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Published for SISSA by Springer

Received: May 3, 2024
Accepted: October 3, 2024

Published: October 31, 2024

The open effective field theory of inflation

Santiago Agüí Salcedo , Thomas Colas and Enrico Pajer

Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, U.K.

E-mail: sa2013@cam.ac.uk, tc683@cam.ac.uk, enrico.pajer@gmail.com

Abstract: In our quest to understand the generation of cosmological perturbations, we face
two serious obstacles: we do not have direct information about the environment experienced
by primordial perturbations during inflation, and our observables are practically limited
to correlators of massless fields, heavier fields and derivatives decaying exponentially in
the number of e-foldings. The flexible and general framework of open systems has been
developed precisely to face similar challenges. Building on previous work, we develop a
Schwinger-Keldysh path integral description for an open effective field theory of inflation,
describing the possibly dissipative and non-unitary evolution of the Goldstone boson of time
translations interacting with an unspecified environment, under the key assumption of locality
in space and time. Working in the decoupling limit, we study the linear and interacting
theory in de Sitter and derive predictions for the power spectrum and bispectrum that depend
on a finite number of effective couplings organised in a derivative expansion. The smoking
gun of interactions with the environment is an enhanced but finite bispectrum close to the
folded kinematical limit. We demonstrate the generality of our approach by matching our
open effective theory to an explicit model. Our construction provides a standard model to
simultaneously study phenomenological predictions as well as quantum information aspects
of the inflationary dynamics.

Keywords: Cosmological models, Effective Field Theories, de Sitter space, Non-Equilibrium
Field Theory

ArXiv ePrint: 2404.15416

Open Access, © The Authors.
Article funded by SCOAP3. https://doi.org/10.1007/JHEP10(2024)248

https://orcid.org/0009-0003-5390-6367
https://orcid.org/0000-0003-3913-8034
https://orcid.org/0000-0002-7921-4479
mailto:sa2013@cam.ac.uk
mailto:tc683@cam.ac.uk
mailto:enrico.pajer@gmail.com
https://doi.org/10.48550/arXiv.2404.15416
https://doi.org/10.1007/JHEP10(2024)248


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Contents

1 Introduction 1
1.1 Summary of the main results 6

2 The open effective field theory of inflation 11
2.1 EFT construction 12
2.2 Open effective functional 15
2.3 Setting up expectations 23

3 Power spectrum 26
3.1 Generating functional 27
3.2 Flat space intuition 29
3.3 Dissipative and noise-induced power spectrum 31

4 Bispectrum 36
4.1 Interactions 36
4.2 Flat space intuition 37
4.3 Non-Gaussianities from dissipation and noises 42

5 Matching 47
5.1 Power spectrum and bispectrum 49
5.2 Langevin equation 51

6 Discussions 56

A Derivation of Seff constraints 59

B de Sitter Keldysh functions 61

C de Sitter power spectrum with derivative noises 62

D Unitary results in Keldysh basis 65

E Langevin equation with derivative noises 69

1 Introduction

A cornerstone of our cosmological model is that cosmological observations on very large scales
are related approximately linearly to correlators of perturbations at the beginning of the hot
big bang. The initial conditions for cosmological perturbations must hence be determined
by some yet unknown dynamics that took place before our universe reheated into a thermal
bath of standard model particles. As the leading proposal for such unknown dynamics,
inflation combines a prolonged phase of accelerated expansion with the quantum generation
of vacuum fluctuations. This framework for the primordial universe forces us to grapple
with a series of peculiar features. First, we are in practice not able to directly probe time

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evolution, but we only see its final outcome at the end of inflation. Even when focusing on
late time observables, we find further severe restrictions to the quantities that are practically
measurable. Indeed, we can only directly access observables related to fields that are massless
or very light compared to the typical scale of the system, namely the Hubble scale of inflation.
Light fields are not generic, and their existence can typically be traced back to some symmetry
of the theory. The quintessential example is the graviton [1], whose masslessness is tied to
general covariance. Because primordial perturbations have been measured to be scalar in
nature, at least to a few percent precision [2], we know there must be more an additional
massless scalar active during inflation. This can be interpreted as curvature perturbations or
as the Goldstone boson of spontaneously broken time translations [3], whose approximate
masslessness is tied to an internal shift symmetry crucially combined with an approximately
linear background time evolution [4]. Heavier fields must necessarily decay with time, and
their existence can at best be established indirectly, e.g. through their effect on light fields.
The same applies to derivatives of light fields, all of which decay exponentially in the number
of e-foldings of inflation. In turn, this tells us that even for light fields, we are practically
unable to probe correlators involving conjugate momenta.

The fact that we can only probe a restricted part of the whole system, namely the
late-time limit of correlators of light fields, has important implications for the theoretical
framework within which we should model inflation. We can choose to model the whole system,
including all fields that might decay at late time, and use the framework appropriate for a
closed system. Alternatively, it might be advantageous to focus on modelling the subsystem
that we do observe, namely the massless curvature perturbations, and instead work within
the framework of an open system. The dichotomy between open and closed systems manifests
itself in various ways. First, conserved quantities can be precisely tracked and leveraged in
a closed system, but their constraints on the observable dynamics are much more relaxed
for open systems. Second, in an open system it is essential to keep track of the correlation
with the rest of the full system, henceforth referred to as the environment. While these
observations apply also to classical evolution, e.g. in a statistical or stochastic theory, they
are especially important when dealing with a quantum system. The way a quantum system
predicts probabilities is very different in a closed system, where all possible histories are
accounted for, from an open system, where information can be lost into the environment.

This work stems from recognising that a satisfactory and comprehensive understanding of
quantum dynamics during inflation, which necessarily relies on effective models, must begin
in the general and flexible framework of open quantum systems. This seems to be a necessary
starting point both on the theoretical and the observational side. On the observational
side, sizeable environmental effects on curvature perturbations lead to new signals to be
searched for in the data, as we will discuss in the rest of this paper. On the theoretical side,
we need to understand how the flat-space rules of Effective Field Theories (EFTs) and the
Wilsonian paradigm translate to an accelerated spacetime where perturbations are created
out of the vacuum, energy is not conserved, length and time scales are dynamical notions,
and information is redistributed globally.

To be more specific, let us mention some concrete setups. First, let us consider the most
vanilla model of inflation we can think of, with a single scalar field and a gentle slow-roll
dynamics. Even in this case, due to the Hubble expansion, the curvature perturbations we

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measure do not experience the same vacuum as they would in flat space. Instead, they interact
with an approximately thermal bath of field excitations at the de Sitter temperature H/2π.
This bath contains all degrees of freedom in the theory and hence also heavy ones. To have
control over the regime of validity of a proposed EFT and to correctly estimate corrections
to our predictions, one would like to estimate the interactions with the environment and
judge if and when they are negligible. In the simplest case, one expects that excitations of
heavy fields are suppressed by a Boltzmann factor (see e.g. [5, 6]). However, a variety of
possibilities have been considered and devised to counteract this suppression, e.g. through a
large breaking of de Sitter boost invariance [7], a chemical potential [8–12], or a tachyonic
mass [13]. More generally, large classes of models feature some additional mechanism for
producing particles in excess of the thermal de Sitter radiation. An idea that goes back a long
way is that of warm inflation [14–16], where the presence of a radiation bath is ensured by a
continuous source that counteracts the Hubble expansion. In many setups the environment
is populated non-thermally. A much studied example is the tachyonic production of gauge
fields of one chirality induced by a coupling ϕFF̃ to a slowly rolling field ϕ [17]. Here, the
universe is filled with gauge field excitations of wavelength of order Hubble, which are diluted
and redshifted away by the expansion but are also continuously produced. Tens of additional
possibilities have been studied over the years, but we refrain from providing a comprehensive
list. The upshot is that, for both minimal and non-minimal setups, one would like to have a
description of the dynamics of curvature fluctuations that allows for possible interactions
with the omnipresent environment. Given such a description one can then more accurately
specify the regimes in which such interactions matter or can be safely neglected.

As the number of possible environments during inflation is clearly infinite and their
properties are completely model dependent, one might wonder how to make concrete progress
in devising a unified description and in identifying some minimal set of characteristic predic-
tions. To this end, it is natural to employ the machinery of EFTs for open quantum systems.
First, we recognise that EFTs have no magical powers: for a generic setup, there need not
be a clear distinction between system of interest and environment, either because the two
are strongly coupled to each other or because they cannot be cleanly separated in terms
of the charges available in the problem. Rather, it is in the presence of a clear separation,
for example dictated by symmetries, and a hierarchy of scales in the problem that EFTs
become useful as an organising principle. Because of this, we would like to consider here
cases where an unknown environment during inflation can be separated from the sector of
curvature perturbations around Hubble crossing because the former is characterised by faster
evolution, i.e. high frequency, and short distance perturbations, i.e. high momenta. In this
regime, one expects that only a handful of properties of the environment need to be specified
to accurately describe the evolution of curvature perturbations, in contrast to the much more
fine-grained description that would be needed in the study of the whole system including
environment. The handful of properties about the environment are specified via Wilsonian
effective couplings in an open EFT description of the system.

An essential feature of an EFT is that only a finite number of interactions need to be
considered for any given finite desired precision. This finiteness is very often1 ensured by the

1Other possibilities exists. Some non-local EFTs may have different organisational principles, such as the
excitations of Fermi surfaces or in the partial re-summation of certain non-localities.

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assumption that the EFT dynamics is local in space and time. Here we make this assumption
and our discussion and results are contingent on this crucial property. Models where the
environment can mediate interactions around the Hubble scale are beyond the scope of our
description. Notice that this of course is not a peculiarity of open systems: also in standard
EFTs, integrating out degrees of freedom that are active at the scales of interest leads to
intractable non-local interactions and the EFT is powerless without the specific input of a
high-energy completion. These situations simply lack a useful hierarchy of scales and there
is no natural organisation to be provided by the EFT.

For inflation described as a closed system, a powerful organising principle has been
identified in the Effective Field Theory of Inflation (EFToI) [3]. Two pioneering investigations
have studied the generalisation of this approach to the case of an open system and our work
builds upon them. The first one by [18] studied the situation in which the Goldstone boson of
time translations π is coupled to noise fluctuations in the environment and additionally may
experience dissipative evolution. The workhorse of their approach is a stochastic generalisation
of the classical equations of motions known as the Langevin equation. This allows one to model
a very generic open dynamics and to identify characteristic signatures. An aspect of this
approach is that functional methods related to Lagrangians and Hamiltonians are set aside in
favour of working directly with the equations of motion. While this can provide a systematic
description, it comes with a few shortcomings. First, symmetries and their associated charges
and currents are naturally discussed in the context of actions and Lagrangians. Second, this
approach might naively appear fundamentally different from the usual techniques used to
study inflation in the context of closed systems, where one employs the operator or path
integral description of the Schwinger-Keldysh or in-in formalism. The irony is that the
Schwinger-Keldysh was developed to study open systems to begin with, and there is no need
to abandon it if we wish account for dissipation, fluctuations and associated phenomena.

Indeed, recent work [19–21] has generated renewed interest in the Schwinger-Keldysh
formalism within the high-energy community, much of which in the context of hydrodynamics.
This work has clarified some of the general rules needed to ensure that a putative open
effective theory indeed arises by integrating out the environment in a consistent, i.e. unitary,
local and causal theory for the full closed system. Similarly to hydrodynamics, for inflation too
we are interested in identifying the system through the techniques of spontaneous symmetry
breaking. Very generally, one can think of the hydrodynamical modes as describing the
dynamics of small local departures from global thermodynamical equilibrium, with possibly
some conserved charge. Similarly, the system of interest in cosmology is the Goldstone boson
of time translations that are spontaneously broken by the time evolution of some inflaton
sector, which selects a preferred foliation of spacetime. Investigation of the open EFT for
the Goldstone boson in this context was initiated in [22] (see also the related work [23]).
There, quadratic theory for π in flat spacetime was derived and studied. Our work continues
their investigation by moving to cosmological spacetimes and including interactions and the
production of non-Gaussianities. For the benefit of the busy reader we provide a summary
of our main results in section 1.1.

In it interesting to compare and contrast our approach here to previous works in the
literature. We start with the recently proposed in-out formalism for cosmological correlators
proposed in [24]. In that work, it was noticed that for unitary Hamiltonian evolution of

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a closed system, cosmological correlators in de Sitter can be computed in the same in-out
formalism used for scattering amplitude in flat space, namely using Feymann rules that only
invoke a single time order Feynman propagator. This is in contrast to the more cumbersome
in-in formalism where four different bulb-bulk propagators and two different bulk-boundary
propagators appear in different combinations depending on the 2V ways to label the V

vertices of a diagram as coming from the left time evolution or right anti-time evolution.
For IR-finite interactions the in-out formalism provides a sizable simplification and facilitate
the derivation of a series of results such as the correlators equivalent of the wavefunction
recursion relations of [25], cutting rules for correlators completely analogous to those of
Cutkosky and Veltman and a natural definition of a de Sitter S-matrix. The equality between
in-in and in-out fails in the presence of an open system and therefore applies only to the
complement of the theories we consider in this work.

Open quantum system techniques have been used for inflation also for a completely
different reason, namely for their ability to re-sum certain secular divergences and to repair
the validity of perturbation theory in the presence of time logarithms [26, 27]. This has been
used extensively to study the regime of stochastic inflation [28, 29] and associated IR issues
arising in de Sitter [30, 31]. We should stress that our motivation to study open system is
completely independent and therefore the models we study are all IR finite and do not require
any IR resumation. In this sense, it is remarkable that a curved spacetime has different
inequivalent ways to favour an open system description. We will reference other studies
employing open system techniques in cosmology in due course.

Nomenclature. There are a lot of variations in the terminology used by different communities
to describe open and non-equilibrium quantum dynamics.2 While the high-energy and particle
physicists often define unitarity through conservation of the state normalisation, that is

d
dt

Tr[ρ] = 0, (1.1)

where ρ is the density matrix of the system, quantum optics and condensed matter communities
might prefer to consider unitary evolution as3

dρ

dt
= −i [H, ρ] with H† = H. (1.2)

In this article, to avoid confusion, we define unitary time evolutions through the existence
of a unitary evolution operator U(t, t0) such that

ρ(t0) → ρ(t) = U(t, t0)ρ(t0)U†(t, t0), (1.3)

with

U†(t, t0)U(t, t0) = U(t, t0)U†(t, t0) = Id, (1.4)
2We thank C. Burgess for discussions on this issue.
3The above two definitions of (non-)unitarity can be reconciled by absorbing the eventual change in the

state’s normalisation unveiling non-unitary dynamics into a non-Hamiltonian evolution, leading to the complete-
positive and trace preserving dynamical maps used in quantum optics [32]. Ultimately, the common property
is the loss of probability in the observed sector and the associated entropy production of open dynamics.

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and initial condition U(t0, t0) = Id. We further consider density matrices which are (i)
normalised Trρ = 1, (ii) Hermitian ρ† = ρ and (iii) positive definite ρ > 0.

Not all physical evolution can be written under the form of eq. (1.3). Whenever some
degrees of freedom are experimentally inaccessible, initially available information can get
distributed in the unknown environment and eventually lost. Such a possibility is studied in
the context of open quantum systems (see [32] for a reference textbook) and maps pure to
mixed states. While information quantifiers such as purity γ ≡ Tr[ρ2] or entanglement entropy
Sent = −Tr[ρ log ρ] are conserved under eq. (1.3), they can vary over time when considering
open dynamics. These variations precisely encode the lack of probability conservation in
open systems due to the leakage of information between observed degrees of freedom and
their surroundings, which is the object of the current article. In this case, the dynamical
evolution of the open system is given in terms of an open effective functional which denotes
the full functional of the fields π± that weighs the in-in path integral,

Seff [π+,π−] = Sπ [π+]− Sπ [π−] + SIF [π+,π−] , (1.5)

where Sπ is the unitary action and non-unitary dissipative evolution is attributed to the
influence functional SIF [33].

Notation and conventions. We employ the Keldysh rotation of the doubled fields defined by

πr = π+ + π−
2 and πa = π+ − π− ⇔ π± = πr ±

1
2πa . (1.6)

The fonts (π,α,β,γ, δ) and (π, α, β, γ, δ) denote fields and EFT parameters before and after
canonical normalisation, respectively, as discussed in section 2.3. The mapping from one
convention to the other is made through

π ≡ f2
ππ , f4

π ≡ α0 − 2α1 (1.7)

and

c2
s ≡ α0

f4
π

, γ ≡ 2γ1
f4

π

, βi ≡
βi

f4
π

for i = 1 to 8, (1.8)

α2 ≡ α2
f4

π

, γ2 ≡ γ4
f4

π

, δi ≡
δi

f4
π

for i = 1 to 6. (1.9)

The mass dimensions of these parameters are

[π] = E, [fπ] = E, [cs] = E0, [γ] = E, [β1] = E2, [β2] = [β4] = E0, (1.10)
[α2] = E0, [γ2] = E, [β6] = [β8] = E0, [β3] = [β7] = E, [β5] = E2, (1.11)
[δ1] = E3, [δ5] = [δ2] = E, [δ4] = [δ6] = E0. (1.12)

1.1 Summary of the main results

Starting from eq. (1.5), we construct the most generic open effective functional Seff [πr,πr]
compatible with (1) unitarity of the UV theory; (2) the spontaneous symmetry breaking of

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time-translations; and (3) locality in time and space of the open effective theory. Condition (1)
provides a set of non-perturbative relations known as non-equilibrium constraints [19, 21, 34]

Seff [πr,πa = 0] = 0 , (1.13)
Seff [πr,πa] = −S∗eff [πr,−πa] , (1.14)

ℑmSeff [πr,πa] ≥ 0. (1.15)

The symmetry requirement further restricts the available dynamics. Breaking the time-
translation symmetry leads to two Stueckelberg fields π+ and π− in the unitary case. Non-
unitary effects further break the symmetry group explicitly to its diagonal subgroup, such
that only πr transforms non-linearly under time-translations and boosts [22, 35]. More in
detail, for ϵr ∈ R and Λµ

r ν ∈ SO(1, 3)

πr(t, x) → π′r(t, x) = πr

(
Λ0

r µxµ + ϵr,Λi
r µxµ

)
+ ϵr + Λ0

r µxµ − t , (1.16)

πa(t, x) → π′a(t, x) = πa

(
Λ0

r µxµ + ϵr,Λi
r µxµ

)
. (1.17)

Finally, locality in time and space ensures the existence of an IR-stable power counting
scheme that can be truncated to the desired level of accuracy.

