_{x4}=-k_{1}+k_{4}-k_{14}-x$ (85)

For $u_{y}$ and $u_{z}$ terms we just replace $x$ in $u_{x}$ terms by $y$ and
$z$. Again, the poles are located at physical momenta consistent with the
interpretation that if initial state has physical particles then processes
like decays can give rise to physical poles. The final expression for the
classical Four-point function is the sum of (70) and (75) + permutations.

####  3.7.2 Quantum correlator

We recall the mode expansion

$\zeta_{\vec{k}}(\tau)=\frac{\Delta_{\zeta}}{\sqrt{2k^{3}}}(u_{k}(\tau)a_{\vec{k}}+u^{*}_{k}(\tau)a^{\dagger}_{-\vec{k}}),u_{k}(\tau)=(1+ik\tau)e^{-ik\tau}$
(86)

From which we find

$\zeta^{\prime}_{k}(\tau)=\frac{\Delta_{\zeta}k^{2}\tau}{\sqrt{2k^{3}}}(a_{\vec{k}}e^{-ik\tau}+a^{\dagger}_{-\vec{k}}e^{ik\tau})$
(87)

We will need the following 2 point functions-

$\langle\zeta^{\prime}_{p}(\tau_{1})\zeta^{\prime}_{q}(\tau_{2})\rangle=(2\pi)^{3}\delta(\vec{p}+\vec{q})\frac{p\Delta_{\zeta}^{2}\tau_{1}\tau_{2}}{2}e^{ip(\tau_{2}-\tau_{1})}\equiv(2\pi)^{3}\delta(\vec{p}+\vec{q})G^{\prime}_{p}(\tau_{1},\tau_{2})$
(88)

$(2\pi)^{3}\delta(\vec{p}+\vec{k})G_{k}(\tau)\equiv\langle\zeta^{\prime}_{p}(\tau)\zeta_{k}(0)\rangle=(2\pi)^{3}\delta(\vec{p}+\vec{k})\frac{\Delta_{\zeta}^{2}\tau}{2k}e^{-ik\tau}=(2\pi)^{3}\delta(\vec{p}+\vec{k})\langle\zeta_{k}(0)\zeta^{\prime}_{p}(\tau)\rangle^{*}$
(89)

At order $\lambda^{2}$, we have-

$\displaystyle\langle\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\zeta_{k_{4}}\rangle=-2Re\int^{0}_{-\infty}d\tau_{2}\int^{\tau_{2}}_{-\infty}d\tau_{1}\bigg{(}\langle\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\zeta_{k_{4}}H(\tau_{2})H(\tau_{1})\rangle$
(90) $\displaystyle-\langle
H(\tau_{1})\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\zeta_{k_{4}}H(\tau_{2})\rangle\bigg{)}$
(91)

We define

$\vec{k}_{12}=\vec{k}_{1}+\vec{k}_{2}=-\vec{k}_{34}\hskip
14.22636pt\vec{k}_{t}=\sum_{i=1}^{4}\vec{k}_{i}$ (92)

This becomes, suppressing momentum conserving delta functions and permutations
for brevity,

$=-2Re\int^{0}_{-\infty}\frac{d\tau_{2}}{H\tau_{2}}\int^{\tau_{2}}_{-\infty}\frac{d\tau_{1}}{H\tau_{1}}\bigg{(}G^{*}_{k_{1}}(\tau_{2})G^{*}_{k_{2}}(\tau_{2})G^{*}_{k_{3}}(\tau_{1})G^{*}_{k_{4}}(\tau_{1})G^{\prime}_{k_{12}}(\tau_{2},\tau_{1})\\\
\\\
-G_{k_{1}}(\tau_{1})G_{k_{2}}(\tau_{1})G^{*}_{k_{3}}(\tau_{2})G^{*}_{k_{4}}(\tau_{2})G^{\prime}_{k_{12}}(\tau_{1},\tau_{2})\bigg{)}+\textrm{perm.}$
(93)

$=-2Re\frac{\Delta_{\zeta}^{10}k_{12}}{32k_{1}k_{2}k_{3}k_{4}}\int^{0}_{-\infty}\frac{d\tau_{2}}{H\tau_{2}}\int^{\tau_{2}}_{-\infty}\frac{d\tau_{1}}{H\tau_{1}}\tau_{1}^{3}\tau_{2}^{3}\bigg{(}e^{i(k_{1}+k_{2})\tau_{2}+(k_{3}+k_{4})\tau_{1}+{k_{12}}(\tau_{1}-\tau_{2})}\\\
\\\
-e^{-i(k_{1}+k_{2})\tau_{1}+(k_{3}+k_{4})\tau_{2}+{k_{12}}(\tau_{2}-\tau_{1})}\bigg{)}$
(94)

And therefore the full 4 point function becomes, after including all possible
contractions

$\begin{split}\frac{4\lambda^{2}\Delta^{10}_{\zeta}H^{-2}k_{12}}{k_{1}k_{2}k_{3}k_{4}}\left(\frac{2}{k_{t}^{3}(k_{3}+k_{4}+k_{12})^{3}}+\frac{6}{k_{t}^{4}(k_{3}+k_{4}+k_{12})^{2}}+\frac{12}{k^{5}_{t}(k_{3}+k_{4}+k_{12})}\right.\\\
\\\
\left.+\frac{1}{(k_{1}+k_{2}+k_{12})^{3}(k_{3}+k_{4}+k_{12})^{3}}\right)+\textit{permutations}\end{split}$
(95)

We see the presence of additional poles besides the one for total energy, but
this is no physical pole, consistent with our expectations from a Bunch Davies
vacuum.

##  4 The origin of singularities

We must clarify what we aim to understand from the above calculations. We wish
to probe the interplay between the nature of the initial state and the
structure of in-in correlators, through the momentum dependence of the
singularity structures. What we mean by this is identifying the configuration
of momenta for which these correlation functions become singular. The
functional form of the singularity isn’t fixed, and as we will see in a later
section, depends crucially on the nature of interactions. We have assumed the
initial conditions are set at $\tau=-\infty$, and in section 8 we see how
things change at a finite early time cutoff.

The authors of Green:2022fwg studied this in the context of understanding
whether the initial conditions of inflation were classical perturbations in an
ensemble or quantum zero point fluctuations of the BD vacuum. The pole
structure was linked to physical processes-such as particle decay in the
initial (classical) state producing poles at physical momenta. However, the
results of Shukla:2016bnu exhibit the physical poles despite considering
$\alpha$-mode insertions in an initial $\alpha$-vacuum. This displays how the
notion of calling a state ’classical’ based on the particle content is subtle,
as the $\alpha$ vacuum-while excited for a Bunch Davies observer, is devoid of
$\alpha$-particles. We will also see how the signatures of initial state
effects reflect in the modification to Maldacena’s consistency criteria(which
is already modified in the presence of higher derivative operators
Creminelli:2003iq )-in particular, non-Bunch Davies initial states produce
severely enhanced deviations from the consistency criteria.