The open EFT of inflation. We work in de Sitter space in the decoupling limit with
at most one derivative per field. Using the canonically normalised fields πr and πa, we
obtain the most generic open effective functional. At leading order in the slow-roll expansion,
the quadratic order reads

S
(2)
eff =

∫
d4x

{
a2π′rπ′a − c2

sa2∂iπr∂iπa (1.18)

− a3γπ′rπa + i
[
β1a4π2

a − (β2 − β4) a2π′2a + β2a2 (∂iπa)2
] }

,

and the cubic order

S
(3)
eff = 1

f2
π

∫
d4x

{[
4α2−

3
2(c

2
s−1)

]
aπ′2r π′a+

1
2(c

2
s−1)a

[
(∂iπr)2 π′a+2π′r∂iπr∂iπa

]
(1.19)

+
(
4γ2−

γ

2

)
a2π′2r πa+

γ

2a2 (∂iπr)2 πa

+i
[
(2β7−β3)a2π′rπ′aπa+β3a2∂iπr∂iπaπa+2(β4+β6−β8)aπ′rπ′2a

−2β4a∂iπr∂iπaπ′a−2β5a3π′rπ2
a−2β6aπ′r(∂iπa)2

]
+δ1a4π3

a+(δ5−δ2)a2π′2a πa+δ2a2(∂iπa)2πa−δ4a(∂iπa)2π′a+(δ4−δ6)aπ′3a

}
.

Primes denote time derivatives with respect to the conformal time η = −1/(aH) where a is
the scale factor and H the Hubble parameter. While the standard effective field theory of
inflation [3] is recovered in the unitary limit, the above open effective functional also captures
non-unitary effects such as dissipation and diffusion of the pseudo-Goldstone boson in an
unknown surrounding environment. For instance, the first line of eq. (1.18) corresponds to
the usual unitary dynamics with the kinetic term and an effective speed of sound cs, whereas

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the second line of eq. (1.18) captures dissipation (controlled by γ) and noise fluctuations
(controlled by β1, β2 and β4). Unitary time evolution of the EFT can be recovered as discussed
in section 2.2.2, where we develop a classification of the EFT operators. Just as in the usual
EFToI [3], non-linearly realised boosts relate non-unitary operators at different orders such
as the dissipation parameter γ[−aπ′r − π′2r /2 + (∂iπr)2 /2]πa [18]. We discuss in section 2.3
the new energy scales that characterise non-unitary evolution and the associated heuristic
estimate of primordial non-Gaussianities.

The power spectrum. The open effective field theory of inflation provides theoretical
predictions for standard cosmological observables such as the power spectrum (section 3)
and the bispectrum (section 4). At leading order, curvature perturbations are related to
the pseudo-Goldstone boson by ζ = −Hπ/f2

π . Symmetry requirements ensure the existence
of a nearly scale invariant power spectrum

∆2
ζ(k) ≡

k3

2π2 Pζ(k) with ⟨ζkζ−k⟩ = (2π)3δ(k + k′)Pζ(k). (1.20)

Considering the first noise term of eq. (1.18), which is controlled by β1, we obtain the
dissipative power spectrum and give an exact expression in eq. (3.51). In the large and
small dissipation regimes that result reduces to

∆2
ζ(k) ∝


β1
H2

H4

f4
π

√
H

γ
+O[(γ/H)−3/2], γ ≫ H,

β1
H2

H4

f4
π

+O(γ/H), γ ≪ H.

(1.21)

The observational constraint ∆2
ζ = 10−9 can be easily obeyed by imposing hierarchies

between the various scales of the problem. While we never assume this in our discussion,
one could further impose thermal equilibrium of the surrounding environment. Then, the
power spectrum in the large dissipation regime scales as

∆2
ζ ∝ T

H

H4

f4
π

√
γ

H
, (1.22)

which reproduces the warm inflation results [14, 15, 36–38]. Analytical results for the two
other noise directions π′2a and (∂iπa)2 can be found in section 3.

The bispectrum. To discuss interactions, one can use the same treatment as in the standard
in-in formalism [39], and we derive Feynman rules in section 4. The rest of this section is
devoted to the computation of the contact bispectrum

⟨ζk1ζk2ζk3⟩ = −H3

f6
π

⟨πk1πk2πk3⟩ ≡ (2π)3δ(k1 + k2 + k3)B(k1, k2, k3). (1.23)

The bispectrum is generated by the cubic operators in eq. (1.19), both in flat space and in
de Sitter. The flat-space results are instructive as computations can easily be carried out
analytically. The generic structure of the contact bispectrum is given by

B(k1, k2, k3) = f(EFT)Polyn (eγ
1 , eγ

2 , eγ
3)

Singγ

, (1.24)

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where f(EFT) is a rational function of the EFT coefficients (and possibly the kinematics
for spatial derivative interactions), and Polyn are elementary symmetric polynomials of
the energy variables

eγ
1 = Eγ

1 + Eγ
2 + Eγ

3 , eγ
2 = Eγ

1 Eγ
2 + Eγ

2 Eγ
3 + Eγ

1 Eγ
3 eγ

3 = Eγ
1 Eγ

2 Eγ
3 , (1.25)

where we used the dispersion relation appropriate for this dissipative system, namely

Eγ
k ≡

√
c2

sk2 − γ2/4 . (1.26)

Moreover, Singγ is a place holder for the singularity structure

Singγ =
∣∣∣∣Eγ

1 + Eγ
2 + Eγ

3 + 3
2 iγ

∣∣∣∣2 ∣∣∣∣−Eγ
1 + Eγ

2 + Eγ
3 + 3

2 iγ

∣∣∣∣2
×
∣∣∣∣Eγ

1 − Eγ
2 + Eγ

3 + 3
2 iγ

∣∣∣∣2 ∣∣∣∣Eγ
1 + Eγ

2 − Eγ
3 + 3

2 iγ

∣∣∣∣2 . (1.27)

This singularity structure captures most of the specificities of the non-unitary dynamics.
Physically, it represents 3 ↔ 0 (all pluses) and 2 ↔ 1 (mixed signs) interactions mediated
by the real particles present in the environment. Fluctuations alone would generate folded
singularities because the state of the system differs from the Bunch Davies vacuum in that
real particles are present also on sub-Hubble scales. On the other hand, dissipation induces
a finite memory of the past. This regularises the folded divergences [18, 40], by effectively
moving the folded pole into complex kinematics. The singularity is not located in the physical
plane and the bispectrum remains finite over the whole dynamical range. This is illustrated
in the left panel of figure 1 where we observe an enhanced but finite signal near to folded
region. The singularity structure Singγ exhibits two different behaviours depending on the
magnitude of the dissipation coefficient γ:

• In the strong dissipation regime, the 3iγ/2 term of eq. (1.27) always dominates and the
signal peaks in the equilateral shape where k1 ≃ k3 ≃ k3 (orange region in figure 1).

• In the small dissipation regime, Singγ can become small in the folded region where
k2 + k3 ≃ k1 and the signal predominantly peaks in the isofolded4 configuration where
k2 ≃ k3 ≃ k1/2 (blue region in figure 1).

Beyond the flat space case, analytical results are hard to reach in full generality and we
mostly rely on numerical results, as in section 4.3. Just as in flat space, different behaviours
emerge in the large (γ ≫ H) and small (γ ≪ H) dissipation regime, see the right panel of
figure 1. For large dissipation the signal peaks in the equilateral configuration, as already
noted in [18]; for small dissipation the signal reaches an extremum near the folded region. At
first sigh, this smoking gun of open dynamics might seem to be degenerate with other classes
of models, which also lead to signal in the folded triangles, such as non-Bunch Davies initial
states [40–46]. A crucial difference is that dissipation regulates the divergence by smoothing

4Here we differentiate between folded singularities, which are a one-parameter family of configurations
modulo rescaling, k1 + k2 = k3, and permutations thereof, from the isofolded configuration, which represents a
single kinematic modulo rescaling, k2 ≃ k3 ≃ k1/2 and permutations thereof.

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γ  1

γ  0.025

γ  5H

γ  0.2H

Figure 1. Shape function S(x2, x3) ≡ (x2x3)2[B(k1, x2k1, x3k1)/B(k1, k1, k1)] for the contact bispec-
trum generated by the operator π3

a in Minkowski (left) and in de Sitter (right). Left: the bispectrum
is given by eq. (1.24) with singularities controlled by Singγ in eq. (1.27) (details in section 4.2). The
singularity is resolved such that the bispectrum remains finite for any physical configuration. We
observe the equilateral (orange) to folded (blue) transition of the shape function as the dissipation
parameter γ decreases. There is an enhancement of the signal close to the isofolded configuration in
the low dissipation regime. Right: the qualitive features of the signal remain the same in de Sitter.
At large dissipation (orange), the signal peaks in the equilateral configuration. At small dissipation
(blue), it is enhanced near the folded region. Two aspects are confirmed by analytical arguments but
are not clearly visible in the plot: (i) dissipation regulates the folded divergence and (ii) consistency
relations still hold in the squeezed limit x3 ≪ x2 = 1.

the peak and displacing it from the edge of the triangular configurations, leading to finite
values of the bispectrum for any physical configuration. In particular, it implies no divergence
in the squeezed limit of the bispectrum k1 ≃ k2 ≫ k3. Small values of γ/H may eventually
lead to an intermediate peak due to the regularised folded singularity, but consistency relations
hold [47–54] and the squeezed limit goes to zero due to the symmetries of the theory.

UV-models and matching. Our open effective field theory of inflation captures all single-
clock models that display a local and possibly dissipative dynamics. One such explicit
UV-model was recently studied in [55]. Here, in section 5 we show that our formalism
provides an accurate low-energy effective description of its dynamics. The model in question
contains, in addition to the inflaton field ϕ, a massive scalar field χ with a softly-broken
U(1) symmetry

S =
∫

d4x
√
−g

[1
2M2

PlR − 1
2 (∂ϕ)2 − V (ϕ)− |∂χ|2 + M2 |χ|2

−∂µϕ

f
(χ∂µχ∗ − χ∗∂µχ)− 1

2m2
(
χ2 + χ∗2

) ]
. (1.28)

This model exhibits a narrow instability band in the sub-Hubble regime, during which particle
production occurs. We demonstrate the equivalence of the non-linear Langevin equation

π′′ + (2H + γ) aπ′ − ∂2
i π ≃ γ

2ρf

[
(∂iπ)2 − 2πξπ′2

]
− a2m2

f

(
1 + 2πξ

π′

aρf

)
δOS , (1.29)

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used in [55] to the open effective field theory of inflation we discuss here, which for this
UV model reduces to

Seff =
∫

d4x

[
a2π′rπ′a − c2

sa2∂iπr∂iπa − a3γπ′rπa + iβ1a4π2
a

+ (8γ2 − γ)
2f2

π

a2π′2r πa + γ

2f2
π

a2 (∂iπr)2 πa − 2i
β5
f2

π

a3π′rπ2
a + δ1

f2
π

a4π3
a

]
. (1.30)

Outline. The rest of this paper is organized as follows. In section 2, we develop the bottom-
up construction of an open effective field theory for single-clock inflation working from the
beginning in the decoupling limit. In section 3, we derive the propagators of the theory and
extract the power spectrum. Section 4 is devoted to the perturbative treatment of interactions
through the computation of the contact bispectrum. Lastly, in section 5, we perform an
explicit matching of our open EFT to a specific UV completion. Conclusions are gathered in
section 6 followed by a series of appendices collecting further details of our calculations.

2 The open effective field theory of inflation

The starting point to study an open system is a choice of the degrees of freedom that we
want to describe, usually known as the system, and those that we consider unobservable,
usually known as the environment. Given our focus on single-clock inflation, our system
will consists of a scalar field π to be identified with a Goldstone boson of time translations,
to be defined shortly. We aim to construct an EFT for π in the presence of non-unitary
time evolution induced by interactions with the environment. A pure state evolving unitarily
admits a wavefunction representation

Ψ [π; η0] =
∫ π

Ω
Dπ+eiSeff [π+]. (2.1)

This is the common starting point of most studies of inflation and the field-theoretic wave-
function Ψ has been the object of intense recent scrutiny [25, 56–70] (see [71, 72] for reviews
and further references). In contrast, a mixed state is described by a density matrix ρ̂red,
whose elements can be determined by a path integral along the in-in contour, a.k.a. the
closed-time path. A useful bookkeeping trick to work with an in-in contour is to double the
number of fields to π+ and π−, such that

ρred
[
π,π′; η0

]
=
∫ π

Ω
Dπ+

∫ π′

Ω
Dπ−eiSeff [π+,π−]. (2.2)

Here Ω represents the choice of initial state and can be in principle an initial mixed density
matrix. In the rest of this work we will assume that in the infinite past the system started
in the Bunch-Davies state, and hence was a pure state even when reduced to the π sector.
The functional Seff contains several contributions. First one can identify a contribution
representing time evolution according to a Hermitian Hamiltonian. This corresponds to
interactions in Seff that do not mix the two branches of the path integral, which take the
general form Sπ [π+]−Sπ [π−]. Second, we can identify non-unitary (non-Hamiltonian) effects,
which encode energy and information losses and gains. These appear in the interactions across

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the two branches of the path integral. The part of Seff representing these contributions is
sometimes referred to as the Feynman-Vernon influence functional SIF [π+,π−]. In summary,
the open effective functional Seff [π+,π−] admits a Hermitian and a non-Hermitian part

Seff [π+,π−] = Sπ [π+]− Sπ [π−] + SIF [π+,π−] . (2.3)

Building the recent literature on non-equilibrium open EFTs [19, 21, 22, 34, 35, 73, 74], our
goal here is to construct and study Seff [π+,π−] for single-clock inflation.

2.1 EFT construction

In this subsection, we introduce and discuss a series of constraints on the open effective
functional Seff [π+,π−]. Some of these constraints follow from general principles, such as
conservation of probabilities, others are specific to systems with a hierarchy of scales and
to the study of single-clock inflation.

2.1.1 Non-equilibrium constraints

A first set of constraints follow from requiring that an open quantum system arises from a closed
system undergoing Hermitian time evolution upon tracing over the environment [19, 21, 32, 34]

Seff [π+,π+] = 0 , (2.4)
Seff [π+,π−] = −S∗eff [π−,π+] , (2.5)

ℑmSeff [π+,π−] ≥ 0 . (2.6)

These conditions follow respectively from the defining conditions of a density matrix:

1. Normalisation, Trρ̂red = 1,

2. Hermiticity, ρ̂†red = ρ̂red,

3. Positivity, ⟨ϕ| ρ̂red |ϕ⟩ ≥ 0 for all |ϕ⟩ in the Hilbert space.

A derivation of these constraints following [34] is given in appendix A. The induced structure
holds non-perturbatively and, in particular, remains valid at all loop orders. As we will see
below, these conditions strongly reduce the number of operators one may include in the EFT.

As the number of terms in Seff [π+,π−] grows rapidly with the number of fields and
derivatives, it is convenient to choose an organising principle. A convenient basis to work
with5 is known as the Keldysh basis (sometimes referred to as the retarded/advanced basis)

πr = π+ + π−
2 and πa = π+ − π− ⇔ π± = πr ±

1
2πa . (2.7)

5Hermitian time-evolution is most manifest in the +/− basis, where it corresponds to the absence of mix
terms between the + and − branches. Conversely, the causality structure simplifies in the r-a basis, where
the πa fields don’t propagate [75]. In the literature, sometimes, the following equivalent notation is used
where πr ≡ πcl and πa ≡ πq [75, 76]. While in some physical situations, this “classical-quantum” notation
can help counting powers of ℏ, this is not the case for inflation and so we prefer to avoid this potentially
misleading nomenclature.