However, one needs to be careful about the physical implications of these
singularity structures. We have seen the inevitability of disconnected
diagrams in correlation functions. Consider a flat space correlator with 3
field insertions and at cubic order. If the external state is a single
particle state of fixed momentum, then its natural contribution to the
S-matrix element is schematically

$\langle p|T\phi(k_{1})\phi(k_{2})\phi(k_{3})|p\rangle\to^{LSZ}\langle
p,k_{1},k_{2}|p,k_{3}\rangle$ (96)

where it is understood that crossed elements can be generated too; we have
just written down a possible configuration. Now, at first order, it is easy to
see that this amplitude will contain disconnected subgraphs where some of the
particles are mere spectators- only $3$ out of $5$ particles can interact at a
cubic vertex, and the remaining two will act as spectators which don’t
participate in the interaction. These diagrams will produce factors of
$\delta(\textrm{subset of }p,k_{i})$, and will therefore only have support at
isolated points in phase space. Generating fully connected contributions will
require us to go to higher order in perturbation theory, where there are
enough vertices to contract with every external leg-but these are subleading
effects. It then becomes very tricky to find measurable imprints of these pole
structures. In fact, the whole discussion is subtle because it is unclear how
to translate from flat space intuition of ’particles’ and their Feynman
diagrams because the notion of particles is itself subtle in curved spaces.
For example, it is entirely possible to switch to a different picture of
particles, and the resulting correlators will have to then be interpreted
differently. As an example which will be discussed in detail later, if we
consider an initial state which is an excited Bunch Davies state such that it
happens to be the vacuum for an $\alpha$-observer, then the correlator can be
computed in two ways. We can write the state in terms of Bunch Davies
excitations and compute the correlator that will now display disconnected
pieces because of the initial state particles, see 3.3. Alternatively, we can
go to a frame where there are no excitations-this frame corresponds to
choosing the $\alpha$-vacuum basis for mode expansion of our fields. Then, the
diagrammatic interpretation of the involved terms will be entirely different,
and we will see no disconnected pieces simply because now the interactions at
a vertex don’t involve initial state particles. Schematically, the two ways to
compute the correlator amount to

${}_{\alpha}\langle
0|\zeta^{\alpha}\zeta^{\alpha}\zeta^{\alpha}|0\rangle_{\alpha}\hskip
8.53581pt\textrm{or}\hskip 8.53581pt_{BD}\langle\big{(}0|e^{\int
d^{3}pa_{p}a_{-p}})|\zeta^{BD}\zeta^{BD}\zeta^{BD}|\big{(}e^{\int
d^{3}pa^{\dagger}_{p}a^{\dagger}_{-p}}|0\big{)}\rangle_{BD}$ (97)

Where we have indicated what mode functions i.e. what particle interpretation
is being used. $a_{p}$ are Bunch Davies operators. Note that the second way of
calculating requires to include contractions representing interactions of in-
state particles at a vertex, whereas the first doesn’t. One expects that
eventually some kind of resummation renders the two approaches equivalent, but
it is not obvious.

The take-home message is this-once we begin considering excited initial
states, then because only finitely many of them can interact at a finite order
in perturbation theory (determined by the nature of the coupling and the order
at which we are interested), correlation functions will be plagued by
disconnected diagrams. Many pole structures will be obscured in observation
because of the fact that their kinematic support lies on isolated points in
phase space.

The disconnectedness of these diagrams clouds the interesting pole structure
that comes from genuine interaction. If, then, we were in a situation where
there are no real particles in the initial state, then one would expect to be
free from these disconnected diagrams. This is in fact what happens whenever
the initial state is a vacuum state of the perturbation modes-and when this
vacuum state is ’excited’ relative to a Bunch Davies vacuum in a manner that
all momenta are excited, one sees the presence of genuine, physically
measurable physical poles. We would like to understand, then, to what extent
do pole structures encode information about the initial state. Let us first
understand their origin.

###  4.1 Singularities and interacting vertices: flat space vs curved space

Let us elaborate on the remarks of the previous section, drawing analogy from
flat space results to see explicitly how the pole structure for in-in
correlators is different from in-out and its interpretation in terms of
physical processes. We will also see the precise origin of pole structures and
the role of mode mixing. Our object of interest will be the structure of
kinematics at a vertex in perturbation theory, since it is integrals over
these vertices that produce analytic structures.

In flat space, we typically calculate time ordered correlators via Wick’s
theorem. Let us first see how kinematics are constrained at a vertex in a
simple $\lambda\phi^{3}/3!$ interaction in flat space. If we calculate the
time ordered correlator, then the leading contribution to the 3 point
correlator in perturbation theory is of the type

$\langle T\phi(x_{1})\phi(x_{2})\phi(x_{3})\rangle_{int}\sim\lambda\int
d^{4}xD_{F}(x_{1}-x)D_{F}(x_{2}-x)D_{F}(x_{3}-x)$ (98)

where $D_{F}(x)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{ipx}/(p^{2}+i\epsilon)$ is the
Feynman propagator, the use of which is tied to the presence of time ordering
on the left. Let us focus on the integral over the vertex, where the momenta
in the propagator’s expression combine into a delta function, which tells us
that four-momentum is conserved at every vertex. Since the $p_{i}$ are being
integrated over, and noting that $D_{F}(x)=D_{F}(-x)$, it is clear that it
really doesn’t matter if we choose to call a momentum $p$ or $-p$. In fact,
being a vertex or a correlation function, it doesn’t immediately correspond to
any physical process. Therefore, we need to specify additional information to
specify the correct kinematic constraints for any physical process-this is
precisely the LSZ prescription. This expression depends on $x_{i}$-the points
of insertion. While the LHS is agnostic to their relative order because of
time ordering within which everything commutes; the RHS needs greater care.
Let us see how LSZ systematically teaches us what to do. Let us say that the
field insertions at $x_{i}$ correspond to one particle states of momentum
$k_{i}$ in the asymptotic region, where the theory is assumed to be free(this
is one of the assumptions that break down in curved space). For concreteness,
let us say that $x_{1},x_{2}$ correspond to insertions in the far past(in-
state), and $x_{3}$ to an insertion in the far future(out-state)-this produces
a $\delta^{(4)}(p_{1}+p_{2}-p_{3})$ from the vertex integral. To see
explicitly the interplay between presence of in/out states and time ordering,
note that in a typical flat space in out correlator, everything including
initial and final states is in time order; LSZ then converts it to an
expression for $\langle f|i\rangle$, where a relative sign between momenta is
induced because $i\int
d^{4}xe^{ipx}(\partial_{\mu}\partial^{\mu}+m^{2})\phi=\sqrt{2\omega_{p}}(a_{\infty}-a_{-\infty})$.
This expression is, formally

$\displaystyle|f\rangle=\sqrt{2k_{1}\cdot
2k_{2}..}a^{\dagger}_{k_{1}}(\infty)a^{\dagger}_{k_{2}}(\infty)..|\Omega\rangle,|i\rangle=\sqrt{2p_{1}\cdot
2p_{2}..}a^{\dagger}_{p_{1}}(-\infty)a^{\dagger}_{p_{2}}(-\infty)..|\Omega\rangle$
(99) $\displaystyle\langle f|i\rangle=\sqrt{2k_{1}\cdot 2k_{2}\cdot\cdot\cdot
2p_{1}2p_{2}..}\langle\Omega|a_{k_{1}}(\infty)a_{k_{2}}(\infty).....a^{\dagger}_{p_{1}}(-\infty)..|\Omega\rangle$
(100)

And everything is naturally time ordered, and we use Feynman propagators to
evaluate this expression via Wick’s theorem. However, in an in-in correlator,
we only have $|i\rangle$ state available, and therefore a correlation function
is of the form

$\langle i|\mathcal{O}(t)|i\rangle\sim\langle
0|a_{p_{1}}(-\infty)a_{p_{2}}(-\infty)..\mathcal{O}(t)a^{\dagger}_{p_{1}}(-\infty)a^{\dagger}_{p_{2}}(-\infty)...|0\rangle$