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In the r-a basis, the conditions on the open effective functional Seff read

Seff [πr,πa = 0] = 0 , (2.8)
Seff [πr,πa] = −S∗eff [πr,−πa] , (2.9)

ℑmSeff [πr,πa] ≥ 0. (2.10)

2.1.2 Time-translation symmetry breaking

Symmetries further restrict the number and structure of EFT operators. The general idea
follows from the recent Schwinger-Keldysh coset construction [35] which aims at studying
symmetry breaking patterns in out-of-equilibrium systems. Let’s consider a schematic
microscopic theory SUV[π, χ] that is invariant under a symmetry group G. Then, the
Schwinger-Keldysh action SUV[π+, χ+] − SUV[π−, χ−] for the closed system is invariant
under G+ × G−. Integrating out the χ field leads to a symmetry breaking pattern under
which Seff [π+,π−] becomes invariant under a smaller subgroup GIR (which is often the
diagonal subgroup transforming π+ and π− simultaneously [35]). Lastly, one might consider
spontaneous symmetry breaking associated to the choice of state, such as the dynamical
Kubo-Martin-Schwinger (KMS) symmetry for thermal states [21]. We don’t discuss this
possibility here because cosmological environments are often nonthermal [77, 78].

As in the EFToI in the decoupling limit [3, 18, 79], our open EFT has a single degree of
freedom, the Nambu-Goldstone boson of the spontaneous breaking of time translation by the
inflaton background. We follow the pioneering work of [22], which first studied this symmetry
breaking pattern. Consider a UV-action that is invariant under independent time-translations
labelled by ϵ± realised non-linearly as

π+(t) → π′+(t) = π+(t + ϵ+) + ϵ+, π−(t) → π′−(t) = π−(t + ϵ−) + ϵ−, (2.11)

and acting linearly on additional environment fields. Upon tracing over the environment, the
open effective functional Seff [π+,π−] is not in general invariant under general ϵ± translations.
More precisely, the effective functional remains invariant under the translation (see the
left-hand panel of figure 2) [22]

ϵ+ = ϵ− = ϵr , (2.12)

while the translations

ϵ+ = −ϵ− = ϵa

2 (2.13)

are explicitly broken (see the right-hand panel of figure 2). In this way, out of two time-
translational symmetries of the microscopic action, we are left with a single diagonal sub-
group ϵ+ = ϵ−.

Let us consider the transformation of πr and πa under the diagonal subgroup of time
translations and boosts. In the Keldysh basis, the ϵr-transformations read [22] (see left-hand

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Figure 2. The ϵr and ϵa transformations on the closed time path, where time is running from
left to right in both contours and the arrow represent path ordering (time ordering in |in⟩ and
anti-time-ordering in ⟨in|). The ϵr transformation translates the “+′′ and “−′′ variables in the same
direction while ϵa transformation does it in the opposite directions. While the ϵr transformation is
preserved, the ϵa one is explicitly broken due to dissipative effects [22].

panel of figure 2)6

πr(t, x) → π′r(t, x) = πr(t + ϵr, x) + ϵr, (2.14)
πa(t, x) → π′a(t, x) = πa(t + ϵr, x), (2.15)

whereas the Λr-transformations follow

πr(t, x) → π′r(t, x) = πr

(
Λ0

r µxµ,Λi
r µxµ

)
+ Λ0

r µxµ − t, (2.16)

πa(t, x) → π′a(t, x) = πa

(
Λ0

r µxµ,Λi
r µxµ

)
, (2.17)

where we introduced Λµ
r ν ∈ SO(1, 3). The important point is that πr non-linearly realises

time-translations and boosts whereas πa transforms linearly, just as ordinary matter [22].

2.1.3 Locality

In this subsection, we discuss locality of Seff in time and space. In the most general case, tracing
over the environment yields an unwieldy non-local effective functional that is intractable
unless one knows the exact UV-completion. Instead, just like for standard EFTs, a dramatic
simplification takes place in the presence of a separation of scales. Here we focus on precisely
this possibility: we envisage that the typical length and time scales characterising the
environment are much shorter than the Hubble time and the Hubble radius at which we
compute cosmological correlators. This hierarchy ensures that our open EFT is local in
space and time, i.e. it features operators that are the product of fields at the same spacetime
point and a finite but arbitrary number of derivatives thereof. Not all UV-models display
such hierarchy. For example, the much studied models of inflation featuring the tachyonic

6One can check that Seff [πr,πa] is left invariant by the transformation given in eq. (2.14). To see it
explicitly, one can consider the expansion

πr(t, x) → πr(t, x) + ϵr [1 + π̇r(t, x)] + O
[
(ϵr)2] ,

πa(t, x) → πa(t, x) + ϵrπ̇a(t, x) + O
[
(ϵr)2] .

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production of gauge modes engendered by a ϕFF̃ coupling [17] would give a non-local
open EFT for π because the gauge fields are mostly produced around Hubble crossing. See
also [77, 80–84] for additional examples where the open EFT could be non-local and/or
non-Markovian. The fact that our open EFT is not useful to describe these models is nothing
new or specific to open systems: the same would happen for a standard EFT if one tried
to integrate out very light or massless fields.

So we want to fucus on models that exhibit a hierarchy of scales. One such model
was constructed and studied in detail recently in [55] and we were able to match our EFT
description to that UV completion, as we will discuss in section 5. We further discuss the
physical implications of locality in section 6.

Now that we motivated a local open EFT, we want to write down all possible local
operators compatible with the non-equilibrium constraints and symmetries. We can do this
by using invariant combinations as fundamental building blocks [35]. The open effective
functional can be constructed out of πa, t + πr and their derivatives ∂µπa and

Pµ = ∂µ(t + πr) = δ0
µ + ∂µπr (2.18)

just as one usually do in e.g. the EFToI [22]. Working at leading order in the derivative
expansion, we now restrict ourselves to operators with at most one derivative per field. At
last, following [18], we work in the decoupling limit where there is a unique dynamical
degree of freedom π up to slow-roll corrections (which remains a valid description despite
the presence of the environment, see appendix D of [18]). Notice that one might also like
to derive our open EFToI starting in unitary gauge and specifying a theory of gravity that
is invariant under spatial diffeomorphisms but not time diffeomorphisms, as in the original
derivation [3]. While doing this order by order in metric perturbations is straightforward,
it is not clear to us how the non-perturbative and diffeomorphism invariant structure of
gravity is organised when fields are doubled. For example, one seems to have infinitely many
choices to covariantise indices using g+

µν , g−µν or combinations thereof. We hope to come
back to this issue in future research.

2.2 Open effective functional

We now aim at writing the most generic local open effective functional compatible with the
non-equilibrium constraints and the above symmetries, with at most one derivative per field.
In addition to the derivative expansion, a useful way to organise the open effective functional
is in powers of πa such that Seff =

∫
d4x

√
−gLeff with

Leff =
∞∑

n=1
Ln with Ln = O(πn

a) (2.19)

where we used the unitarity condition (2.8) to notice that Leff starts from the first order term
in πa. Notice that we are not saying that operators with more powers of πa are suppressed,
and we are not invoking any argument about the ℏ expansion, which is not justified in
general in this context [76]. Instead we are just using the organisation in powers of πa to
neatly groups the many different terms. Restricting ourselves to cubic operators for practical
applications, we focus on L1, L2 and L3 for which we illustrate the general procedure, the
next order being at best quartic in πa.

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On total derivatives. Before developing the explicit construction, we wish to discuss
the status of total derivatives whose removal greatly simplifies the discussion [22]. In the
inflationary context, working with the conformal time η ∈ (−∞, 0], it might be a concern
that integration-by-part (IBP) might generate boundary terms [63, 85] and might not be
as harmless as in flat space. A way to avoid this issue is obviously to avoid IBP. Yet, it
is instructive to see how this issues manifests itselft in our open EFT and contrast it to
the usual EFToI. The boundary conditions of the in-in path integral impose πa = 0 at the
boundary. Consequently, any boundary term proportional to πa vanishes. Non-vanishing
boundary terms must contain at least one time derivative acting on πa. They correspond
to operators with more than one derivative acting on the field variables in the bulk, which
we neglected by focusing on the leading EFT operators. For this reason, we safely discard
total derivatives in this work.

2.2.1 An effective functional for the Goldstone boson

Let us construct Ln order by order in πa following the above principles.

L1 functional. Let us illustrate the procedure by first considering L1. We aim at using the
building blocks defined around eq. (2.18) to construct invariant combinations that are linear in
πa. The only option consists in multiplying πa and P µ∂µπa = (−π̇a + ∂µπr∂µπa) by powers of

(PµP µ + 1) = −2π̇r + (∂µπr)2 , (2.20)

leading to [22]7

L1 =
∞∑

n=0
(PµP µ + 1)n [γnπa − αnP µ∂µπa] . (2.21)

The EFT coefficients γn and αn are in general functions of t+πr which have to be real because
of the conjugate condition (2.9). In the slow-roll regime, one can assume time-independence
for the EFT coefficients at leading order in slow-roll [22].

Background evolution and tadpoles

Before describing the dynamics of the fluctuations, let us connect with the standard background
evolution of the EFToI by discussing tadpole cancellation [3, 86]. As mentioned below
eq. (2.19), the unitarity condition (2.8) imposes that Leff starts linear in πa. Therefore, the
only available tadpoles are [18, 22]

Seff ⊃
∫

d4x
√
−g [γ0(t)πa + α0(t)π̇a] , (2.22)

which leads to the tadpole cancellation

−γ0 + α̇0 + 3Hα0 = 0. (2.23)

7The sign in front of the γn term is chosen for later convenience.

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Let us compare this result to the usual construction. In the standard EFToI approach, the
background FLRW evolution is obtained from varying

SEFToI =
∫

d4x
√
−g

[1
2M2

PlR − Λ(t)− c(t)g00
]

(2.24)

with respect to the g00 and gii components of the metric, leading to

3M2
PlH

2 = c(t) + Λ(t) , (2.25)
−M2

PlḢ = c(t). (2.26)

One can also reintroduce diffeomorphism invariance through the Stückelberg trick t → t + π

and then vary the action eq. (2.24) with respect to π. This leads to the tadpole cancellation [3]

Λ̇ + ċ + 6Hc = 0 . (2.27)

We see that this is not a new equation but instead follows from combining eqs. (2.25) and (2.26).
This is to be expected because Einstein’s equation imply the conservation of the total stress-
energy tensor ∇µTµν = 0. Rotated in the Keldysh basis, the above two tadpoles lead to8

− Λ(t)|t→t+πr+πa
2
+ Λ(t)|t→t+πr−πa

2
= −Λ̇(t)πa +O(π2

a), (2.28)

and

− c(t)g00
∣∣∣
t→t+πr+πa

2
+ c(t)g00

∣∣∣
t→t+πr−πa

2
= ċ(t)πa + 2c(t)π̇a +O(π2

a). (2.29)

We identify

α0(t) ≡ 2c(t) and γ0(t) ≡ ċ(t)− Λ̇(t), (2.30)

such that eqs. (2.22) and (2.27) are equivalent.

The main physical outcome of the above discussion is the following. As noticed in [18],
there is no new tadpole for this class of local dissipative models of inflation. The background
evolution is fixed by the slicing and probes the global energy density, which does not distinguish
the contributions of the inflaton from those of the unknown environment.9 It is only at
the level of the fluctuations that the distinction between system and environment becomes
relevant, as we can disentangle observable degrees of freedom associated to the hydrodynamical
direction π and unobservable degrees of freedom that have been integrated out.

Now we have fixed the background dynamics and removed the tadpoles, L1 takes the
explicit form

L1 = −α0∂µπr∂µπa −
∞∑

n=1
αn

[
−2π̇r + (∂µπr)2

]n
(−π̇a + ∂µπr∂µπa)

+
∞∑

n=1
γn

[
−2π̇r + (∂µπr)2

]n
πa. (2.31)

8Notice that these terms are manifestly unitary. They obviously pass the unitary test described below in
section 2.2.2 if one accounts for the slow-roll suppressed terms such as Λ̈πrπa.

9It would be interesting to further investigate if there exists a slicing where system and environment are
distinguishable from the background dynamics, for instance through different charges.

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Notice that only α0, α1 and γ1 provide quadratic terms in π relevant for the dispersion
relation of the Goldstone mode [22]. The α0 term is the usual kinetic term written in the
Keldysh basis, see the basis transformation (2.7). The tadpole cancellation discussed above
leads to the relation α0(t) = 2c(t) = 2M2

Pl|Ḣ|. The α1 term generates a non-trivial speed
of sound accompanied by higher order operators controlling the appearance of equilateral
non-Gaussianities [3]. In contrast to the α0 and α1 terms, the γ1 term has no unitary
counterpart and leads to a dissipative term in the πr equation of motion, as we will see in
section 5. Interestingly, the dissipation term π̇rπa is accompanied by a cubic interaction
(∂µπr)2 πa, as first noted in [18], such that the combination is invariant under Lorentz boosts.

Let us explicitly consider the quadratic and cubic contributions. Indeed, in addition
to the expansion in powers of πa, one can classify

Leff =
∞∑

n,m=1
L(m)

n with L(m)
n = O(πm, πn

a , πm−n
r ) (2.32)

where m labels the number of field operators. The quadratic contributions are

L(2)
1 = (α0 − 2α1) π̇rπ̇a − α0∂iπr∂iπa − 2γ1π̇rπa (2.33)

where α0 and α1 control the unitary kinetic term with a non-trivial speed of sound and γ1
encodes the linear dissipation of the system onto its environment. At cubic order, L1 reads

L(3)
1 =(4α2 − 3α1) π̇2

rπ̇a + α1 (∂iπr)2 π̇a + 2α1π̇r∂iπr∂iπa

+ (4γ2 − γ1) π̇2
rπa + γ1 (∂iπr)2 πa (2.34)

where the first line corresponds to parts of the unitary operators π̇3 and (∂iπ)2π̇ and the
second line to the non-linear dissipation induced by the non-linearly realised symmetries,
as discussed in [18].

L2 functional. Following [22], let us now construct L2, which is quadratic in πa. Just
as above, working with operators containing at most one derivative, in the slow-roll limit,
we obtain

L2 = i
[
β1π

2
a + β2 (∂µπa)2 + β3 (−π̇a + ∂µπr∂µπa)πa + β4 (−π̇a + ∂µπr∂µπa)2 + · · ·

]
(2.35)

where the third and fourth terms are obtained from P µ∂µπaπa and (P µ∂µπa)2 respectively
and the dots represent higher order terms obtained by multiplying the first four terms by
arbitrary powers of (P µPµ + 1) = −2π̇r + (∂µπr)2. The action being at least quadratic
in πa, there is no tadpole contribution. The i in front directly follows from the conjugate
condition (2.9). While the term proportional to β1 is the standard noise term appearing in
the Langevin equation, see section 5, we observe the presence of derivative corrections such
as (∂µπa)2 and π̇2

a in the β2 and β4 terms which make the noise scale-dependent.
There exists a positivity condition on the β’s coefficients due to eq. (2.10) which imposes

ℑmSeff [πr,πa] ≥ 0. In flat space, making use of the derivative expansion which tells us
that ω2, k2 ≪ |β1/β2,4| (the quadratic term in β3 can be written as a total derivative and

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removed), the authors of [22] concluded that β1 dominates in L2, such that the positivity
constraint imposes

β1 > 0. (2.36)

We will see this constraint emerging from a different point of view in section 3, where it
follows from requiring the positivity of the power spectrum. This different requirement further
imposes constraints on β2 and β4. This positivity constraint on the noise kernel directly
translates into consequences for the non-Gaussian signal if we multiply this operator by
higher powers of (P µPµ + 1) = −2π̇r + (∂µπr)2.

Note that if the positivity constraints ℑmSeff [πr,πa] ≥ 0 seems easy to satisfy for odd
powers of πa as long as the Wilsonian coefficients are real, the case of even powers is much
less trivial. For even powers of πa, one can ask if there exists combination of operators
which are not positive definite or total derivatives. So far, we have not identified any of
these terms even considering higher-derivative operators, which may guarantee, at least
in principle, the boundedness of even powers of πa. The second concern is related to the
presence of non-linear interactions involving πr. If one considers the simplest noise term
iβ1π

2
a, one can multiply this operator by (P µPµ + 1) = −2π̇r + (∂µπr)2 to obtain equally

valid operators. The cubic term π̇rπ
2
a is unbounded and one must impose a (non-unitary)

perturbativity bounds to ensure convergence.
As above, let us explicitly consider the quadratic and cubic contributions for L2. The

three quadratic noise are controlled by

L(2)
2 = i

[
β1π

2
a − (β2 − β4) π̇2

a + β2 (∂iπa)2
]

, (2.37)

where we removed the total derivative related to β3 at quadratic order. At cubic order,
we obtain

L(3)
2 = i

[
− (β3 − 2β7)π̇rπ̇aπa + β3∂iπr∂iπaπa + 2(β4 + β6 − β8)π̇rπ̇

2
a

− 2β4∂iπr∂iπaπ̇a − 2β5π̇rπ
2
a − 2β6π̇r(∂iπa)2

]
, (2.38)

where we needed to introduce the Wilsonian coefficients β5,β6,β7 and β8 associated to the
higher-order operators included in the dots of eq. (2.35), obtained from multiplying the first
four terms in eq. (2.35) by (P µPµ + 1) = −2π̇r + (∂µπr)2. The physical interpretation of
these terms is discussed below in section 2.2.2.