We must therefore resort to using Wightman functions, and this is precisely
where destructive interference between modes can occur. For comparison,
consider the difference between a sample in-out and in-in correlator. First,

$-i\lambda\delta(\sum\vec{k}_{i})\int dt\langle
k,out|T\phi_{k_{1}}(t)\phi_{k_{2}}(t)\phi_{k_{3}}(t)|p,p^{\prime},in\rangle\\\
\supset\langle
a_{k}(\infty)\phi_{k_{1}}\rangle\langle\phi_{k_{2}}a^{\dagger}_{p}(-\infty)\rangle\langle\phi_{k_{3}}a^{\dagger}_{p^{\prime}}(-\infty)\rangle\\\
\\\ =-i\lambda\delta(\sum\vec{k}_{i})\int
dt\delta(\vec{k}+\vec{k}_{1})\delta(\vec{k}_{2}-\vec{p})\delta(\vec{k}_{3}-\vec{p^{\prime}})e^{i(k-p-p^{\prime})t}=-i\lambda\delta^{(4)}(k-p-p^{\prime})$
(101)

Each contraction here is a Feynman propagator. The $1/2k$ normalizations
cancel in the propagator and the definition of the external states.

To see what happens without an out state, let us compute the in-in correlator
at equal-time insertions for this example. The first difference in the vertex
integrals is the replacement

$\int_{\infty}^{\infty}dt^{\prime}\to 2Im\int^{t}_{-\infty}dt^{\prime}$ (102)

where $t$ is the time of insertion. With this replacement, the delta function
for energies becomes a pole, with the expression(here $p$ is $p^{0}$)

$\int_{\infty}^{\infty}dt^{\prime}e^{i(p_{1}+p_{2}+p_{3})t^{\prime}}=\delta(p_{1}+p_{2}+p_{3})\to
2Im\int^{t}_{-\infty}dt^{\prime}e^{i(p_{1}+p_{2}+p_{3})t^{\prime}}=\frac{2}{p_{1}+p_{2}+p_{3}}$
(103)

The question now is, in flat space, can we have physical poles like
$(k_{1}+k_{2}-k_{3})^{-1}$ in equal time in-in correlators. Clearly, these
poles would correspond to an energy conserving delta function for a physical
process $k_{1},k_{2}\to k_{3}$. But these, as we saw, arise after carrying out
LSZ and placing some insertions in the far past and some in the far future-
this cannot be done in an equal time in-in correlator. This was also observed
in Green:2022fwg where it was shown that LSZ killed all poles which didn’t
correspond to physical processes in flat space.

So then how do physical poles arise in inflation? First, note that in a
general backgrounds the modes are complicated functions $u_{k}(t)e^{\pm(\omega
t-\vec{k}\cdot\vec{x})}$. Secondly, in the in-in correlators, one replaces the
Feynman propagator $D_{F}(t_{1},t_{2})$ with the Wightman function, $G_{W}$.
Recall that the flat space wightman function and feynman propagator are
related as

$\displaystyle D(t)=\langle
0|\phi_{k}(t)\phi_{k^{\prime}}(0)|0\rangle=(2\pi)^{3}\delta(\vec{k}+\vec{k^{\prime}})\frac{1}{2k}e^{-ikt}=D(-t)^{*},$
(104) $\displaystyle D_{F}(t)=\theta(t)D(t)+\theta(-t)D(t)^{*}$ (105)

We see that while the Feynman propagator contains both mode functions
formally, it only ever has support on one of them-whichever is picked by time
ordering. In contrast, the classical Green’s function and the Wightman
function for a Bugolyubov transformed mode is

$\displaystyle G_{c}(k.t)=\frac{-i}{2k}(e^{ikt}+e^{-ikt})$ (106)
$\displaystyle
G_{W}^{BG}(0,\tau)=(2\pi)^{3}\delta(\vec{k}+\vec{k}^{\prime})(Au_{k}(0)+Bu_{k}(0))(A^{*}u_{k}^{*}e^{ik\tau}+B^{*}u_{k}e^{-ik\tau})$
(107)

where $u_{k}=\frac{H}{\sqrt{2k^{3}}}(1+ik\tau)$. Both of these contain both
frequencies of modes, and it is natural to expect that they will generate a
similar pole structure in correlation functions. Note a number of things. Time
ordering automatically ensures all these Wightman functions are just Feynman
propagators, and the relative sign of the mode functions in the integral boils
down to the existence of both in and out states.

To see this explicitly, suppose we instead calculated the (physically
nonsensical) time ordered quantity $\langle
0|\phi_{1}\phi_{2}\phi_{3}|k,p,p^{\prime},in\rangle$ where there’s no out
state. Then, the first (time ordered) contraction above is replaced by
$\langle a_{k}(\infty)\phi_{k_{1}}(t)\rangle\sim
e^{ikt}\to\langle\phi_{k_{1}}(t)a^{\dagger}_{k}(-\infty)\rangle\sim e^{-ikt}$,
where the physical meaning changes from creation at the vertex to
annihilation. This new correlator evaluates to
$-i\lambda\delta^{(4)}(k+p+p^{\prime})$, which ofcourse cannot be physically
reached, but has the same energy-momentum dependence as the Bunch Davies in-in
correlator where we would see a $1/(k+p+p^{\prime})$ pole for mode insertions
with momenta $k,p,p^{\prime}$. We see clearly the role of the presence of an
out-state in reaching physical singularities, and the presence of unphysical
total energy singularities in the presence of only in states. The relationship
between time ordering and the presence of out states can also be seen by
noting that $\langle a_{k}(\infty)\phi(t)\rangle$ doesn’t vanish because of
how time ordering places the $a$ so that it acts on the bra. If we had an in-
state on the left, this would be replaced by $\langle
a_{k}(-\infty)\phi(t)\rangle$ which would be killed by time ordering as it
would take $a$ to the right and annihilate the vacuum. Thankfully, this such a
correlator wouldn’t be time ordered anyway and therefore we’d be forced to use
Wightman functions instead of the Feynman propagator, and they don’t vanish.

Now, consider a correlation function with a highly occupied state $N$, of the
form $\int d^{4}x\langle N|T\phi\phi\phi|N\rangle$. By construction, this
correlator is a physical process with external states put on shell. Therefore,
integrating over the vertex will produce physical poles depending on which
mode of the initial state we use to contract with the operators inside the
correlator. We can pick out 2 $k_{i}^{\prime}s$ from the in state and one from
the out state and get a delta function that conserves the in-out momentum,
which in the in-in case would produce a physical pole. In the language of flat
space Feynman rules, these physical delta functions arise because of the fact
that contractions with in and out states induce exponentials with different
signs of momenta-indeed this is due to the implicit LSZ that has been carried
out already where we know that the exponentials for in/out states are
hermitian conjugates of each other.