L3 functional. One can carry on this construction to access L3. We have

L3 = δ1π
3
a + δ2(∂µπa)2πa + δ3 (−π̇a + ∂µπr∂µπa)π2

a + δ4 (−π̇a + ∂µπr∂µπa) (∂νπa)2

+ δ5 (−π̇a + ∂µπr∂µπa)2 πa + δ6 (−π̇a + ∂µπr∂µπa)3 + · · · , (2.39)

where the δ3 term originates from (P µ∂µπa)π2
a, the δ4 term from (P µ∂µπa)(∂νπa)2, the δ5

from (P µ∂µπa)2 πa and the δ6 term from (P µ∂µπa)3. As above, the dots represent higher
order terms obtained by multiplying the first terms by arbitrary powers of (P µPµ + 1) =
−2π̇r + (∂µπr)2. As we will see below, the interpretation of these terms is ambiguous, as

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they can either be associated to unitary or non-unitary operators depending on how they
relate to contributions from L1. For this reason, we develop in section 2.2.2 a classification
of these terms.

As above, let us explicitly consider the quadratic and cubic contributions for L3. Since
these contributions are at least cubic in πa, there is no quadratic contribution. If we restrict
ourselves to the cubic order, we obtain

L(3)
3 = δ1π

3
a + (δ5 − δ2)π̇2

aπa + δ2(∂iπa)2πa − δ4(∂iπa)2π̇a + (δ4 − δ6)π̇3
a, (2.40)

the δ3 cubic contribution being a total derivative.

Summary

Under the assumptions specified in section 2.1, the most generic second order open effective
functional (up to total derivatives) is

L(2) = (α0 − 2α1) π̇rπ̇a − α0∂iπr∂iπa

− 2γ1π̇rπa + i
[
β1π

2
a − (β2 − β4) π̇2

a + β2 (∂iπa)2
]

, (2.41)

where the EFT coefficients are chosen to match the notations of [22]. The first line corresponds
to the usual unitary dynamics which the kinetic term and an effective speed of sound. The first
term of the second line controlled by γ1 corresponds to the dissipation due to the surrounding
environment. At last, the βi coefficients control the diffusion (noise-induced) process.10

At cubic order, the most generic open effective functional (up to total derivatives) writes

L(3) = (4α2 − 3α1) π̇2
rπ̇a + α1 (∂iπr)2 π̇a + 2α1π̇r∂iπr∂iπa (2.42)

+ (4γ2 − γ1) π̇2
rπa + γ1 (∂iπr)2 πa

+ i
[
− (β3 − 2β7)π̇rπ̇aπa + β3∂iπr∂iπaπa + 2(β4 + β6 − β8)π̇rπ̇

2
a (2.43)

− 2β4∂iπr∂iπaπ̇a − 2β5π̇rπ
2
a − 2β6π̇r(∂iπa)2

]
+ δ1π

3
a + (δ5 − δ2)π̇2

aπa + δ2(∂iπa)2πa − δ4(∂iπa)2π̇a + (δ4 − δ6)π̇3
a, (2.44)

where (2.42) originates from L(3)
1 , (2.43) originates from L(3)

2 and (2.44) from L(3)
3 .

In de Sitter, in terms of the conformal time and the scale factor a = −1/(Hη), the open
effective functional up to cubic order reads

S
(2)
eff =

∫
d4x

{
(α0 − 2α1) a2π′rπ

′
a − α0a2∂iπr∂iπa (2.45)

− 2a3γ1π
′
rπa + i

[
β1a4π2

a − (β2 − β4) a2π′2a + β2a2 (∂iπa)2
] }

,

10Note that if the environment obeys thermal equilibrium, the KMS symmetry imposes a relation between
the γi and the βj [21].

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and

S
(3)
eff =

∫
d4x

{
(4α2−3α1)aπ′2r π

′
a+α1a(∂iπr)2π′a+2α1aπ′r∂iπr∂iπa (2.46)

+(4γ2−γ1)a2π′2r πa+γ1a2 (∂iπr)2πa

+i
[
−(β3−2β7)a2π′rπ

′
aπa+β3a2∂iπr∂iπaπa+2(β4+β6−β8)aπ′rπ′2a

−2β4a∂iπr∂iπaπ
′
a−2β5a3π′rπ

2
a−2β6aπ′r(∂iπa)2

]
+δ1a4π3

a+(δ5−δ2)a2π′2a πa+δ2a2(∂iπa)2πa−δ4a(∂iπa)2π′a+(δ4−δ6)aπ′3a
}

.

2.2.2 Classification of the EFT operators

The above construction exhibits a wide zoology of terms compared to its unitary counterpart:
5 free parameters in L(2) compared to only 1 in the standard EFToI [3]; 13 free parameters
in L(3) compared to only 1 in [3]. While some operators describe faithful non-unitary effects
generated by the presence of additional degrees of freedom, others are simply a consequence
of writing unitary interactions in the Keldysh basis. In this section, we develop a procedure
to distinguish unitary from non-unitary operators.

Recovering the EFToI. In this work, we study the decoupling limit of π interacting with
an unknown environment. Therefore, we expect our open effective functional Seff to be able
to reproduce in a certain limit the EFToI [3]. This limit defines the unitary direction of
the parameter space of the theory. Let us first consider the quadratic terms of the EFToI
which reads in the Keldysh basis

1
2
[
π̇2

+ − c2
s(∂iπ+)2

]
− 1

2
[
π̇2
− − c2

s(∂iπ−)2
]
= π̇rπ̇a − c2

s∂iπr∂iπa. (2.47)

It can be matched with the first line of eq. (2.41) for

c2
s = α0

α0 − 2α1
. (2.48)

We can go to the next order in the EFToI and consider cubic operators. Starting with π̇3,
the unitary interaction in the Keldysh basis reads

π̇3
+ − π̇3

− = 3π̇2
rπ̇a + 1

4 π̇
3
a. (2.49)

Comparing with the terms in eqs. (2.42) and (2.44) which include π̇2
rπ̇a and π̇3

a, the unitary
combination in eq. (2.49) imposes the relation among the EFT coefficients11

δ4 − δ6 = 1
12 (4α2 − 3α1) . (2.50)

A similar procedure follows for (∂iπ)2π̇. In the Keldysh basis, this vertex reads

(∂iπ+)2π̇+ − (∂iπ−)2π̇− = (∂iπr)2π̇a + 2∂iπr∂iπaπ̇r +
1
4 π̇a(∂iπr)2. (2.51)

11This relation defines an orthogonal direction δ4 − δ6 = −12 (4α2 − 3α1) intrinsically related to non-unitary
effects. It would be interesting to explore whether information quantifiers such as the purity only depends on
the value of the EFT coefficients along this orthogonal direction.

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Comparing it to the operators appearing in eqs. (2.42) and (2.44), it specifies the unitary
direction12

δ4 = −1
4α1. (2.52)

One can then be reassured that in a certain limit, the current constructions reduces to
the usual EFToI of [3].

Unitary and orthogonal directions. What about the other operators? Our open effective
functional has more Wilsonain coefficients than the EFToI. Does it imply that all the
remaining operators are intrinsically related to non-unitary effects? The symmetry structure
of the theory allows one to answer this question is a systematic manner. In the limit where
non-unitary effects are absent, eq. (2.3) reduces to the unitary effective action

Seff [π+,π−] = Sπ [π+]− Sπ [π−] . (2.53)

This restriction is obtained by restoring the ϵa symmetry explicitly broken by the non-unitary
effects (see Right panel of figure 2) [22]. Indeed, in the unitary limit, the two branches of
the path integral must transform equally under ϵ±

π±(t, x) → π′±(t, x) = π±(t + ϵ±, x) + ϵ±. (2.54)

One can impose this by acting on Seff with the ϵa symmetry given by ϵ0
+ = −ϵ0

− = ϵa/2.
Expressing eq. (2.54) in the Keldysh basis and expanding linear order in ϵa we obtain

πr(t, x) → π′r(t, x) = πr(t, x) + ϵa

2 π̇a(t, x) +O
(
ϵ2
a

)
(2.55)

πa(t, x) → π′a(t, x) = πa(t, x) + ϵa [1 + π̇r(t, x)] +O
(
ϵ2
a

)
. (2.56)

Unitary combinations of operators must leave Seff invariant under the above transformation.
Let us illustrate this procedure with the kinetic terms π̇rπ̇a and ∂iπr∂iπa appearing in

eq. (2.41) that have been identified as being unitary through the comparison with the EFToI.
Under the ϵa transformation eqs. (2.55) and (2.56), we notice they lead to total derivatives.
Hence, Seff made of these terms is ϵa-invariant, indicating they can be encountered in a
unitary theory as one would expect. On the contrary, the quadratic dissipative term π̇rπa

and diffusive terms π2
a, π̇2

a and (∂iπa)2 are not invariant. Consequently, they have no unitary
counterpart and represent genuine non-unitary effects.

For cubic interactions the effective action is invariant under ϵa only for the specific
combinations identified in eqs. (2.50) and (2.52). Indeed, one can check the combinations
3π̇2

rπ̇a + π̇3
a/4 and (∂iπr)2π̇a + 2π̇r∂iπr∂iπa + (∂iπa)2π̇a/4 are invariant. From these, one

recovers the usual cubic interactions π̇3 and (∂iπ)2π̇ expressed in the Keldysh basis. Any
deviation from these fine-tuned combinations would be associated to non-unitary dynamics.
In particular, notice that unitary combinations only involve odd powers of πa. Hence, any
even powers of πa always relate to diffusive/noise processes [22].

12The orthogonal direction is then defined by δ4 = −4α1. It would again be interesting to consider if the
entanglement tracers depends on the combination (4δ4 + α1) alone.

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Unitarization procedure. It interesting to understand if we can complete an operator is
order to obtain a unitary combination. A proposal to unitarize an operator is the following.
Let us consider an operator O (πr,πa) made of field insertions of πr, πa and their (at most
single) derivatives. The combination of operators

U [O (πr,πa)] = O
(
π+
2 ,π+

)
+O

(
π−
2 ,−π−

)
−O

(
π−
2 ,π−

)
−O

(
π+
2 ,−π+

)
(2.57)

is unitary by construction. For operators such as π2
a (the diffusion operators) that are

intrinsically non-unitary, the above combination vanishes. Also note that some operators
such as π̇rπa (the dissipation operators) are unitarized into total derivatives and so do not
add any net contribution to the open effective functional. Lastly, some of the contributions
obtained out of eq. (2.57) eventually violate symmetries of the problem (e.g. the fact that
πr non-linearly realises time-translations and boosts) and must then be discarded. This
classification allows us to rewrite Wilsonian coefficients in terms of unitary and orthogonal
directions. An interesting avenue would consist of exploring which non-unitary directions
maximise the entangling dynamics between system and environment.

2.3 Setting up expectations

The EFT coefficients are dimensionful quantities, such that [α0] = [α1] = E4, [γ1] = E5,
[β1] = E6 and [β2] = [β4] = E4. One can then ask what are the relevant scales controlling
the physics and the regime of validity of our EFT description.

2.3.1 Energy scales and canonical normalisation

To treat the problems of the scales, we first canonically normalise the fields such that they
have dimension of energy E. This canonical normalisation relates the original Wilsonian
coefficients to quantities of physical interest such as the speed of sound cs, the dissipation
scale γ or the fluctuations of the environment β1. We first define the energy scale

f4
π ≡ α0 − 2α1 . (2.58)

From this we construct the canonically normalised field

π ≡ f2
ππ. (2.59)

Notice the use of different Greek fonts to distinguish normalised and un-normalised fields. It
follows that at leading order, the curvature perturbations are given by the relation

ζ = −H

f2
π

π. (2.60)

In terms of the canonically normalised variables, the quadratic action takes the form

S
(2)
eff =

∫
d4x

{
a2π′rπ′a − c2

sa2∂iπr∂iπa (2.61)

− a3γπ′rπa + i
[
β1a4π2

a − (β2 − β4) a2π′2a + β2a2 (∂iπa)2
] }

.

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The rescaled coefficients are

c2
s ≡ α0

f4
π

, γ ≡ 2γ1
f4

π

, βi ≡
βi

f4
π

for i = 1 to 8. (2.62)

The dimensions of the parameters appearing above are

[π] = E, [fπ] = E, [cs] = E0, [γ] = E, [β1] = E2, [β2] = [β4] = E0. (2.63)

Expressed in terms of the canonically normalised variables, the cubic action becomes

S
(3)
eff = 1

f2
π

∫
d4x

{[
4α2−

3
2(c

2
s−1)

]
aπ′2r π′a+

1
2(c

2
s−1)a

[
(∂iπr)2 π′a+2π′r∂iπr∂iπa

]
(2.64)

+
(
4γ2−

γ

2

)
a2π′2r πa+

γ

2a2 (∂iπr)2 πa

+i
[
(2β7−β3)a2π′rπ′aπa+β3a2∂iπr∂iπaπa+2(β4+β6−β8)aπ′rπ′2a

−2β4a∂iπr∂iπaπ′a−2β5a3π′rπ2
a−2β6aπ′r(∂iπa)2

]
+δ1a4π3

a+(δ5−δ2)a2π′2a πa+δ2a2(∂iπa)2πa−δ4a(∂iπa)2π′a+(δ4−δ6)aπ′3a

}
,

where we defined the rescaled coefficients13

α2 ≡ α2
f4

π

, γ2 ≡ γ4
f4

π

, δi ≡
δi

f4
π

for i = 1 to 6 , (2.65)

with dimensions

[α2] = E0, [γ2] = E, [β6] = [β8] = E0, [β3] = [β7] = E, [β5] = E2,

[δ1] = E3, [δ5] = [δ2] = E, [δ4] = [δ6] = E0. (2.66)

From now on, we work in this canonical basis and use the rescaled action given in eqs. (2.61)
and (2.64).

2.3.2 Heuristic estimates

We are now in the position to carry out a rough estimate of the non-Gaussianities sourced by
the cubic operators of eq. (2.64). For simplicity, we set β2 = β4 = 0 and focus on the leading
noise term of eq. (2.61) controlled by β1. The estimate relies on the following rules:

• We can approximate πr from the amplitude of the primordial power spectrum ∆2
ζ , ac-

counting for the canonical normalisation and the leading-order relation with ζ such that

πr ∼ f2
π

H
∆ζ . (2.67)

• We can estimate spatial derivatives by the spatial momenta. The value of spatial
derivatives in different directions is, on average, the same by isotropy:

∂iπr,a ∼ kπr,a. (2.68)
13Notice that α2 can be related to the EFToI [3] parameter M4

3 = −f4
πα2.

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Adiabatic perturbations of momentum k freeze at a scale factor a∗ that depends on the
dissipation parameter γ [18]. We derive this freezing time by comparing the early and
the late time limit of the power spectrum in section 3. This leads to the relation

csk ∼ a∗H

√
H + γ

H
, (2.69)

where we use the expression (H + γ) as a shorthand reminder of a quantity that scales
to leading order in γ → ∞ as γ and to leading order in γ → 0 as H. While freezing
still occurs at wavelengths around (sound) horizon crossing at low dissipation, it is
displaced to sub (sound) horizon wavelength at large dissipation.

• The characteristic frequencies of the retarded πr and advanced πa components are
estimated in section 3 to be

π′r,a ∼ aHπr,a. (2.70)

The fact that a derivative acts similarly on πr and πa follows from consistency under
integration by part, for instance considering

a2π′rπ′a = −a2 [π′′r + 2aHπ′r
]
πa. (2.71)

Similarly consistency under integration by parts should apply to all other interactions.

• The retarded component πr evolves according to a dynamics sourced by the advanced
component πa and controlled by the environment noise β1 [75]. This sourced dynamics
implies that πr and πa are not of the same amplitude, their ratio being controlled by

πr

πa
∼ β1

H(H + γ) . (2.72)

This relation can be obtained from evaluating the equation of motion for πr at a = a∗
as done in section 3.

These prescriptions imply some hierarchies among the quadratic operators. Comparing
the kinetic terms a2π′rπ′a and c2

sa2∂iπr∂iπa and the linear dissipation a3γπ′rπa with the noise
a4β1π2

a, we observe that

a2π′rπ′a
a4β1π2

a

∼ H

H + γ
,

c2
sa2∂iπr∂iπa

a4β1π2
a

∼ 1,
a3γπ′rπa

a4β1π2
a

∼ γ

H + γ
. (2.73)

This illustrates the dynamical regimes of a driven-dissipative harmonic oscillator. At low
dissipation, a3γπ′rπa is negligible and the system is mostly controlled by the other three
operators. At large dissipation, we enter the overdamped regime in which a2π′rπ′a becomes
subdominant compared to the other contributions.

To estimate the size of non-Gaussianities, we can first approximate the ratio between
the cubic operators in eq. (2.64) and the dominant quadratic terms in eq. (2.61). Based on
the above estimate, a choice of operator that is valid both at large and small dissipation
is a4β1π2

a (or equivalently c2
sa2∂iπr∂iπa), leading to

fNL∆ζ ∼ L3
a4β1π2

a

. (2.74)

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Then we can compare our analytical prediction for the dependence on γ of the amplitude
of non-Gaussianities in the equilateral configuration with numerical results obtained in
section 4, see e.g. figure 11.