This is the state of affairs. In the presence of particles in the external
states, the possibility of physical pole arises because they may either
annihilate or be created at a vertex. In an in-out correlator, the relative
sign between the time of insertions for in and out states(which is in practice
consistently accounted for via LSZ) between these processes reflects in
physical singularities. In an in-on correlator, operationally, this reflects
in the fact that $\langle a\phi_{p}(t)\rangle\sim e^{ikt}$ whereas
$\langle\phi_{p}(t)a^{\dagger}\rangle\sim e^{-ikt}$. We will see that physical
poles will come from contractions schematically of the type

$\langle...p..|\zeta_{1}\zeta_{2}\zeta_{3}|...p..\rangle\supset\langle
p\zeta_{1}\rangle\langle\zeta_{2}\zeta_{3}p\rangle$ (108)

where we indicate the terms which contract together at a vertex(the second
term), and the term that spectates(the first term). In the absence of
particles in the initial state, or the inability to perform LSZ; correlation
functions will only depend on the Wightman functions between field insertions,
and in time dependent backgrounds it is possible-in fact necessary- to have
Wightman functions that include both modes(as opposed to the Feynman
propagator which picks only one based on time ordering) and therefore produce
physical singularities through their interference terms, despite the absence
of features like time ordering and LSZ in in-in correlators. In curved space,
we will use the trick of performing calculations in a frame where there are no
particles, but the Wightman functions include both modes and consequently help
achieve the same physical singularities as one would expect in the presence of
particles.

The occurrence of physical poles in cases where this can’t be done and
therefore initial state particles interacting at a vertex cannot be avoided is
discussed in 7. With particles in the initial state, an in-in correlator
receives contributions from terms where one of the initial state particles is
contracted with the (cubic)vertex insertion along with 2 of the field
insertion, while the third spectates. This contraction generates a physical
pole at the right momentum support for the spectator. In section 3.5 the
occurrence of this pole in the presence of a single particle in the initial
state was studied in detail.

##  5 Choice of interaction and their imprint on singularities

The role of singularities of the S-matrix in flat space has a clear physical
meaning and origin. The role of poles and branch cuts are easy to interpret
from their physical meaning, which lies in the Kallen-Lehmann decomposition of
correlators. The nature of singularities is determined by the nature of
interactions considered and the kind of processes they involve(for instance,
tree level exchanges generates simple poles whereas loops are generally
accompanied by branch cuts). Correlation functions are expected to be
constrained by Lorentz invariance, and in flat space for example we have delta
function singularities indicating Poincare invariance that conserves momenta.
In deSitter, the correlation functions are similarly constrained by the
deSitter isometries which are conveniently those of an Euclidean CFT
Creminelli:2011mw ; Kehagias:2012pd ; Antoniadis:2011ib . We will, for
completeness, briefly discuss the nature of singularities with different
interactions in deSitter. The point is to emphasize that while the functional
form of singularities can change depending on the choice of interaction, the
object we are trying to probe is the momentum structure encoded in the
singularities, which carries a clear physical signature of the choice of
quantization. Let us begin by noting that far outside the horizon, the
behaviour of a scalar field $\phi$ that has a very small coupling to the
inflaton is of the form

$\phi\sim\tau^{\Delta}\hskip
14.22636pt\Delta\equiv\frac{3}{2}\bigg{(}1-\sqrt{1-\frac{4m^{2}}{9H^{2}}}\bigg{)}$
(109)

In this section we will focus on the case of Bunch Davies modes in a Bunch
Davies initial state. Next, note that the correlators must be functions of the
de Sitter invariant distance-

$d(x_{i},x_{j})=\frac{|\vec{x}_{i}-\vec{x}_{j}|^{2}}{\tau_{i}\tau_{j}}-\bigg{(}\frac{\tau_{i}}{\tau_{j}}+\frac{\tau_{j}}{\tau_{i}}\bigg{)}\to\frac{|\vec{x}_{i}-\vec{x}_{j}|^{2}}{\tau_{i}\tau_{j}}|_{\tau\to
0}$ (110)

Note that, in particular, this means that the spatial dependence of the
correlation function is specified automatically once the time dependence of
the field(which is related to its mass) is specified at late times
Creminelli:2011mw . The Fourier transform of this correlator contains the
momentum dependence we are interested in, and this tells us that once the late
time behaviour of the field is specified, the momentum dependence is specified
too. Let us see how this works with the simple example of an interaction
$\sqrt{-g}\lambda\phi^{3}/6$. For a general $\Delta$, the time dependence of
the 3 point function will be fixed as
$\sim\tau_{1}^{\Delta}\tau_{2}^{\Delta}\tau_{3}^{\Delta}$, and therefore a
generic position space function can only take the form

$\langle\phi_{1}\phi_{2}\phi_{3}\rangle\sim\prod\frac{\tau_{i}^{\Delta}}{|\vec{x}_{i}-\vec{x}_{j}|^{\Delta}}$
(111)

Let us see what happens for different values of $\Delta$. For a field that has
mass $m=\sqrt{2}H$ we have a conformally coupled scalar with $\Delta=1$. The
Fourier transform of $\prod 1/|\vec{x}_{i}-\vec{x}_{j}|$ goes as
$1/(k_{1}k_{2}k_{3})$, and indeed in momentum space we find with a full in-in
calculation, upto a 3-momentum conserving delta function

$\langle\phi_{k_{1}}\phi_{k_{2}}\phi_{k_{3}}\rangle=\frac{\pi\lambda
H^{2}}{8}\tau_{*}^{3}\frac{1}{k_{1}k_{2}k_{3}}$ (112)

where $\tau_{i}=\tau_{*}$ is imposed for an equal time correlator. Note that
the result is dependent on $\tau_{*}$. To study the limit $\Delta\to 0$, which
is more subtle because now the late time behaviour of fields isn’t a simple
power law, one can use the fact that $\lim_{\Delta\to 0}\tau^{\Delta}\sim
1+\Delta\log\tau$ and anticipate a logarithmic dependence on the late time,
and therefore, on momenta. This is to be expected, since the field has no mass
of it’s own and therefore the only kinematic quantity available to make the
logarithm dimensionless is a linear combination of momenta. The classical
scale invariance of the massless theory is broken when one imposes a
regulator(in the form of a finite cutoff $\tau_{*}$), and this breaking is
captured in the logarithmic piece. Maldacena’s consistency requirement
directly relates this nongaussianity to a breaking of scale invariance, as we
shall later see. It is important to note that the massless case is subtle,
because the limits $\tau\to 0,\Delta\to 0$ don’t commute, and a naively
$\tau$-independent field outside the horizon acquires a logarithmic time
dependence due to the cubic interaction. This logarithmic dependence on time
then fixes the momentum dependence because of the nature of dS invariant
distances. The in-in calculation in momentum space yields, as expected

$\langle\phi_{k_{1}}\phi_{k_{2}}\phi_{k_{3}}\rangle=\frac{H^{2}\lambda}{12(k_{1}k_{2}k_{3})^{3}}\bigg{(}k_{1}k_{2}k_{3}-\sum
k_{i}k_{j}^{2}+\sum k_{i}^{3}(-1+\gamma+\log(k_{t}\tau_{*}))\bigg{)}$ (113)