Like in the EFToI, one of the amplitudes of the unitary vertices in eq. (2.64) is controlled
by the speed of sound cs. The associated non-Gaussianities can be estimated through the
above prescriptions, leading for instance to

L3 ⊃ (c2
s − 1)
2f2

π

a (∂iπr)2 π′a → fNL ∼ (c2
s − 1)
c2

s

. (2.75)

This matches the usual expectation from [3]. We can then consider the two dissipative
vertices (∂iπr)2πa and π′2r πa controlled by γ and related to the quadratic dissipation π′rπa

through non-linearly realised boosts. Using the above prescriptions, we find for the first
operator the linear-in-γ scaling

L3 ⊃ γ

f2
π

a2(∂iπr)2πa → fNL ∼ 1
c2

s

γ

H
, (2.76)

in agreement with the results of [18, 55]. The second vertex can be estimated by

L3 ⊃ γ

f2
π

a2π′2r πa → fNL ∼ γ

H + γ
, (2.77)

which correctly reproduces the numerical results of figure 11. Noise terms quadratic in πa

can also be estimated in the same manner, leading for instance to

L3 ⊃ iβ5
f2

π

a3π′rπ2
a → fNL ∼ β5

β1
. (2.78)

Noticeably, this ratio of the noise amplitudes β1 and β5 is independent of the dissipation
parameter γ and leads to approximately constant fNL for any value of γ/H, as seen in the
specific model studied in [55] and further discussed in section 5. At last, cubic operators
in πa also source a bispectrum signal such as

L3 ⊃ δ1
f2

π

a4π3
a → fNL ∼ δ1

β2
1
(H + γ). (2.79)

Again our estimates correctly reproduce the numerical results of figure 11 which plateaus
at low dissipation and grows linearly at large dissipation for fixed values of β1 (red curve
in figure 11). As we discussed in section 5, the ratio of δ1 over β2

1 controls the amplitude
of the non-Gaussian noise compared to its Gaussian counterpart. Similar estimates can be
obtained for all cubic operators of eq. (2.64). This heuristic derivation correctly accounts
for all contributions of figure 11, hence providing a valuable insight to the rich physics of
the open EFT of inflation.

3 Power spectrum

Once the open effective functional is known, we can ask how one can derive a new set of
Feynman rules. To this end, first we need to solve the free theory, as it is the one that
determines the propagators. In this section, we obtain the generating functional of the

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correlation functions of the free theory of the canonically normalised fields πa and πr. We
start from the quadratic action (2.61) and introduce sources Ja and Jr. The path integral
over πa and πr results in a Gaussian partition function Z[Ja, Jr]. The functional derivatives
of Z[Ja, Jr] yield the correlators between πa and πr. Of special interest are the two-point
functions ⟨πr(η1)πr(η2)⟩ and ⟨πr(η1)πa(η2)⟩. The former is dubbed the Keldysh propagator
and encodes fluctuations of the system mediated by the presence of the environment [75].
The latter is the retarded Green function of the system, which encodes the propagation of
perturbations sourced by the environment.

3.1 Generating functional

The open effective functional in eq. (2.61) can be integrated by parts to

S
(2)
eff =

∫
d4x

{
− 1

2
(
a2π′r

)′
πa − 1

2
(
a2π′a

)′
πr +

a2c2
s

2
(
πa∂2

i πr + πr∂2
i πa

)
− a3γ

2 π′rπa + a3γ

2 π′aπr +
3a3Hγ

2 πaπr

+ i

[
a4β1π2

a + (β2 − β4)
(
a2π′a

)′
− a2β2πa∂2

i πa

]}
, (3.1)

where we defined H ≡ a′/a. This integration by parts can be done thanks to the boundary
condition on πa. Even though some terms now seem to break invariance under the non-linear
transformation of πr, there is a balance between coefficients such that the action is still
invariant. The form of eq. (3.1) allows us to write the action as a bilinear on the fields

S
(2)
eff = −1

2

∫
d4x (πr, πa)

(
0 D̂A

D̂R −2iD̂K

)(
πr

πa

)
, (3.2)

the matrix being a second order differential operator acting to the right, which is made of

D̂A ≡ a2∂2
η + a3

(2H
a

− γ

)
∂η − a2c2

s∂2
i − 3a3Hγ, (3.3)

D̂R ≡ a2∂2
η + a3

(2H
a

+ γ

)
∂η − a2c2

s∂2
i , (3.4)

D̂K ≡ a4β1 + a2(β2 − β4)
(
∂2

η + 2H∂η

)
− a2β2∂2

i . (3.5)

The following path integral of the free theory computes the diagonal of the density
matrix of the system

ρred [π, π; η0] =
∫ π

Ω
Dπr

∫ 0

Ω
Dπa exp

{
iS

(2)
eff [πr, πa]

}
. (3.6)

Upon the introduction of sources, we obtain the generating function

Z[Jr, Ja] =
∫ π

Ω
Dπr

∫ 0

Ω
Dπa exp

{
− i

2

∫
d4x (πr, πa)

(
0 D̂A

D̂R −2iD̂K

)(
πr

πa

)

+
∫

d4x (Jrπr + Jaπa)
}

(3.7)

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Completing the square for πa and πr allows us to factorise the dependence on the sources
Ja and Jr. This is done by introducing a shift in the path integral(

πr

πa

)
=
(
Πr

Πa

)
+
∫

d4y

(
A11(x, y) A12(x, y)
A21(x, y) 0

)(
πr

πa

)
(3.8)

Demanding that the terms linear in Πr/a vanish, we find14

− i

2

(
0 D̂A

D̂R −2iD̂K

)(
A11(x, y) A12(x, y)
A21(x, y) 0

)
+ 1

2

(
δ(x − y) 0

0 δ(x − y)

)
=
(
0 0
0 0

)
, (3.9)

where the Dirac delta δ(x − y) is a tensor density. These equations can be rewritten in a
covariant form such that it becomes straightforward to recognise the advanced and retarded
propagators

D̂R(x)A12(x, y) = −iδ(x − y) , A12(x, y) = −iGR(x, y) , GR(x0 < y0) = 0, (3.10)
D̂A(x)A21(x, y) = −iδ(x − y) , A21(x, y) = −iGA(x, y) , GA(x0 > y0) = 0. (3.11)

A fundamental property of the retarded and advanced propagator is that they are mapped
to each other under the exchange of the arguments:

GR(x, y) = GA(y, x) . (3.12)

The Keldysh propagator is obtained from the matrix element A11(x, y) = −iGK(x, y), which
obeys the differential equation

D̂R(x)A11(x, y) = 2D̂K(x)GA(x, y). (3.13)

Notice that GK is thus not a Green’s function of some equation of motion. We choose to
symmetrise in x ↔ y such that

GK(x, y) = i

∫
d4z

√
−g(z)GR(x, z)D̂K(z)GA(z, y)

+ i

∫
d4z

√
−g(z)GR(y, z)D̂K(z)GA(z, x) (3.14)

where we have used the property (3.12) to write A11(x, y) in the most symmetric way.15

Note that causality is implemented in a natural way as the retarded propagator requires
x0 > z0 and the advanced propagator requires z0 < y0.

This leaves the partition function to be

Z[Jr,Ja] = exp
{
− i

2

∫
d4x

∫
d4y (Jr(x),Ja(x))

(
GK(x,y) GR(x,y)
GA(x,y) 0

)(
Jr(y)
Ja(y)

)}
(3.15)

14There exists a similar equation where we integrate by parts on D̂A/R/K which yields the same information.
15The generalisation to non-local noises is straightforward from here by upgrading the local D̂K(z) to

a non-local D̂K(z1 − z2), where now we have the integrand to be GR(x, z1)D̂K(z1 − z2)GA(z2, y) with the
appropriate factors of the square root of the metric.

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where we used the fact that Z[0, 0] = 1 in the closed time contour from trace normalisation [75].
The two-point function reduces to

⟨πr(x1), πr(x2)⟩ =
δ2

δJr(x1)δJr(x2)
Z[Ja, Jr]

∣∣∣∣∣
Jr,a=0

= −iGK(x1, y2). (3.16)

Evaluating the two-point function at coincident times in Fourier space provides an expression
for the power spectrum of the system

⟨πr(k, η0)πr(k′, η0)⟩ = P π
k (η0)(2π)3δ(k + k′) with P π

k (η0) = −iGK(k; η0, η0). (3.17)

3.2 Flat space intuition

At linear order, the dissipative theory introduces new ingredients compared to the unitary
case, which are the dissipation coefficient γ and the three noises controlled by β1, β2 and β4.
Before discussing the inflationary case, it is instructive to get familiar with the formalism
through some explicit flat space computation. In flat space, the open effective functional reads

S
(2)
eff [πr, πa] =

∫
d4x

(
πr πa

)( 0 −1
2
(
∂2

t + γ∂t − c2
s∂2

i

)
−1

2
(
∂2

t − γ∂t − c2
s∂2

i

)
i
[
β1 − (β2 − β4)∂2

t + β2∂2
i

])(πr

πa

)
.

(3.18)

The equations of motion for the propagators read(
∂2

t ± γ∂t + c2
sk2
)

GR/A(k; t1, t2) = δ(t1 − t2) , (3.19)

and, from eq. (3.14),

GK(k; t1, t2) = 2i

∫
dt′GR(k; t1, t′)

[
β1 − (β2 − β4)∂2

t′ + β2k2
]

GA(k; t′, t2). (3.20)

The dynamics is easily solved in frequency space, where

GR/A(k;ω) = − 1
ω2 ± iγω − c2

sk2 = − 1
(ω− − ω+)

[ 1
ω − ω−

− 1
ω − ω+

]
(3.21)

with

ω± ≡ −i
γ

2 ± Eγ
k and Eγ

k ≡

√
c2

sk2 − γ2

4 . (3.22)

Two comments are in order. First, notice that dissipation is incompatible with linearly
realised Lorentz boosts. Instead, boosts are non-linearly realised only on πr. Second, there is
no need to introduce any iϵ in the retarded-advanced propagators because the correct position
of the poles is already determined by the non-vanishing dissipation. Related to this, in the
presence of dissipation the retarded Green’s function has only a finite memory of the past,
decaying exponentially in time. As a consequence, time integrals involving GR/A already
converge without the need to specify any small rotation into the complex plane.

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We assume Eγ
k to be real (damped regime), keeping in mind that the overdamped regime

for which Eγ
k ∈ iR can be obtained by analytic continuation. Performing the temporal

inverse Fourier transform, we obtain

GR/A(k; τ) =
∫ dω

2π
eiωτ GR/A(k;ω) = ∓

sin
[
Eγ

k τ
]

Eγ
k

e−
γ
2 τ θ (±τ) , (3.23)

where for GR/A(k; t1, t2) we defined τ ≡ t1 − t2. Notice that the presence of dissipation
automatically ensures convergence, without the need for an iϵ. The Keldysh component is
also easily obtained in frequency space where

GK(k;ω) = 2GR(k;ω)D̂K(k;ω)GA(k;ω) = 2i
β1 + (β2 − β4)ω2 + β2k2

(ω2 − c2
sk2)2 + γ2ω2

. (3.24)

One can analyse the pole structure and obtain the real-space propagator from

GK(k; τ) = 2i

∫ dω

2π

[
β1 + (β2 − β4)ω2 + β2k2] e−iωτ∏4

i=1(ω − ωi)
(3.25)

with the poles satsfying ω2 = −ω1, ω3 = −ω∗1, ω4 = −ω∗1 and

ω1 = Eγ
k + i

γ

2 . (3.26)

A long but straightforward derivation leads to

GK(k; τ) = i
e−

γ
2 τ

c2
sk2

[2
γ

[
β1 + c2

sk2(β2 − β4) + k2β2
]
cos

(
Eγ

k τ
)

+
[
β1 − c2

sk2(β2 − β4) + k2β2
] sin (Eγ

k τ
)

Eγ
k

]
(3.27)

In the coincident time limit, we obtain the dissipative power spectrum

P π
k = 2β1

γc2
sk2 + 2(β2 − β4)

γ
+ 2β2

c2
sγ

. (3.28)

Note that the vacuum contribution to the power spectrum, the usual P π
k = 1/2Eγ=0

k term,
is absent because of the exponential decay in time caused by dissipation. Nevertheless,
with hindsight, one could still recover the standard vacuum Minkowski power spectrum
by taking both β1 and γ to zero while keeping their ratio fixed. From the fact that the
equal-time power spectrum must be non-negative we conclude that the β’s must be positive.
This computation highlights several features of the formalism. While GR(k, τ) encodes
the dynamics but is oblivious to the state of the system, GK(k, τ) captures the state of
the environment by probing the statistics of fluctuations [75]. The final outcome is an
interplay between the dissipation of the system into its surrounding and the fluctuations of
the environment getting imprinted onto the observable sector. Crucially, these two effects
cannot be easily disentangle from one another.

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3.3 Dissipative and noise-induced power spectrum

Back to cosmology, we now extend the previous computation to account for the expanding
spacetime. Embedded in the inflating background, the retarded and Keldysh differential
operators become

D̂R = 1
H2η2

[
∂2

η −
2 + γ

H

η
∂η − c2

s∂2
i

]
(3.29)

and

D̂K = 1
H2η2

[
β1

H2η2 − (β2 − β4)
(

∂2
η − 2

η
∂η

)
+ β2∂2

i

]
. (3.30)

From now on, we set cs = 1 for simplicity and discuss the effect of cs at the end of the
computation.

Scaling dimensions. Before deriving the propagators of the theory, let us briefly comment
on the two homogeneous solutions of D̂R in Fourier space (mode functions). Obeying the
dynamical equation (

∂2
η −

2 + γ
H

η
∂η + k2

)
πk = 0, (3.31)

the two homogeneous solutions are given by

πk(η) ∝ ηνγ H(1)
νγ

(−kη) and ∝ ηνγ H(2)
νγ

(−kη) , (3.32)

where

νγ ≡ 3
2 + γ

2H
, (3.33)

and H(1) and H(2) are Hankel functions. Selecting positive frequency mode functions of the
form e−ikη in the asymptotic past, −kη ≫ 1, one can safely discard one of the two solutions.
At late times, where −kη ≪ 1, the dissipative mode functions acquire scaling solutions

πk(z = −kη) = O+z∆+
[
1 +O(z2)

]
+O−z∆−

[
1 +O(z2)

]
, (3.34)

where O+ and O− depend on γ and H, and the scaling dimensions are

∆+ = 0 and ∆− = 3 + γ

H
. (3.35)

Comparing this result to the well-known relation for closed systems ∆+ +∆− = d, involving
representations related by a shadow transform, we note that dissipation acts in the same
way as a continuation to an non-integer number of dimensions. It would be interesting to
investigate if this suggests a useful regularisation procedure for loop contributions, different
from the one used in [62, 87, 88]. These relations crucially depart from the unitary theory case.
Yet, let us stress it would be misleading to derive the power spectrum simply by squaring these
mode functions. Indeed, in an open theory, dissipation is inextricably linked to the fluctuations
generated by the environment [32]. Instead, the power spectrum is derived as follow.

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Retarded Green function. Eq. (3.29) determines the equations of motion obeyed by
the Green function(

∂2
η1 −

2 + γ
H

η1
∂η1 + k2

)
GR(k; η1, η2) = H2η2

1δ(η1 − η2). (3.36)

The normalisation is fixed by the continuity of the retarded Green function at coincident
time and its first derivative discontinuity is controlled by the time-dependent prefactor of
the ∂2

η1 term in eq. (3.36), that is

GR(k; η2, η2) = 0, and ∂η1GR(k; η1, η2)
∣∣∣
η2=η1

= H2η2
2. (3.37)

This imposes

GR(k; η1, η2) =
π

2H2(η1η2)
3
2

(
η1
η2

) γ
2H

ℑm
[
H

(1)
3
2 + γ

2H

(−kη1)H(2)
3
2 + γ

2H

(−kη2)
]

θ(η1 − η2) (3.38)

which is consistent with [18, 55]. It turns out to be sometime convenient to express the
retarded Green function in terms of Bessel functions of the first kind

GR(k; η1, η2) =
π

2
H2

k3

(
z1
z2

)νγ

z3
2
[
Yνγ (z1)Jνγ (z2)− Jνγ (z1)Yνγ (z2)

]
(3.39)

where zi ≡ −kηi.

Keldysh propagator. We now turn our attention to the computation of the Keldysh
propagator of the theory from which the observables such as the power spectrum can be
derived.16 The Keldysh function is obtained from

GK(k; η1, η2) = i

∫
dη′GR(k; η1, η′)D̂K(η′)GA(k; η′, η2) + (η1 ↔ η2) (3.40)

= i

∫
dη′GR(k; η1, η′)D̂K(η′)GR(k; η2, η′) + (η1 ↔ η2). (3.41)

Notice that this equation has precisely the structure one would expect for the power spectrum
of a field obeying the Langevin equation with source fluctuations possessing a power spectrum
D̂K(η′), as we will see in section 5.2. Injecting eq. (3.30) into the above expressions, we
obtain three contributions

GK
1 (k; η1, η2) = i

β1
H4

∫ η2

−∞

dη′

η′4
GR(k; η1, η′)GR(k; η2, η′) + (η1 ↔ η2) (3.42)

GK
2 (k; η1, η2) = i

β4 − β2
H2

∫ η2

−∞

dη′

η′2
GR(k; η1, η′)

(
∂2

η′ −
2
η′

∂η′

)
GR(k; η2, η′) + (η1 ↔ η2)

(3.43)

GK
3 (k; η1, η2) = i

β2k2

H2

∫ η2

−∞

dη′

η′2
GR(k; η1, η′)GR(k; η2, η′) + (η1 ↔ η2) , (3.44)

16Note that we do not deduce a noise-free power spectrum out of the mode functions derived above contrarily
to [18, 55]. Even in non-equilibrium settings where the fluctuation-dissipation relations do not strictly apply,
the two effects are indissociable [89, 90], which is captured in the formalism below.

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which correspond to the three noise terms appearing in eq. (3.30). Here we assumed η2 ≤ η1
without loss of generality. The power spectrum is obtained from the coincident time limit
η1 = η2 = η0.

Let us compute the first contribution eq. (3.42), which corresponds to the noise term
of [18, 55] as we will make explicit in section 5. Injecting eq. (3.39) in eq. (3.42), we obtain
the Keldysh propagator

GK
1 (k; η1, η2) = i

π2β1
4k3 (z1z2)νγ

{
Yνγ (z1)Yνγ (z2)A(1)

νγ
(z2) + Jνγ (z1)Jνγ (z2)C(1)

νγ
(z2) (3.45)

−
[
Jνγ (z1)Yνγ (z2) + Jνγ (z2)Yνγ (z1)

]
B(1)

νγ
(z2)

}
+ (1 ↔ 2)

where

A(1)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ J2

νγ
(z′) (3.46)

B(1)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Jνγ (z′)Yνγ (z′) (3.47)

C(1)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Y 2

νγ
(z′) (3.48)

are complicated functions given explicitly in eqs. (B.4), (B.5) and (B.6) respectively.

Power spectrum. Considering that ζ = −Hπ/f2
π where π is the canonically normalised

field, the reduced power spectrum

∆2
ζ(k) ≡

k3

2π2 Pζ(k) with ⟨ζkζ−k⟩ = (2π)3δ(k + k′)Pζ(k) (3.49)

is obtained in the coincident time limit of the Keldysh propagator given in eq. (3.45), such that

Pζ(k) =
π2β1
8k3

H2

f4
π

z2νγ

[
Y 2

νγ
(z)A(1)

νγ
(z) + J2

νγ
(z)C(1)

νγ
(z)− 2Jνγ (z)Yνγ (z)B(1)

νγ
(z)
]

. (3.50)

In the super-Hubble regime z ≪ 1, the power spectrum freezes and we recover the result
from [55], that is

∆2
ζ(k) =

1
4

β1
H2

H4

f4
π

22νγ
Γ (νγ − 1) Γ (νγ)2

Γ
(
νγ − 1

2

)
Γ
(
2νγ − 1

2

) (3.51)

which is indeed dimensionless. Keeping in mind that νγ ≡ 3
2 + γ

2H , one can expand this
result in the small and large dissipation regime leading to

Dissipative power-spectrum

∆2
ζ(k) ∝



β1
H2

H4

f4
π

+O
(

γ

H

)
, γ ≪ H,

β1
H2

H4

f4
π

√
H

γ

[
1 +O

(
H

γ

)]
, γ ≫ H.

(3.52)

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The observational constraint ∆2
ζ = 10−9 is easily obtained by imposing hierarchies between

the various scales of the problem. Note that if one further imposes thermal equilibrium of
the environment such that the fluctuation-dissipation relation holds, the dynamical KMS
symmetry imposes β1 = 2πγT where T is the environment temperature [22], such that in
the large dissipation regime (γ ≫ H)

∆2
ζ ∝ T

H

H4

f4
π

√
γ

H
(3.53)

which reproduces the warm inflation expectation [14, 15, 36–38].

The two other contributions given in eqs. (3.43) and (3.44) follow accordingly, the details
of which are deferred in appendix C. They also lead to equally valid scale invariant power
spectra on super-Hubble scales

∆2
ζ(k) ⊃



15
32(β4 − β2)

H4

f4
π

22νγ
Γ (νγ − 2) Γ (νγ)2

Γ
(
νγ − 3

2

)
Γ
(
2νγ − 1

2

) , Eq. (3.43),

3
16β2

H4

f4
π

22νγ
Γ (νγ − 2) Γ (νγ)2

Γ
(
νγ − 3

2

)
Γ
(
2νγ − 3

2

) , Eq. (3.44),
(3.54)

which expand in the large dissipation limit (γ ≫ H) to

∆2
ζ(k) ⊃


(β4 − β2)

H4

f4
π

√
H

γ

[
1 +O

(
H

γ

)]
, Eq. (3.43),

β2
H4

f4
π

√
γ

H

[
1 +O

(
H

γ

)]
, Eq. (3.44).

(3.55)

The obtained contributions may again satisfy the observational constraint ∆2
ζ = 10−9 by

imposing some hierarchy between the various scales (more stringent for the last contribution
due to the

√
γ/H enhancement in the spatial derivative case).

Inclusion of the speed of sound. As we focused on the decoupling limit, we were able
to choose units such that cs = 1. However, if we aim in the future to include coupling to
gravity, or to look in details at non-linearly realised boosts, we need to include a speed of
sound for the Goldstone mode that may differ from unity. The speed of sound will appear in
various observables computed in this work. For the power spectrum we find

∆2
ζ(k) ⊃



1
4c3

s

β1
H2

H4

f4
π

22νγ
Γ (νγ − 1) Γ (νγ)2

Γ
(
νγ − 1

2

)
Γ
(
2νγ − 1

2

) ,

15
32c3

s

(β4 − β2)
H4

f4
π

22νγ
Γ (νγ − 2) Γ (νγ)2

Γ
(
νγ − 3

2

)
Γ
(
2νγ − 1

2

) ,

3
16c5

s

β2
H4

f4
π

22νγ
Γ (νγ − 2) Γ (νγ)2

Γ
(
νγ − 3

2

)
Γ
(
2νγ − 3

2

) ,

(3.56)

the different powers on cs differentiate between time and spatial derivatives. The speed of
sound also enters the discussion of horizon exit, as it takes z = −kη to z = −cskη.

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Imprint of the perturbations. Before closing the discussion on the power spectrum, let
us further comment on the freezing of the adiabatic fluctuations on large scales. The early
time behaviour of the power spectrum (3.50) is given in the sub-Hubble regime z ≫ 1 by

Pζ(k; η) =
1
2k3

β1
H2

H4

f4
π

z

νγ − 1 , (3.57)

which is O(z) suppressed at early time compared to the usual closed system Bunch-Davies
vacuum solution. In the heuristic estimates of section 2.3, we made use of the fact that
most of the perturbations are imprinted on the statistics around a particular number of
e-foldings characterised by z∗ ≡ −cskη∗ = csk/(a∗H). The characteristic scale of z∗ given
in eq. (2.69), can be derived by matching the early and late time behaviours of the power
spectrum given in eqs. (3.57) and (3.51) respectively. From this analysis, we conclude the
curvature perturbations freeze around

z∗ = 22νγ−2 Γ3(νγ)
Γ
(
νγ − 1

2

)
Γ
(
2νγ − 1

2

) ≈


1 γ ≪ H,
√

π

2

√
γ

H
γ ≫ H,

(3.58)

summarized through

z∗ ∼

√
γ + H

H
. (3.59)

While the freezing still occurs around (sound) horizon crossing when dissipation is small,
we recover the known fact that freezing occurs sooner in the large dissipation regime, on
sub (sound) horizon scales [18, 55].

Once z∗ is known, we can inject it in the equation of motion for the Goldstone mode πr

to estimate the characteristic frequency17 and the difference in amplitude compared to the
advanced component πa. We consider the equation of motion for the retarded component
πr obtained from the quadratic action in eq. (2.61), that is

π′′r + 2aH

(
1 + γ

2H

)
π′r + c2

sk2πr = 2ia2β1πa. (3.60)

Eq. (3.60) is sourced term on the right-hand side by the environment noise controlled by
β1. Let us first estimate the characteristic frequency of the system, π′r ∼ aωπr where ω is
to be determined from eq. (3.60). We can exploit eq. (3.59) to estimate spatial derivatives
at freezing time, leading to c2

sk2 ∼ a2H(H + γ). It implies that[
ω2 + (3H + γ)ω + H (H + γ)

]
πr ∼ 2iβ1πa. (3.61)

At small dissipation, there is no doubt that ω is controlled by H as one recovers the usual
closed system dynamics (up to the peculiarity of being sourced by the noise β1). At large
dissipation, the interplay between the two characteristic timescales H and γ renders the
estimate less straightforward. Eq. (3.61) describes a forced damped harmonic oscillator. In

17As discussed around eq. (2.70) in heuristic estimate of section 2.3, consistency under integration by part
implies that πr and πa have the same characteristic frequency.

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this situation, any solution has two contributions: a transient one, with a lifetime T ∼ 1/γ

after which it decays away and a stationary one, with a much lower characteristic frequency.
In this case, the lowest frequency is given by the Hubble parameter H. Therefore, ω ∼ H

in both regimes, as stated in eq. (2.70). Combining both estimates of c2
sk2 and ω with

eq. (3.60), we finally obtain the ratio between πr and πa stated in eq. (2.72), which completes
the derivation of the heuristic rules used in section 2.3.

4 Bispectrum

Beyond the Gaussian statistics, higher-point functions are computed following the usual
perturbative approach. Once the propagators are known, we can derive a new set of Feynman
rules from which we construct correlators order by order in perturbation theory. In this
section, after reviewing the standard in-in treatment of interactions, we study the structure
of the bispectrum (three-point function in Fourier space) both in flat space where analytic
results are easily obtained and in de Sitter, which exhibits a specific phenomenology.

4.1 Interactions

Interactions are treated as in the familiar in-in approach, see [39] for a review. This provides
a comforting unified treatment for the cases of an open and closed system. The only small
difference from some references is that we find it convenient to work in the Keldysh basis, πr,a

instead of the π± basis. Expectation values of Q(η) ≡ π(η, x1) · · ·π(η, xn) are defined through

⟨Q⟩ =
∫

DπrDπa [πr(η, x1) · · ·πr(η, xn)] eiSeff [πr,πa]
∣∣∣
πa(η0)=0

(4.1)

where initial conditions lie on the boundaries of the path integral. Notice that since for
cosmology we are only interested in the product of fields rather than their momentum
conjugate, we are effectively only probing the diagonal of the density matrix prepared by the
Schwinger-Keldysh path integral. Once the generating functional is known, ⟨Q⟩ is extracted
out of functional derivatives

⟨Q⟩ = δ

δJr(η, x1)
· · · δ

δJr(η, xn)
Z[Jr, Ja]

∣∣∣∣
Jr,a=0

(4.2)

just as we did in eq. (3.16) for the two-point function. The Feynman rules are derived as in [39]
and lead to figure 3. There are two propagators that are −iGK(k; η, η′) (continuous line)
which connects πr(k; η) to πr(k; η′) and −iGR(k; η, η′) (continuous-to-dashed line), which
connects πr(k; η) to πa(k; η′). Notice that the latter is directional, with the continuous line
being attached to the πr(k; η) insertion and the dashed part to the πa(k; η′).18 If there is an
operator gπm

r πn
a appearing in Leff , it leads to a ig contribution per vertex with m continuous

legs and n dashed legs. Then, diagrams evaluation follows the exact same rules as in [39].
As seen from eq. (4.1), external legs connecting to the conformal boundary η0 → 0− are
continuous, corresponding to πr insertions. An example is given in figure 4 for a contact
bispectrum. One can easily be convinced of these Feynman rules by recovering some known
results in flat space as we do in appendix D.

18There is no propagator connecting πa(k; η) to πa(k; η′) which is a consequence of the causality structure
of the closed time contour [75].

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Figure 3. Feynman rules in the Keldysh basis.

Figure 4. The two diagrams to compute the interaction given in eq. (4.3). Left: diagram corresponding
to the vertex π̇2

r π̇a, resulting in eqs. (4.5). Right: diagram corresponding to the vertex π̇3
a, resulting

in (4.6).

4.2 Flat space intuition

Before turning our attention to primordial cosmology, it is instructive to discuss the generic
structure of the contact three-point functions in Minkowski in the presence of dissipation.
We derive these results following the Feynman rules enumerated in figure 3, the propagators
being given in eqs. (3.23) and (3.27). We now need to consider both unitary and non-unitary
operators presented in eqs. (2.42), (2.43) and (2.44). For the sake of clarity, we consider the
case where β2 = β4 = 0 but the results can be extended to accommodate for non-zero β2
and β4 if desired. We also set cs = 1 for simplicity. In appendix D, we further discuss how
to recover the unitary results in the absence of dissipation and noise.

π̇3 interactions. Let us first consider

Lint = −α

3!
(
π̇3

+ − π̇3
−

)
= −α

2

(
π̇2

r π̇a + 1
12 π̇3

a

)
, (4.3)

where the minus sign in front comes from the Lorentzian signature, assuming α > 0.19 In
the unitary shift symmetric case, the operators π̇3 and (∂iπ)2π̇ do not generate any contact
bispectrum in Minkowski due to the time reversal symmetry t → −t and π → −π (one can
check explicitly that there is zero contribution to the bispectrum, each diagram vanishing
independently). Dissipation spontaneously breaks this symmetry and we observe that the
diagrams now lead to a non-zero contribution to the bispectrum.

19In the EFT construction given in eqs. (2.42), (2.43) and (2.44), this case corresponds to (4α2 − 3α1)/f6
π =

−α/2 and (δ4 − δ6)/f6
π = −α/24 which indeed lies in the unitary direction.

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Let us consider the bispectrum

⟨πk1πk2πk3⟩ ≡ (2π)3δ(k1 + k2 + k3)Bπ(k1, k2, k3). (4.4)

We have two contact diagrams to consider presented in figure 4, Bπ(k1, k2, k3) = Dπ̇3
1 + Dπ̇3

2 .
The first one leads to

Dπ̇3
1 = α

2

∫ t0

−∞(1±iϵ)
dt
[
∂tG

K(k1; t0, t)
][

∂tG
K(k2; t0, t)

][
∂tG

R(k3; t0, t)
]
+5 perms. (4.5)

Upon injecting eqs. (3.23) and (3.27) in eq. (4.5) and summing over the six possible per-
mutations, we observe that Dπ̇3

1 = 0. The second diagram is made of the πa components
only, and reads

Dπ̇3
2 = α

24

∫ t0

−∞(1±iϵ)
dt
[
∂tG

R(k1; t0, t)
][

∂tG
R(k2; t0, t)

][
∂tG

R(k3; t0, t)
]
+5 perms. (4.6)

Under the same procedure, this diagram leads to a non-zero contribution to the bispectrum
such that

Bπ(k1, k2, k3) =
αγ

2
Poly6 (e

γ
1 , eγ

2 , eγ
3)

Singγ

, (4.7)

where we defined the singularity structure

Singγ =
∣∣∣∣Eγ

1 + Eγ
2 + Eγ

3 + 3
2 iγ

∣∣∣∣2 ∣∣∣∣−Eγ
1 + Eγ

2 + Eγ
3 + 3

2 iγ

∣∣∣∣2
×
∣∣∣∣Eγ

1 − Eγ
2 + Eγ

3 + 3
2 iγ

∣∣∣∣2 ∣∣∣∣Eγ
1 + Eγ

2 − Eγ
3 + 3

2 iγ

∣∣∣∣2 , (4.8)

remembering that Eγ
k =

√
c2

sk2 − γ2/4. This singularity structure captures most of the
specificities of the non-unitary dynamics. It emerges from time integrals of the form∫ t0

−∞(1±iϵ)
dte±iEγ

1 (t0−t)e±iEγ
2 (t0−t)e±iEγ

3 (t0−t)e−
3
2 γ(t0−t) , (4.9)

which follow from the structure of the propagators given in eqs. (3.23) and (3.27). Physically,
it represents 3 ↔ 0 and 2 ↔ 1 interactions mediated by the environment. Fluctuations
generate folded singularities while dissipation displaces the pole and regularises the divergence.
Consequently, the singularity is not located in the physical plane and the bispectrum remains
under perturbative control over the whole kinematical space. As we will see below, this
singularity structure is generic and does not depend on the details on the interactions. The
details of the particular interaction are imprinted into Polyn, which is a nth-order polynomial
of the elementary symmetric polynomials

eγ
1 = Eγ

1 + Eγ
2 + Eγ

3 , eγ
2 = Eγ

1 Eγ
2 + Eγ

2 Eγ
3 + Eγ

1 Eγ
3 eγ

3 = Eγ
1 Eγ

2 Eγ
3 . (4.10)

For this specific case,

Poly6 (e
γ
1 ,eγ

2 ,eγ
3)= 243γ6−792γ4eγ

2+396γ4 (eγ
1)

2+576γ2 (eγ
2)

2−1088γ2eγ
2 (e

γ
1)

2

+272γ2 (eγ
1)

4+1024γ2eγ
1eγ

3+512(eγ
1)

2 (eγ
2)

2−384(eγ
1)

4
eγ

2 (4.11)
−1024eγ

1eγ
2eγ

3+64(eγ
1)

6+512eγ
3 (e

γ
1)

3+768(eγ
3)

2
.

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Figure 5. The shapes of accessible triangles fulfilling the momentum conservation δ(k1 + k2 + k3).
The triangles are parameterised along x2 ≡ k2/k1 and x3 ≡ k3/k1 such that x3 < x2 < 1 and
x2 + x3 > 1, the two conditions to construct closed triangles. A region of particular interest for the
scope of this article is the folded region where x2 + x3 ≃ 1, which interpolates between the squeezed
and isofolded points. The singularity structure discussed in eq. (4.8) peaks close to the folded region,
and provides a smoking gun of dissipative dynamics.