This is where the role of derivative interactions becomes important. For a
massless field, the apparent singularity at $\tau_{*}\to 0$ can be avoided by
considering only derivative interactions. The mathematical origin is simple-
the interaction hamiltonian in conformal time is of the form
$\int\frac{\lambda}{6\tau^{4}H^{4}}\phi^{3}$ which is ill defined at $\tau=0$.
In the above 3 point correlator, the origin of the logarithm is from integrals
of the form $\int e^{ik_{t}\tau}/\tau$ which are singular at the point
$\tau=0$. If we had sufficient powers of $\tau$ in the numerator, then this
problem can be cured. With a purely cubic interaction, the Wightman functions
entering the in-in formula are
$\int\frac{d\tau}{\tau^{4}}\prod_{i}\langle\phi_{i}(0)\phi(\tau)\rangle\sim\int\prod_{i}(\frac{1}{2k_{i}^{3}}(1-ik_{i}\tau)e^{ik_{i}\tau})$
where $\tau$ is at the vertex being integrated over in the interaction
hamiltonian. If, however, we consider the Wightman function for
$\langle\zeta\zeta^{\prime}\rangle\sim\sqrt{k}\tau e^{ik\tau}$, then we see
that the numerator’s factors of $\tau$ can compensate for the ill defined
$\tau\to 0$ behaviour of the Hamiltonian. With an interaction like
$\zeta^{\prime 3}$, the interaction hamiltonian is
$\int\frac{\lambda}{6H\tau}\zeta^{\prime 3}$, and the late time divergence is
softer. Then, the in in calculation will contain integrals of the type
$\int\frac{d\tau}{\tau}\prod_{i}\langle\phi_{i}(0)\phi(\tau)\rangle\sim\int
d\tau\tau^{2}\prod_{i}e^{ik_{i}\tau}$, which has no log divergences. The
singularities are in fact simple poles of the form $1/k_{t}$. In fact, it
suffices to have an interaction of the type $\zeta\zeta^{\prime 2}$-this is
indeed a leading cubic interaction in Hamiltonian, where the $1/\tau^{2}$ in
the denominator of the interaction hamiltonian can get compensated by two
powers of $\tau$ coming from $\zeta\zeta^{\prime}$ contractions; the remaining
$\zeta\zeta$ contraction is free of any $1/\tau$ divergences.

Thus, we see that considering derivative interactions softens the behaviour of
the $\tau\to 0$ limit of the correlators, and one needn’t conduct any explicit
regularization. We see that one needs a minimum of two derivatives in the
interaction to get rid of this regulator requirement, and for simplicity we
consider a $\zeta^{\prime 3}$ interaction.

Now, the crucial point we wish to emphasize. While the nature of singularities
might be different in their functional forms (logarithmic, simple poles, or a
combination); the momentum dependence of these singularities for the Bunch
Davies case is always $(k_{1}+k_{2}+k_{3})$. What we aim to understand is-
having fixed an interaction and therefore the functional form of the
singularity- does the nature of this momentum dependence change once we allow
different initial states and/or mode functions? The answer, as we see, is
quite generically yes.

##  6 Comments on the Maldacena consistency criteria

The Maldacena consistency criteria is a consequence of frequency-dependent
freezing times of various modes. Consider the three point function in the
squeezed limit $k_{1}<<k_{2},k_{3}$. In this case, the mode $\zeta_{k_{1}}$
freezes in (becomes a classical variable) before the modes
$\zeta_{k_{2},k_{3}}$ have crossed the horizon. It then serves as a background
over which the latter modes propagate. In effect, the 3 point function reduces
to

$\langle\zeta_{2}\zeta_{3}\rangle_{\zeta_{1}}\approx\langle\zeta_{1}\zeta_{2}\rangle_{0}+\zeta_{1}\frac{\partial}{\partial_{\zeta_{1}}}\langle\zeta_{1}\zeta_{2}\rangle_{0}$
(114)

The fact that $\zeta_{1}$ has frozen in can be accounted for by a rescaling of
coordinates-in particular-

$ds^{2}=dt^{2}-a^{2}e^{2\zeta_{1}}(dx^{2}+dy^{2}+dz^{2})\implies
x\to(1+\zeta_{1})x$ (115)

Thus, the squeezed limit can be see to be accounted for by spatial
reparametrizations. Then, in position space one can account for invariance
under spatial reparametrizations as

$\langle\zeta(x)\zeta(0)\rangle_{\zeta_{1}}=\langle\zeta(x+\zeta_{1}x)\zeta(0)\rangle_{0}$
(116)

We can now Taylor expand the RHS too, to get a term of the form
$\zeta_{1}x\frac{d}{dx}\langle\zeta\zeta\rangle$ and then multiply both sides
with $\zeta_{1}$ before averaging. Converting to momentum space, this yields

$\lim_{k_{1}<<k_{2},k_{3}}\langle\zeta_{1}\zeta_{2}\zeta_{3}\rangle\approx\langle\zeta_{k_{1}}\zeta_{-k_{1}}\rangle
k\frac{d}{dk}\langle\zeta_{k_{2}}\zeta_{k_{3}}\rangle$ (117)

Now, for a nearly scale invariant power spectrum,
$\langle\zeta\zeta\rangle\sim k^{-3+n_{s}}$, and therefore we derive a
consistency equation

$\lim_{k_{1}<<k_{2},k_{3}}\langle\zeta_{1}\zeta_{2}\zeta_{3}\rangle=-n_{s}\langle\zeta_{k_{1}}\zeta_{-k_{1}}\rangle\langle\zeta_{k_{2}}\zeta_{k_{3}}\rangle$
(118)

Note the crucial step where we absorbed the background field into the metric,
therefore finding ourselves a coordinate system where the power spectrum is
insensitive to the background waveCreminelli:2004yq . This step was valid
because at leading order, $\zeta$ interactions come from the nonlinearities of
the metric itself, without any reference to the features of the potential.
However, the interaction we have considered comes from a potential with
derivative interactions $\dot{\phi}^{4}$. It is known Creminelli:2003iq that
nongaussianities from such interactions are suppressed by
$\frac{k_{L}^{2}}{k_{S}^{2}}$ in the squeezed limit, where $S,L$ are the large
and small momenta respectively( the subscripts are conventionally chosen to
denote short and large distances). The basic idea is that in the squeezed
limit, the relevant interactions are those where one of the insertions has no
derivatives, and this is the insertion that’s chosen to be the short
wavelength mode. Explicitly, the relevant action at cubic order is
Creminelli:2011rh

$S=\int d^{4}x\epsilon a^{3}\bigg{(}(1+3\bar{\zeta})\zeta^{\prime
2}-(1+\bar{\zeta})(\partial_{i}\zeta)^{2}\bigg{)}$ (119)

where the insertion that’s taken to be the squeezed wavelength has been made
explicit. Any corrections to this will therefore be from derivative
corrections, and it was argued in Creminelli:2011rh that the corrections are
atleast quadratic in $(k_{L}/k_{S})$. There are many ways to see this,
starting from nothing that the power spectrum at late time recieves genuine
corrections only from second derivatives, because
$\langle\zeta_{1}\zeta_{2}\rangle=P(k_{1})(1+k^{2}\tau^{2})$ and therefore if
one was to organise this expression as a power series in $k\tau$, the leading
correction at $\tau\to 0$ would come from $\partial_{\tau}^{2}$, not before.
Yet another way to see this is to note that schematically, the Wightman
functions $\langle\zeta\zeta\rangle$ and $\langle\zeta\zeta^{\prime}\rangle$
differ by a factor of $k^{2}\tau$ and therefore in the in-in formalism, we
replace the former with the latter(as is what happens if we switch from
inflation’s leading interaction to our), we get additional powers of
$(k_{L}/k_{S})^{2}$. On more general grounds, focusing on the momentum
dependence and ignoring the prefactors, the bispectrum is always of the form

$\langle\zeta_{1}\zeta_{2}\zeta_{3}\rangle\sim\frac{1}{(k_{1}k_{2}k_{3})^{3}}f(k)$
(120)

And in the squeezed limit where we denote $k_{1}\to k_{L},k_{2,3}\to k_{S}$,
we find that in the case where $\zeta$ is only a (scalar) mode of the metric,
$f(k)\to k_{L}^{3}$ therefore restoring the consistency criteria. However,
with an interaction like ours, $f(k)$ includes term of the nature
$k_{L}^{2}k_{S}$ because now the short wavelength can be put onto any of the
three insertions in the interaction hamiltonian without being ignored unless
we wish to kill the entire contribution. This produces the required
$(k_{L}/k_{S})^{2}$ suppression. As has been emphasized in Creminelli:2011rh ,
$f(k)\to 0$ as $k_{L}^{2}$, not $k_{L}$. On the other hand, without higher
derivative features in the potential, $f(k)$ is a constant in $k_{L}$. For an
explicit example, consider our eq(9), where the bispectrum is

$\Delta\langle\zeta_{1}\zeta_{2}\zeta_{3}\rangle\sim\frac{1}{(k_{1}k_{2}k_{3})(k_{1}+k_{2}+k_{3})^{3}}\to
P(k_{s})P(k_{L})\bigg{(}\frac{k_{L}}{k_{S}}\bigg{)}^{2}$ (121)

And therefore we see that the consistency condition is violated by an amount,
as has been argued on general grounds,
$\left(1+\bigg{(}\frac{k_{L}}{k_{S}}\bigg{)}^{2}\right)$.

The behaviour of $f(k)$ is more interesting in the case we have physical
poles. The general conclusion is that this time, the behaviour is modified by
a reciprocal quantity-$k_{S}/k_{L}$ which is severely enhanced in the squeezed
limit. In the squeezed limit, for all cases other than choosing purely BD
initial conditions, the consistency criteria fails. The reason is the presence
of physical poles, which generate terms proportional to $k_{1}/k_{3}$ which
swamp the $\vec{k}_{3}\to 0$ limit and spoil the consistency. The spoiling
depends on the angle between the momenta, because we have a constraint
$\sum\vec{k}_{i}=0$ from which we can study the poles in terms of the
magnitudes $k_{i}$ as well as the angles. It is clear that such a structure
wouldn’t occur if we had only physical poles, because then all 3 momenta are
on an equal footing inside the correlator and setting any one to zero can only
generate $k_{1}+k_{2}$ poles, not a $k_{1}-k_{2}$ pole that can potentially
compete with the $k_{3}\to 0$ pole and therefore require to be kept intact
before taking the limit. Since the consistency condititon is linked to spatial
reparametrization invariance, it is tempting to think that OPE, gauge choice
and the BD initial conditions are all intimately linked. In fact, it was
argued in Shukla:2016bnu how the reparametrization invariance is a feature of
Einstein’s equations, and with a non-BD vacuum, we must account for the effect
of it’s contributions to the stress energy tensor’s vacuum expectation value.
Neglecting this contribution means that we have not fully accounted for the
modification to the ADM action-the piece which actually solves Einstein’s
equations. Therefore, we are working with an action that doesn’t yield the
correct equations of motion, and therefore we have lost the freedom to carry
out spatial reparametrizations, and consequently the consistency condition.

We therefore see that while the squeezed limit doesn’t directly contain
information about the analytic structure of the bispectrum because the modes
have been in a sense fixed; it does carry information about the initial states
and the vacuum of the modes in the form of departures from Maldacena
consistency. In particular, a $(k_{L}/k_{S})^{2}$ departure is a signature of
a Bunch Davies initial state with higher order interactions, whereas
everything else produces a large deviation in the form of $k_{S}/k_{L}$ poles.

##  7 Classical vs Quantum

The authors of Green:2020whw ; Green:2022fwg argued that physical poles are a
reflection of a classical initial state. At the level of calculation, it
amounted to using the classical Green’s function,

$G_{c}(0,t)=\frac{-i}{2k}(e^{ik\tau}-e^{-ik\tau})$ (122)

which contains both positive and negative frequency modes, and therefore the
product of these Green’s functions generates poles with varying momentum
dependence depending on the interference terms between the modes. However, the
Wightman function for purely Bunch Davies modes contains only one of these
modes, whereas the one for Bugolyubov transformed states(for example the alpha
vacuum) contains both-

$G_{W}^{BD}(0,\tau)\sim
u_{k}(0)u_{k}^{*}(\tau)e^{ik\tau},G_{W}^{\alpha}(0,\tau)\sim
v_{k}(0)v_{k}^{*}(\tau)=v_{k}(0)(A^{*}u_{k}^{*}(\tau)e^{ik\tau}+B^{*}u_{k}(\tau)e^{-ik\tau})$
(123)

Therefore, we clearly see the link between the appearance of mode mixing in
the case of classical as well as Bogolyubov transformed states, and the
subsequent appearance of physical poles. What about the case of excited
states? The presence of particles in the initial state means that there are
additional possible interactions at a vertex, where some of the field
insertions annihilate not just each other but also these initial state
particles at a vertex. This is drastically different from the case of a vacuum
initial state where only the field insertions annihilate(or create) among
themselves at a vertex, and the only kinematic constraint then is conservation
of their spatial momenta. The next crucial difference is the interplay of
energies(stacked in the exponentials in the mode functions) at the vertex. In
the cases where the initial state particles don’t interact at a vertex but
merely spectate, the vertex represents a creation of particles from the
vacuum, as in the figure (cite figure). Since the vertex factor for creation
at a vertex is accompanied by the mode $e^{ik\tau}$, the creation of three
particles comes with a $\tau-$integral over $e^{i(k_{1}+k_{2}+k_{3})\tau}$,
and produces a total $k$ pole as we have seen. However, the vertex factor that
accompanies an annihilation is quite different, and has the opposite energy as
the mode function for creation. If a particle in the initial state with
momentum $P$ annihilated at a vertex to produce 2 particles of momenta
$k_{1},k_{2}$; then we would find the prefactor at the vertex to be
$e^{i(k_{1}+k_{2}-P)\tau}$, along with a momentum conserving delta function
for the remaining particles which is just an exchange of momentum, and not a
genuine interaction. Together, they produce a physical pole, and it’s origin
is clear-it’s the relative sign of momenta in the mode functions corresponding
to annihilation vs creation. However, because of these same momentum
conserving delta functions, these physical poles have a very restricted
support and will therefore not be observable except at isolated points in
phase space.

An obvious way to get rid of the isolated nature of these singularities is to
average over all possible momenta of initial state particles-this smears out
the phase space where the poles have support. Crucially, this cannot be
accomplished with a state where a only fixed number of momentum states are
occupied, no matter how highly. That is to say, states of the type
$\otimes_{k}(a^{\dagger}_{k})^{n}|0\rangle$ cannot display observable physical
poles unless we consider a sum over all possible momenta at each order in $n$.
Schematically, a correlator like $\langle
P|\zeta_{1}\zeta_{2}\zeta_{3}|P\rangle$ won’t generate observable physical
poles, but a correlator like $\int d^{3}P\langle
P|\zeta_{1}\zeta_{2}\zeta_{3}|P\rangle$ will. Indeed, this is what we saw for
states excited with respect to the Bunch Davies vacuum. As we introduce more
and more particles, we must accept the possibility of more and more possible
disconnected pieces and consequently isolated poles at a given order in
perturbation theory, and it seems unlikely that we will ever be rid of this
problem. However, for some excited states (relative to a Bunch Davies
observer), it is possible to go to a frame where this state is a vacuum, and
there are no particles in the initial state to interact at a vertex. These
states are precisely the Bugolyubov transforms of the Bunch Davies vacuum,
which are excited for a Bunch Davies observer but have zero net momentum. If
we compute these correlators in a frame where the particles actually exist, we
see(3.3) indeed the presence of physical poles corresponding to physical
interactions of initial state particles at a vertex, but it is difficult to
find a clear interpretation due to additional disconnected pieces that must
exist for reasons mentioned above. Instead, calculating them using the
accelerated observer’s basis ensures that these particles are not seen, but
their physics is now captured in the modified mode functions-specifically, we
must now also include negative frequency modes that otherwise appear due to
annihilation of particles at a vertex. The physics of annihilation at a vertex
is now absorbed into the physics of the new mode functions, and the
mathematical correspondence is clear. Figure 3 demonstrates this discussion.
(Here, by ’create/annihilate’ we simply mean the way in which they contract-
either with an $a_{p}$ or with $a^{\dagger}_{p}$).