It is instructive to consider the shape of the bispectrum generated by this interaction.
The momentum conservation in δ(k1 + k2 + k3) in eq. (4.4) forces the momenta to form a
close triangle. As different inflationary models predict maximal signals in different triangular
configurations (see e.g. [42, 91]), the shape function

S(x2, x3) ≡ (x2x3)2 B(k1, x2k1, x3k1)
B(k1, k1, k1)

(4.12)

is an informative probe of the mechanism generating primordial non-Gaussianities. The
variables x2 ≡ k2/k1 and x3 ≡ k3/k1 control the shape of the triangles and are restricted by
δ(k1 + k2 + k3) to the region max(x3, 1 − x3) ≤ x2 ≤ 1. In figure 5, we present the main
shapes of interest in this article. It appears that the singularity structure Singγ presented
in eq. (4.8) exhibits two different behaviour depending the magnitude of the dissipation
coefficient γ. In the strong dissipation regime (right panel of figure 6), the 3

2 iγ appearing
in eq. (4.8) always dominates the bispectrum contribution such that the signal peaks in the
equilateral shape where x2 ≃ x3 ≃ 1. On the contrary, in the small dissipation regime, Singγ

can become small in the folded region where x2 + x3 ≃ 1 such that the signal predominantly
peaks near the isofolded configuration where x2 ≃ x3 ≃ 1/2 (left panel of figure 6).

This type of singularities have already been encountered in cosmology, mostly in the
context of non-Bunch-Davies initial states [40–45]. The main difference with the current
investigation is that, due to the presence of the dissipative environment, the would-be folded
singularity is regularised, i.e. Singγ ̸= 0 whenever γ ̸= 0. This clearly appears in figure 7
where the peak of the shape function as one approaches the folded singularity x2 + x3 ≃ 1
is plotted on the top-right panel. The resolution of the singularity is a useful feature of the
formalism as it allows one to keep perturbative control over all configurations. In particular,
one does not have to introduce an artificial cutoff to handle the dissipative interactions.

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0.2 0.4 0.6 0.8 1.0
x3

0.6

0.7

0.8

0.9

1.0

x2

γ  0.05

S(x2, x3)

1

2

3

4

5

6

0.2 0.4 0.6 0.8 1.0
x3

0.6

0.7

0.8

0.9

1.0

x2

γ  5

S(x2, x3)

0.1

0.3

0.5

0.7

0.9

Figure 6. Shape function of the contact bispectrum generated by π̇3
a in Minkowski given in eq. (4.7).

Left: in the small dissipation regime, the singularity structure Singγ given in eq. (4.8) becomes small
in the folded region x2 +x3 ≃ 1 which enhances the bispectrum. Right: in large dissipation regime, the
imaginary contributions from the dissipation in dominates Singγ such that no bispectrum enhancement
is observed in the x2 + x3 ≃ 1 folded region.

𝐴

𝐵

𝐶

0.5 0.6 0.7 0.8 0.9 1.0

x

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

S
(x
,x

)

A γ = 0.08

γ = 0.12

γ = 0.16

0.0 0.2 0.4 0.6 0.8 1.0

x3

0.0

0.2

0.4

0.6

0.8

1.0

S
(1
,x

3
)

B

γ = 0.08

γ = 0.12

γ = 0.16

0.0 0.1 0.2 0.3 0.4 0.5

x

0.0

0.5

1.0

1.5

2.0

2.5

S
(1
−
x
,x

)

C

γ = 0.08

γ = 0.12

γ = 0.16

Figure 7. Top left: 3d shape function of the contact bispectrum generated by π̇3
a in Minkowski given

in eq. (4.7) for three different values of the dissipation parameter γ ∈ [0.08; 0.12; 0.16]. We observe
the equilateral-to-folded transition of the shape function as the dissipation parameter decreases. Top
right: 2d cut along the direction x2 = x3 = x appearing in red in the 3d plot. The singularity is
resolved such that the bispectrum remains well defined for any triangular configuration and any value
of the dissipation parameter γ. Bottom left: 2d cut along the direction x2 = 1 appearing in black in
the 3d plot. Consistency relations ensure the signal vanishes in the squeezed limit x3 ≪ 1. Bottom
right: 2d cut along the direction x2 = 1− x3 appearing in purple in the 3d plot. Consistency relations
are again observed in the squeezed limit x3 ≪ 1.

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(
4γ2 − γ

2
)

π̇2
rπa Bπ = −3(8γ2−γ)

2γ2
β2

1
f6

π

Poly4(eγ
1 ,eγ

2 ,eγ
3)

Singγ

γ
2 (∂iπr)2 πa Bπ = −8γ

γ2
β2

1
f6

π

K2

k2
1k2

2k2
3

iβ3∂iπr∂iπaπa Bπ = β3
16

β1
f6

π

K2

k2
1k2

2k2
3

Poly8(eγ
1 ,eγ

2 ,eγ
3)

Singγ

2i (β4 + β6 − β8) π̇rπ̇2
a Bπ = β4+β6−β8

8γ
β1
f6

π

Poly6(eγ
1 ,eγ

2 ,eγ
3)

Singγ

−2i
[
β4∂iπr∂iπaπ̇a + β5π̇rπ2

a + β6π̇r(∂iπa)2] Bπ = 3[(2β6−β4)K2−β5]
γ

β1
f6

π

Poly4(eγ
1 ,eγ

2 ,eγ
3)

Singγ

δ1π3
a + δ2(∂iπa)2πa Bπ = −3

4
δ1−δ2K2

f6
π

Poly4(eγ
1 ,eγ

2 ,eγ
3)

Singγ

(δ5 − δ2) π̇2
aπa Bπ = 1

32
δ2−δ5

f6
π

Poly6(eγ
1 ,eγ

2 ,eγ
3)

Singγ

(δ4 − δ6) π̇3
a Bπ = 3γ

16
δ6−δ4

f6
π

Poly6(eγ
1 ,eγ

2 ,eγ
3)

Singγ

Table 1. Non-vanishing contributions to the contact bispectrum from the cubic interactions of
eqs. (2.42), (2.43) and (2.44). Singγ is given in eq. (4.8) and we defined K2 ≡ (k1.k2 +k1.k3 +k2.k3).
The Polyn appearing in the expressions are polynomials of the variables given in eq. (4.10) whose
details are not given for readability reasons.

Other interactions. Other interactions follow the same structure. In table 1, we summarize
the non-vanishing contact bispectra generated from the cubic interactions presented in
eqs. (2.42), (2.43) and (2.44).20 The generic structure is the following

Dissipative bispectrum in Minkowski

Bπ(k1, k2, k3) = f(EFT)Polyn (eγ
1 , eγ

2 , eγ
3)

Singγ

(4.13)

where f(EFT) a rational function of the EFT coefficients (and possibly the kinematics for
spatial derivative interactions), Polyn are polynomials of the variables given in eq. (4.10) and
Singγ is the singularity structure expressed in eq. (4.8).

The simplicity of the structure, which originates from integrals of the form of eq. (4.9),
might suggest the future development of bootstrap techniques for this kind of local dissipative
dynamics. It also suggests the physics is well captured from the interpretation of Singγ

controlling the amplitude of 3 ↔ 0 and 2 ↔ 1 interactions mediated by the environment.

20Note that while (∂iπr)2 π̇a and π̇r∂iπr∂iπa lead to non-vanishing contributions of the generic form given
in eq. (4.13), the combination α1 (∂iπr)2 π̇a + 2α1π̇r∂iπr∂iπa appearing in eq. (2.42) leads to a vanishing
bispectrum, the two contributions perfectly cancelling.

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4.3 Non-Gaussianities from dissipation and noises

Now that we have gained insight on the properties of dissipative dynamics compared to its
unitary counterpart, it is time to apply these techniques to primordial cosmology. Remem-
bering that the curvature perturbations are defined though ζ = −Hπ/f2

π in the spatially
flat gauge, the primordial bispectrum reads

⟨ζk1ζk2ζk3⟩ = −H3

f6
π

⟨πk1πk2πk3⟩ ≡ (2π)3δ(k1 + k2 + k3)B(k1, k2, k3). (4.14)

Quantities of interest to characterise the non-Gaussian signatures are the amplitude of
the signal

fNL(k1, k2, k3) ≡
5
6

B(k1, k2, k3)
P (k1)P (k2) + P (k1)P (k3) + P (k2)P (k3)

, (4.15)

discussed for specific configurations below, and the shape function, already defined in eq. (4.12)
which we reproduce here for clarity

S(x2, x3) ≡ (x2x3)2 B(k1, x2k1, x3k1)
B(k1, k1, k1)

. (4.16)

Here x2 ≡ k2/k1 and x3 ≡ k3/k1 are restricted to the region max(x3, 1 − x3) ≤ x2 ≤ 1.
One can proceed just as in the flat space case presented in section 4.2 to evaluate contact
bispectra. The generic structure of the integrals is

B(k1, k2, k3) = (−i)nK+nR+1 H3

f6
π

g

H4−nd

∫ 0−

−∞(1±iϵ)

dη

η4−nd

D̂({ki}, ∂η)
[
GK/R(k1, 0, η)GK/R(k2, 0, η)GR(k3, 0, η) + 5 perms.

]
(4.17)

where nK counts the number of Keldysh progagators, nR the number of retarded ones and
nd the number of (spatial and temporal) derivatives. D̂({ki}, ∂η) is a differential operator
schematically representing the nth

d spatial and temporal derivatives acting on the propagators.
Note that there is always at least one GR due to the at least linearity in πa inherited from
eq. (2.8). The bulk-to-boundary propagators are

GR(k, 0, η) = −H2

2k3 z3
(

z

2

)−νγ

Γ(z)Jνγ (z) (4.18)

and

GK(k, 0, η) = −i
π

4k3 β1 (2z)νγ Γ(νγ)
[
Yνγ (z)A(1)

νγ
(z)− Jνγ (z)B(1)

νγ
(z)
]

, (4.19)

where we expressed the quantities in terms of z = −kη and νγ = 3
2 + γ

2H , and A
(1)
νγ (z) and

B
(1)
νγ (z) are defined in eqs. (3.46) and (3.47) respectively. It is also useful to consider

∂ηGR(k, 0, η) = H2

2k2 z2
(

z

2

)−νγ

Γ(νγ)
[
zJνγ−1(z)− (2νγ − 3)Jνγ (z)

]
(4.20)

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and

∂ηGK(k, 0, η) = i
π

4k2 β1 (2z)νγ−1 Γ(νγ)
{ [

−zYνγ+1(z) + 2νγYνγ (z) + zYνγ−1(z)
]
A(1)

νγ
(z)

[
−zJνγ+1(z) + 2νγJνγ (z) + zJνγ−1(z)

]
B(1)

νγ
(z)
}

. (4.21)

It is now a matter of evaluating the time integral of eq. (4.17) injecting the above expressions
for the propagators. This task is analytically hard in full generality, therefore, below we mostly
rely on numerical integration and only derive analytical results in some specific regimes.

Numerical results. In this section, we summarise the main phenomenological implications
of the contact bispectra generated by the cubic operators of eqs. (2.42), (2.43) and (2.44).
Rather than an exhaustive investigation, which we postpone for future work, we here highlight
the main features one can expect from the bispectrum signal for this class of models. The
Mathematica code used to perform the analysis is available upon request.

Many of the sixteen cubic operators appearing in eqs. (2.42), (2.43) and (2.44) lead to
similar signatures for the contact bispectrum. Hence, we only display the results for a subset
of these operators chosen in the following manner:

• From L1 given in eq. (2.42), we first consider π̇2
r π̇a and (∂iπr)2 π̇a, which are two

operators appearing in the unitary limit where one recovers the usual EFToI. We
highlight how their signature is modified in the presence of dissipation due to the
modified structure of the propagators.

• From L1 given in eq. (2.42), we also consider π̇2
rπa and (∂iπr)2 πa which are related to

the quadratic dissipation π̇rπa by the non-linear realisation of boosts. These operators
have been discussed in [18], mostly in the large dissipation regime. Here, we complement
that discussion with results at the small dissipation.

• From L2 given in eq. (2.43), we consider ∂iπr∂iπaπa and π̇rπ2
a. The former arises as

the non-linear realisation of boosts acting on a noise term (P µ∂µπa)πa, which is a total
derivative at linear order, that is π̇aπa. The latter has an interesting signature in f eq

NL
as a function γ/H as first noted in [55] and discussed in section 5.

• From L3 given in eq. (2.44), we consider π̇3
a which also appears in the unitary limit

where one recovers the usual EFToI. We also consider π3
a as people might a priori

worry it behaves differently to the others due to the absence of derivatives. Instead, this
term leads to a mostly similar signatures as the other interactions due to the modified
propagators compared to the unitary case.

Moreover, π′2r πa, (∂iπr)2 πa, π′rπ2
a and π3

a are also the four cubic operators appearing in
the matching with the UV completion of [55] discussed in section 5. The shapes of the
contact bispectrum generated by these four operators are displayed in figures 8 and 9 (the
other operators essentially follow the same trend). Just as for the flat space case, different
behaviours emerge in the large (γ ≫ H) and small (γ ≪ H) dissipation regime. While the
former peaks in the equilateral configuration as already noted in [18], the latter reaches
an extremum near the folded region.

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S(x2, x3)

-0.5

0.

0.5

1.0

(a)

S(x2, x3)

0.

0.2

0.4

0.6

0.8

1.0

(b)

S(x2, x3)

0.

0.2

0.4

0.6

0.8

1.0

(c)

S(x2, x3)

0.

0.2

0.4

0.6

0.8

1.0

(d)

Figure 8. Shapes of the bispectrum at large dissipation γ = 5H . In this regime, the signal peaks in the
equilateral configuration x2 = x3 = 1. Consistency relations hold in the squeezed limit x3 ≪ x2 = 1.
a. (∂iπr)2

πa operator; b. π̇2
rπa operator; c. π̇rπ2

a operator; d. π3
a operator.

This smoking gun of open dynamics might seem degenerate with other classes of models
that also lead to a signal in the folded triangles such as non-Bunch Davies initial states [40–46].
A crucial difference, which appears in our numerical treatment and is confirmed analytically
below, is that dissipation regularises the divergence by smoothing the peak and displacing it
from the edge of the triangular configurations, leading to finite values of the bispectrum for
any physical configuration. In particular, it implies no divergence in the squeezed limit of the
bispectrum k1 ≃ k2 ≫ k3, which is displayed in figure 10. Small values of γ/H may eventually
lead to an intermediate peak due to the regularised folded singularity, yet consistency relations
hold [47–54] and the squeezed limit goes to zero because of the symmetries of the theory.
Notice that operators such as π3

a follow this trend despite what one may have naively thought
in the absence of derivatives. This is because of the modified propagators compared to the
free theory, which for instance ensure IR convergence.

The amplitude in the equilateral configuration is controlled by

f eq
NL ≡ 10

9
k6

(2π)4
B(k, k, k)

∆4
ζ

. (4.22)

In figure 11, we display the dependence of f eq
NL on the dissipation parameter γ/H from

numerical integration of the bispectrum B(k, k, k) for the operators considered above. One

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S(x2, x3)

-2

-1

0

1

(a)

S(x2, x3)

-1.0

-0.5

0.

0.5

1.0

(b)

S(x2, x3)

-1.0

-0.5

0.

0.5

1.0

(c)

S(x2, x3)

-0.5

0.

0.5

1.0

(d)

Figure 9. Shapes of the bispectrum at low dissipation γ = 0.001H. In this regime, the signal
peaks near folded configurations x2 + x3 = 1. Consistency relations still hold in the squeezed limit
x3 ≪ x2 = 1. The tiny oscillations are artefacts of the numerical integration over a finite range. a.
(∂iπr)2

πa operator; b. π̇2
rπa operator; c. π̇rπ2

a operator; d. π3
a operator.

0.0 0.2 0.4 0.6 0.8 1.0

x3

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

S
(1
,x

3
)

(∂iπr)
2πa

π̇2
rπa

π̇rπ
2
a

π3
a

(∂iπr)
2π̇a

π̇2
r π̇a

(∂iπr∂
iπa)πa

Figure 10. Shape function along the direction x2 = 1 for different contributions to the contact
bispectrum at large dissipation γ = 5H. Consistency relations ensure the signal vanishes in the
squeezed limit x3 ≪ 1.