To contrast the situation with flat space time ordered in-out correlators,
recall that the frequency of the mode functions there is fixed via LSZ, and
time ordering picks the right frequencies in the Feynman propagators. In
contrast, for in-in correlators, both the classical Green’s function and the
quantum Wightman functions for Bugolyubov states contain both signs of
frequencies. Even in flat space, the Wightman function can have either
frequency depending on the time order of the field insertions
$\langle\phi_{k}(t)\phi_{k^{\prime}}(0)\rangle\sim\delta(\vec{k}+\vec{k^{\prime}})e^{-ipt}/2k=\langle\phi_{k}(0)\phi_{k^{\prime}}(t)\rangle^{*}$;
where the complex conjugation is equivalent to a negative frequency. It is
precisely time ordering that eventually tells us which frequency to choose.
The lack of this luxury with in-in correlators is what is ultimately tied to
physical poles.

Figure 3: (Left) the case of classical or accelerated observer’s mixed mode
functions, (Right) the case of genuine interaction of initial state particle.
The $\pm$ sign is determined by whether the particle is created/annihilated at
the vertex. The latter generates physical poles. The red dashed line
represents a spectator.

##  8 Choosing a cutoff

While carrying out the integrals in the in-in correlators, the lower limit
$\tau\to-\infty$ must be taken with care Chen_2007 ; Chen_2010 . Generally,
the exponential factor at the lower limit is taken to vanish because of it’s
oscillatory nature Ganc:2011dy . The general idea is that Holman:2007na the
initial conditions need to be set when the EFT of inflation can still be
trusted, and the Bunch Davies prescription is singled out by setting them at
$\tau\to-\infty$. However, assuming that there is some cutoff scale $\Lambda$
upto which our EFT can be trusted, then the validity of this EFT is reached at
some $k\sim a(\tau_{0})\Lambda$ where $k$ is a physical momentum scale today-
note that inflation redshifts energies. Scales that were once in the UV become
part of the IR due to the universe’s expansion. Setting a cutoff at a finite
$\tau_{0}$ automatically takes us away from the Bunch Davies prescription, and
we need to introduce more general initial conditions, which yields modes which
are Bogolyubov transformations of Bunch Davies. We would like to understand if
taking the lower limit $\tau\to-\infty$ in our calculations hides any
information about the physical poles seen by an accelerating observer-in
particular, if the singular behaviour at physical energies is still seen. If
we instead keep a cutoff intact, then we see that the integrals containing
physical poles are modified to($k^{\prime}_{j}\equiv k_{t}-2k_{j}$)

$\textrm{Im}\int_{\tau_{0}}^{0}d\tau\tau^{2}e^{ik^{\prime}_{j}\tau}=\textrm{Im}\frac{i(2+e^{ik^{\prime}_{j}\tau_{0}}({k^{\prime}}_{j}^{2}\tau_{0}^{2}+2ik^{\prime}_{j}\tau_{0}-2))}{{k^{\prime}}_{j}^{3}}\hskip
8.53581pt\textrm{(with $\zeta^{\prime 3}$ interaction)}\\\ \\\
=\frac{2}{{k^{\prime}}_{j}^{3}}(1-\cos(k^{\prime}_{j}\tau_{0}))+\frac{\tau_{0}^{2}\cos(k^{\prime}_{j}\tau_{0})}{k^{\prime}_{j}}-2\frac{\tau_{0}\sin(k^{\prime}_{j}\tau_{0})}{{k^{\prime}}_{j}^{2}}$
(124)

We need to check the leading singular behaviour as $k^{\prime}_{j}\to 0$.
Expanding the numerators for small $k^{\prime}_{j}$(but finite $\tau_{0}$) we
find

$\textrm{Im}\int_{\tau_{0}}^{0}d\tau\tau^{2}e^{ik^{\prime}_{j}\tau}|_{k^{\prime}_{j}\to
0}=\frac{2\tau_{0}^{2}}{k^{\prime}_{j}}-\frac{2\tau_{0}^{2}}{k^{\prime}_{j}}+\textrm{terms
regular in $(k^{\prime}_{j}\tau_{0})$}$ (125)

From this expression, it is clear that the limits $k^{\prime}_{j}\to 0$ and
$\tau_{0}\to-\infty$ don’t commute, and in fact taking $\tau_{0}\to-\infty$
before going to the folded configuration $k^{\prime}_{j}\to 0$ displays a
divergence(from the terms regular in $\tau_{0}$ which blow up in this limit),
whereas the opposite order of limits is finite. This transparently
demonstrates that the divergence in the folded limit is an artifact of the
initial conditions being set at $\tau_{0}=-\infty$. The absence of a
divergence can directly be seen by plotting (LABEL:7.1) as a function of
$k^{\prime}_{j}$.

Figure 4: Plot of $f(k^{\prime}_{j})$(see (LABEL:7.1)) as a function of
$k^{\prime}_{j}$. This shows that there is no singularity in the folded limit.
In addition to this there is an oscillatory feature.

Therefore, clearly there is no singularity in the folded limit. This can be
seen directly from Fig 4 where we have plotted (LABEL:7.1) as a function of
$k^{\prime}_{j}$. It has been noted earlier too, for example in Chen_2007 ,
that the divergence in the folded limit can be cured by introducing an early
time cutoff. In Holman:2007na , a similar phenomenon was shown for the
interaction of inflation, where the presence of a cutoff rendered the folded
limit finite but regular in the cutoff, and would therefore be subject to the
same conclusion as ours. For an initial state that is a Bogolyubov transform
of the Bunch Davies vacuum, setting the initial conditions away from $-\infty$
is perhaps reasonable because one can argue that a Bogolyubov transformed
operator basis at $\tau_{0}$ can be thought to be time evolved from a Bunch
Davies basis which was imposed at $\tau\to-\infty$ . However, there is no
first-principles argument for this, and there are other excited states where
this reasoning wouldn’t be straightforward.

In the literature, one will find plentiful examples of both cases - where
initial conditions are either set at $-\infty$ or at a finite cutoff. Since
the primary motive of our analysis is to study and compare the pole structure
of correlators, and trace the mathematical origin of physical poles in quantum
and classical correlators, we will not dwell on this issue further, and treat
our initial conditions set at $-\infty$.

##  9 Conclusions

Our results can be summarized as follows (BD: Bunch Davies, BG-Bogolyubov
transform)

Initial state | Pole structure
---|---
BD vacuum |  $k_{t}$ pole
Excited over BD with finite modes present | disconnected physical poles, connected $k_{t}$ pole
Excited over BD with all modes present | connected $k_{t}$ pole, observable physical pole
Excited over BD but a BG | both physical and $k_{t}$ poles
Excited over BD but coherent |  $k_{t}$ pole
Excited over BD but coherent for some BG | both $k_{t}$ and physical poles
Classical | both $k_{t}$ and physical poles

We find that despite the rich variety of possible initial states and vacua
choice for the modes, the analytic structures of correlators - in particular
the bispectrum - is largely of two types. The case where the initial state is
Bunch Davies has a characteristic total energy pole, whereas all other
possibilities exhibit in addition physical poles. An interesting exception is
that of Coherent states (under the assumption of vanishing one point
function), which are excited states but when built with Bunch Davies
particles, don’t display any physical poles. These poles are restored for
Coherent states relative to Bugolyubov transformed observers. In fact, it
makes more sense to note that vacua are just special(trivial) cases of
coherent states, and therefore the pole structure obtained with a coherent
state is expected to be reproduced for vacua. Departures from the Bunch-Davies
pole structure are also encoded in the sharp $k_{S}/k_{L}$ violation of the
Maldacena consistency criteria(where $k_{L}$ is the small momentum) in the
presence of physical poles, and therefore the squeezed limit of the bispectrum
is a useful observational tool to characterise the initial state. It is
tempting to interpret these as physical scattering/decay processes in the
initial state, as was hinted in Green:2020whw ; but we show that this
interpretation is unclear. For one, coherent states with vanishing one point
functions don’t display any physical poles, despite containing excitations. On
the other hand, an alpha vacuum that serves as the initial state for the modes
as well as inflation exhibits physical poles which cannot be ascribed to
alpha-particles. One could in principle circumvent this apparent discrepancy
by demanding that by particles one means only BD particles, but this is
against the spirit of general covariance and the well known fact that such an
interpretation is not unique. We see that initial state effects on the
bispectrum lead to enhancements in certain momentum configurations; a fact
that has been noted before in the literature. We have attempted to unearth the
connection between the mathematical machinery that leads to physical poles in
a classical calculation vs an in-in calculation, as well as the corresponding
intuition from flat space. We have also made clear the role of derivative
interactions in generating singularities which are simple poles. We believe
making these associations transparent clarifies much of our understanding
about correlation functions in deSitter and the origin of their singularities.

## Acknowledgments

We would like to thank Aleksandr Azatov, Paolo Creminelli, and Sachin Jain for
valuable feedback and discussions. We thank Sounak Sinha for initial
collaboration in the project. DG acknowledges support through the Ramanujan
Fellowship and MATRICS Grant of the Department of Science and Technology,
Government of India.

## Appendix

##  Appendix A Perturbations and perturbation theory

The inflationary background is a quasi-de Sitter spacetime with deviations
from pure Sitter being slow-roll suppressed. In this paper we consider a pure
de Sitter background and analyse correlations of the adiabatic fluctuations
$\zeta$ about the background. Before embarking on calculating correlation
functions of the perturbations, It is essential to clarify the meaning of
perturbations in a general curved background. Since, we are free to perform
arbitrary coordinate transformations we can remove or introduce fictitious
perturbations which makes cosmological perturbation theory
subtleBardeen:1980kt . Consider a homogeneous function $\rho(t)$ which has no
spatial dependence. Now, perform the following coordinate transformation

$x^{\mu}\rightarrow x^{\prime\mu}=x^{\mu}+\epsilon^{\mu}(x)$ (126)

Clearly in the primed coordinate system the function $\rho$ now also has
spatial dependence. The fact is that one has to take into account full set of
perturbations i.e field perturbations + metric perturbations and one can push
the perturbations entirely into the metric or the field configuration but
cannot get rid of them completely.

The field/metric perturbations are defined as follows

$\delta Q(x)=Q(x)-\overline{Q(t)}$ (127)

where $Q(x)$ is any function in real-physical spacetime and $\overline{Q(t)}$
is a background function we associate to $Q(x)$. This is the so called passive
approach to perturbation theory as described inMukhanov:1990me . Here, $Q$ can
represent a scalar, vector or even a tensor quantity. The background function
is supposed to represent the evolution of the unperturbed homogeneous universe
and is fixed i.e after a coordinate transformation
$\overline{Q(t)}\rightarrow\overline{Q(t^{\prime})}$, the function does not
change! Since, the background is time-dependent even the scalar perturbations
are not gauge-invariant. For concreteness let us work out transformation law
for scalar perturbations.

$\delta Q(x)=Q(x)-\overline{Q(t)}$ (128)

Now perform the infinitesimal transformation given in (126) and we have

$\delta
Q^{\prime}(x^{\prime})=Q^{\prime}(x^{\prime})-\overline{Q(t^{\prime})}$ (129)

Notice, that the background function does not change as mentioned above. On
the RHS we can replace $Q^{\prime}(x^{\prime})$ by $Q(x)$ since it is a scalar
function and we have

$\displaystyle\delta
Q^{\prime}(x^{\prime})=Q(x)-\overline{Q(t)}-\epsilon^{0}\dot{\overline{Q(t)}}$
(130)

The dot represents time derivative. Now, we can combine the first and the
second term on the RHS giving the perturbation in the unprimed coordinates and
we get our final expression

$\delta Q^{\prime}(x^{\prime})=\delta Q(x)-\epsilon^{0}\dot{\overline{Q(t)}}$
(131)

It is very clear (as mentioned above) that if the background is not time
dependent then the perturbations are gauge invariant.

Since, the perturbations are gauge dependent, It is useful to work with Gauge-
Invariant variables instead. One of most used variable is the co-moving
curvature perturbation denoted by $\zeta$, which equals the curvature
perturbation in the co-moving gauge defined below.

$\delta\phi=0,\hskip
56.9055ptg_{ij}=a^{2}\left(\left(1+2\zeta\right)\eta_{ij}+\gamma_{ij}\right),\hskip
56.9055pt\partial_{i}\gamma_{ij},\hskip 56.9055pt\gamma_{ii}=0$ (132)

where $a$ is the scale factor. Notice, in this gauge the scalar perturbation
is completely absorbed in the metric as $\zeta$, defined as

$\zeta=\Psi+H\frac{\delta\phi}{\dot{\phi}}$ (133)

by construction $\zeta$ is the curvature perturbation $\Psi$ in the above
defined co-moving gauge i.e $\zeta=\Psi_{|\delta\phi=0}$. It is easy to check
that the quantities on the right hand side of (133) can be computed in any
gauge. We will perform all our calculations using $\zeta$ and for simplicity
we will assume $\frac{\lambda}{3!}\dot{\zeta}^{3}$ interaction(see section 5)
to facilitate comparison with the results of Green:2020whw throughout.

To transition to a quantum theory of these perturbations, one needs to define
a vacuum state and the accompanying basis of operators. We will largely be
concerned with 2 kinds of vacuum states-the Bunch Davies vacuum, and the
$\alpha$-vacuum. Perturbation theory in alpha vacuum is subtle, for reasons
including but not limited to the difficulties in regularization of the energy
momentum tensor. The most common choice for a dS invariant background is the
BD prescription for modes, and when we consider excited states, we will
introduce them as excited for a Bunch Davies observer. We will then see what
happens when there exist observers(related by a Bogolyubov transform) whose
vacuum happens to coincide with some of these excited states-in particular the
$\alpha-$vacuum. We will work with a derivative interaction as it is
mathematically simple, and despite being an irrelevant operator that is
ideally suppressed by some UV energy scale, it’s imprints on nongaussianities
is important-in fact, a theory with these interactions shows greater NG
enhancement than a theory without them Holman:2007na . This is because these
interactions describe the IR behaviour of a high energy theory today, and the
scales accessible at low energies today must have been in the UV in the far
past-this is a direct consequence of the fact that in dS irrelevant