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10−2 10−1 100 101 102

γ/H

100

101

102

f
eq N

L

Planck 2018: f eq
NL = 26± 47

SPHEREX & MegaMapper

(∂iπr)
2πa

π̇2
rπa

π̇rπ
2
a

π3
a

10−2 10−1 100 101 102

γ/H

100

101

102

f
eq N

L

Planck 2018: f eq
NL = 26± 47

SPHEREX & MegaMapper

(∂iπr)
2π̇a

π̇2
r π̇a

π̇3
a

(∂iπr∂
iπa)πa

Figure 11. Amplitude in the equilateral configuration as a function of the dissipation parameter γ/H .
Continuous (dashed) lines are positive (negative) values. Left: the four cubic operators appearing in
the matching with [55] performed in section 5. All the curves well reproduce the results from [55].
The otherwise arbitrary values of the EFT coefficients were chosen to qualitatively reproduce the
results from [55]. Right: new operators of the open EFToI that may lead to specific signatures on
the non-Gaussian signal. Again, numerical values of the EFT coefficients are arbitrary, chosen to
be comparable to the Left panel. All scalings with γ are consistent with the heuristic estimates of
section 2.3.

can use observational constraints f eq
NL = 26 ± 47 from [92] to place bounds on the EFT

parameters in the large dissipation regime. For instance, a numerical fit of the (∂iπr)2πa

contributions leads to f eq
NL ≃ −γ/(4H) which is consistent with the result from [18, 55] and

our heuristic estimate of section 2.3. It naively implies that γ/H < 80 at 68% confidence. Of
course, this is more of a proof of principle than a realistic estimate due to the cumulative
effects of different cubic operators that cannot be disentangled one from another. Yet, it
demonstrates how this class of model can be confronted to data. Such observational bounds
could tighten thanks to future LSS experiments such as SPHEREX [93] or MegaMapper [94],
further constraining on this class of models. One would eventually consider to perform the
same kind of analysis in the small dissipation regime for

f folded
NL ≡ 5

24
k6

(2π)4
B(k, k/2, k/2)

∆2
ζ(k/2)[4∆2

ζ(k/2) + ∆2
ζ(k)]

, (4.23)

for instance using the CMB-BEST pipeline [95]. We leave it for future work.

Analytical discussion. An interesting question is the fate of folded singularities in an
expanding background. In general, a folded singularity appears when two modes resonate
with a third for an infinite amount of time. For this reason, the origin of the divergence
can be derived from the early time oscillating behaviour of the propagators. Hence, let us
consider the simplest cubic operator π3

a and the cubic bispectrum it generates

B(k1, k2, k3) = −6H3

f6
π

δ1
f2

π

∫ 0−

−∞(1±iϵ)

dη

H4η4 GR(k1, 0, η)GR(k2, 0, η)GR(k3, 0, η). (4.24)

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Injecting eq. (4.18) into the above expression and expanding at early time in the sub-Hubble
regime when η → −∞, we observe the time integral reduces to a collection of terms of the form∣∣∣∣∣

∫
dη

e−i(±k1±k2±k3)η

η1+ 3
2

γ
H

∣∣∣∣∣ < ∞ when γ

H
> 0. (4.25)

This is the analogue of the flat space case discussed in eq. (4.9) from which we concluded
the resolution of the singularity in presence of non-vanishing dissipation. In the vanishing
limit, we recover the log-folded singularity familiar to non-Bunch Davies initial states [40–46].
Contrarily to this dramatically uncontrolled divergence along the folded region of figure 5,
dissipation tames the peak and displaces it from the edge of the physical region such that
the folded singularity remains resolved and under perturbative control for any value of the
kinematic variables.

One may wonder if there are implications of the above mentioned singularity on the
squeezed limit of the bispectrum k1 ≃ k2 ≫ k3. As long as πr transforms as the usual pseudo-
Goldstone boson, which non-linearly realises time-translation and boosts, the standard
arguments [48–50] should hold. As a consequence of the singularity being resolved, the
curvature perturbation bispectrum still goes to zero in the squeezed limit with an eventual
intermediate peak in the small dissipation regime, when γ ≪ H.

In additions to the sub-Hubble and squeezed regimes discussed above, analytical results
can also be obtained for specific values of the dissipation parameter γ. Indeed, Hankel
functions with half-integer order parameter νγ are simpler to handle. This happens for
γ/H = 2n with n ∈ N. In this case, propagators decompose into polynomials multiplying
an oscillatory phase. It offers a window of opportunity into computing analytic expressions
for shapes of interest. Yet, beware that the results can significantly depart from the general
case of arbitrary γ so one must be careful before drawing general conclusions about the
behaviour of dissipative dynamics.

5 Matching

For an EFT construction to be of interests, it has to encompass a class of physically motivated
models. For this reason, we consider in this section the matching of our open EFT to a
concrete “ultra-violet” model recently constructed and studied in [55]. This is not the first
model of dissipative dynamics, but it has the distinguishing feature of leading to a local
low-energy dynamics around and below the Hubble scale. This is related to the fact that the
window of instability for particle production involves wavelengths that are parametrically
sub-Hubble. This is in constrast with the much studied model based on the coupling ϕFF̃ [17].
In that case, gauge fields are produced around horizon crossing and hence mediate interactions
at distances and time scales of order Hubble, which appear non-local in the open EFT.

The model in [55] provides an example of a partial UV completion and illustrates the
scope of our EFT. In addition to the inflaton field ϕ, the model features a massive scalar
field χ with a softly-broken U(1) symmetry. The action of the model is given by

S =
∫

d4x
√
−g

[1
2M2

PlR − 1
2 (∂ϕ)2 − V (ϕ)− |∂χ|2 + M2 |χ|2

− ∂µϕ

f
(χ∂µχ∗ − χ∗∂µχ)− 1

2m2
(
χ2 + χ∗2

) ]
. (5.1)

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The last term in eq. (5.1) breaks the U(1) symmetry χ → eiαχ for α ∈ R. An important
parameter for the dynamics of the system is ρ ≡ ϕ̇0(t)/f , which controls the mixing of ϕ

with χ. Here ϕ̇0(t) is the time derivative of the inflaton background ϕ0(t). The hierarchy
of scales for which the treatment of [55] is valid is

f ≫ ρ ≳ M ≫ m ≫ H. (5.2)

The model exhibits a narrow instability band in the sub-Hubble regime, during which particle
production occurs. The amount of particle production is controlled by the dimensionless
parameter

ξ ≃ m4

8HρM2 ≳ O(1). (5.3)

The inflaton fluctuations experience a dissipative dynamics due to the presence of an en-
vironment of χ particles generated by the instability. The inflaton dynamics is effectively
described in terms of a non-linear Langevin equation [55]

π′′ + (2H + γ) aπ′ − ∂2
i π ≃ γ

2ρf

[
(∂iπ)2 − 2πξπ′2

]
− a2m2

f

(
1 + 2πξ

π′

aρf

)
δOS , (5.4)

where we neglected the inflaton potential contribution a2V ′′π which is slow-roll suppressed.
The dissipation parameter γ (with units of mass) is determined by the following combination
of microphysical parameters

γ ≃ ξm4

πMf2 e2πξ. (5.5)

The effect of noise is captured in terms of its two- and three-point statistics

⟨δOS(k, η)δOS(k′, η′)⟩ ≃ (2π)3δ(k + k′)δ(η − η′)H4η4νO , (5.6)
⟨δOS(k, η)δOS(k′, η′)δOS(k′′, η′′)⟩ ≃ (2π)3δ(k + k′ + k′′)δ(η − η′)δ(η − η′′)H8η8νO3 , (5.7)

with the amplitudes of the noises being controlled by

νO ≃ M

m

e4πξ

4π2 and νO3 ≃ e6πξ

π2m2 . (5.8)

The parameters at play are summarized in the first line of table 2.
As we demonstrate below, the low-energy dynamics of this model is equivalently described

in terms of21

Seff =
∫

d4x

[
a2π′rπ′a−c2

sa2∂iπr∂iπa−a3γπ′rπa+iβ1a4π2
a

+(8γ2−γ)
2f2

π

a2π′2r πa+
γ

2f2
π

a2 (∂iπr)2 πa−2i
β5
f2

π

a3π′rπ2
a+

δ1
f2

π

a4π3
a

]
, (5.9)

21The open EFT of eq. (5.9) has no derivatives acting on πa (apart from the standard kinetic term) and up
to one derivative acting on πr.

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Parameters:

UV completion [55] M m f ρ ξ γ νO/νO3

Open EFT (5.9) f2
π cs γ γ2 β1 β5 δ1

Matching: f2
π = ρf

cs = 1 γ = ξm4

πMf2 e2πξ

8γ2 = (1− 2πξ) γ β1 = νO
2ρf

m4

f2

β5 = 2πρfξβ1 6δ1 = ρf m6

f3 νO3

Table 2. Matching of the EFT coefficients appearing in eq. (5.9) to the microphysical parameters
of [55]. The matching is obtained either at the level of the power spectrum and contact bispectra as
in section 5.1 or by deriving the Langevin equation for the fluctuations as in section 5.2.

upon matching the EFT coefficients to the microphysical parameters of the model through the
relations of table 2. The matching can be performed by comparing the power spectrum and
bispectrum obtained from the top-down and bottom-up approaches, as we do in section 5.1.
Alternatively, the matching can also be derived at the level of the Langevin equation. In
section 5.2, we develop a “stochastic unravelling” [32] of our EFT22 which holds at non-linear
order and matches the top-down construction of [55] upon restricting to eq. (5.9) and table 2.

5.1 Power spectrum and bispectrum

First, by noticing that in [55] ζ = −Hπ/(ρf), and comparing this expression with our linear
relation between ζ and π given in eq. (2.59), we obtain f2

π = ρf . Second, one notices that
the speed of sound in this set up is cs = 1 (see eq. (5.4)). This provides a matching for our
first two EFT coefficients. Then, we turn our attention to the computation of the power
spectrum, which in [55] was found to be

⟨ζkζ−k⟩ = (2π)3δ(k + k′) H2

(ρf)2
m4

f2 νO

∫ η

−∞

dη′

(Hη′)4

[
GR(k; η, η′)

]2
. (5.10)

22Indeed, open quantum systems can equivalently be described in terms of a path integral formulation
(Feynman-Vernon influence functional), a dynamical equation for the reduced density matrix (master equation
— which reduces to Lindblad equation in the Markovian limit) or their stochastic unravelling (Langevin equation
or its quantum analogue the stochastic Schrödinger equation). Each formulation has its own benefits [32], the
Langevin equation being well-suited for numerical simulations, the influence functional adapted to describe
relativistic settings in a manifestly covariant manner and the master equations useful for their ability to
implement non-perturbative resummations. See [96] for a cosmology oriented review of these techniques.

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Here we accounted for the rescaling of the Green function by a2(η′) in the definition of [55].
Comparing eq. (5.10) with our results from section 3.3, especially the expressions for the
retarded Green function in eq. (3.39) and for the Keldysh function in eq. (3.42) from which
we extract the power spectrum, we can identify

γ = ξm4

πMf2 e2πξ and β1 = νO
2ρf

m4

f2 , (5.11)

without even performing the integral. Note that we used eq. (5.5) to relate the derived
parameter γ to the UV paremeters of [55].

We now turn our attention the bispectrum expressions found in [55]. We first consider
the (∂iπ)2 term in the right-hand side of the Langevin eq. (5.4). Following [55], it leads to

B(k1, k2, k3) =
γ

H

(
H

ρf

m2

f

)4

ν2
O

∫ dη1

(Hη1)2

∫ dη2

(Hη2)4

∫ dη3

(Hη3)4{
GR(k1; 0, η1)GR(k2; 0, η2)GR(k3; 0, η2) (5.12)

×
[
(k2.k3)GR(k2; η1, η2)GR(k3; η3, η2) + 2 perms.

]}
.

It might not be direct to realise that this expression is strictly equivalent to the contact
bispectrum generated by the (∂iπr)2 πa operator of the open EFT spelled in eq. (5.9).
Following eq. (4.17), this contribution writes

B(k1,k2,k3)=
H3

f6
π

γ

2f2
π

∫ dη1

(Hη1)2

[
(k1.k2)GK(k1,0,η1)GK(k2,0,η1)GR(k3,0,η1)+5 perms.

]
(5.13)

which might indeed be non-trivial to match. Yet, if one accounts for the fact that, based
on eq. (3.42)

GK(k1, 0, η1) = iβ1

∫ dη2

(Hη2)4 GR(k1; 0, η2)GR(k1; η1, η2) (5.14)

and inject in eq. (5.13) the matching for obtained β1 in eq. (5.11), one exactly recovers
eq. (5.12) without even having to perform the time integrals.

Similar identification of the functional forms of the integrals for the π′2 term of eq. (5.4)
with π′2r πa in the open EFT (5.9) leads to

8γ2 = (1− 2πξ) γ. (5.15)

Similarly, the π′δOS term of eq. (5.4) is mapped to π′rπ2
a in the open EFT of eq. (5.9) such that

β5 = 2πρfξβ1. (5.16)

At last, the non-Gaussian component of the noise in eq. (5.4) captured through eq. (5.7)
transforms into the π3

a term of the open EFT (5.9), leading to

6δ1 = ρf
m6

f3 νO3 , (5.17)

which completes the matching summarised in table 2.

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5.2 Langevin equation

The formalism presented in this work is not the only way to address open quantum system
dynamics [32]. Another commonly employed technique to understand the interaction between
system and environment is the Langevin equation. This approach relays on the saddle point
approximation for the path integral that defined our open quantum system (2.2). We can use
it to compute tree-level quantities, such as the contact bispectrum. We will focus on deriving
the Langevin equation for the top-down model [55]. We first start with the derivation of
the homogeneous stochastic dynamics from the path integral (2.2), taking Seff to be the
quadratic functional from section 3 with D̂K set to zero. We then incorporate the operator
iβ1π2

a on the right-hand side of the Langevin equation through a Hubbard-Stratonovich
transformation [97, 98], that introduces a new field, δOS , which is the stochastic noise
that appears in eq. (5.4). This allows us to match the Wilsonian coefficient β1 with the
parameters in [55]. Finally, we explain how we introduce the different contact interaction
terms derived from the influence functional (2.3) into the Langevin equation. This analysis
depends on the powers of πa in each term. Those linear in πa are easy to incorporate, as
the approach is the same as for the D̂R,A terms in the homogeneous Langevin equation.
Those quadratic and cubic in πa require a modification of the HS trick, yielding a coupling
between the system and the noise and self-interactions of the noise (that can be interpreted
as non-Gaussian statistics of the noise).

5.2.1 Linear stochastic dynamics

The diagonal of the density matrix is given by a path integral over the πr component with
boundary condition πr(η0) = π and the πa component with boundary condition πa(η0) = 0,
leading to

ρ[π, π] =
∫ π

BD
Dπr

∫ 0

BD
DπaeiSeff[πr,πa]. (5.18)

Homogeneous solution. We start by considering the quadratic functional obtained by
setting D̂K = 0, that is

Seff[πr, πa] =
∫

d4xa4
(

π′rπ′a
a2 − c2

s

∂iπr∂iπa

a2 − γ
π′rπa

a

)
. (5.19)

Integrating by parts, we can write the influence functional as linear in πa (the boundary
terms vanish because they are linear in πa(η0)) such that

Seff[πr, πa] =
∫

d4xa4
[
−∂η

(
a2π′r

)
a4 + c2

s

∂2
i πr

a2 − γ
π′r
a

]
πa. (5.20)

We can now consider the path integral over πa as enforcing the constraint

ρ[π, π] =
∫ π

BD
Dπr δ

[
−∂η

(
a2π′r

)
a4 + c2

s

∂2
i πr

a2 − γ
π′r
a

]
. (5.21)

This greatly simplifies the calculation of n-point functions of the πr component. If we want
to compute correlation functions of πr, it is then enough to solve the equation of motion

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defined by the constrain, that is

π′′k + (2H + γ) aπ′k + c2
sk2πk = 0, (5.22)

which defines the differential operator D̂R. Comparing with eq. (5.4), we see that we were right
to call γ the same in both equations and that cs = 1. We can rebuild the retarded propagator
from section 3 by looking at the Green function for the homogeneous equation of motion

∂2
ηGR(k; η, η′)− 1

η

(
2 + γ

H

)
∂ηGR(k; η, η′) + c2

sk2GR(k; η, η′) = H2η2δ(η − η′) (5.23)

with boundary conditions GR(k; η < η′) = 0 and

GR(k; η, η) = 0 , ∂ηGR(k; η, η′)
∣∣∣∣
η=η′

= H2η2. (5.24)

It finally leaves to the previously found result given in eq. (3.39), which we reproduce here
for clarity

GR(k; η1, η2) =
π

2H2 (η1η2)
3
2

(
η1
η2

) γ
2H

Im
[
H

(1)
3
2 + γ

2H

(−cskη1)H(2)
3
2 + γ

2H

(−cskη2)
]

. (5.25)

Quadratic noise. The derivation of the Langevin equation from the open effective functional
relies on rewriting all terms to be linear in πa. When we introduce the quadratic terms in
πa like iβ1π2

a, this rewriting cannot be achieved by integration by parts. Rather, we have to
introduce an auxiliary Gaussian field δOS to rewrite quadratic terms in πa in a linear form.23

This procedure is known as the Hubbard-Stratonovich transformation [97, 98]. We here focus
on the term iβ1π2

a, which is the one usually considered in the literature. We further discuss
the inclusions of the derivative noises appearing in eq. (2.37) in appendix E.

Consider the action for the iβ1π2
a term. This term can be written as the outcome of

a path integral over a Gaussian field δOS

exp
(
−
∫

d4xa4β1π2
a

)
= N0

∫
D[δOS ] exp

[∫
d4xa4

(
−δO2

S

4β1
+ iδOSπa

)]
, (5.26)

with N0 being the normalisation constant. In this way, the path integral that defines the
density matrix of the system is linear in πa

ρ[π, π] = N0

∫
D[δOS ]

∫ π

BD
Dπr

∫ 0

BD
Dπa (5.27)

× exp
{∫

d4xa4
[
−δO2

S

4β1
+ i

(
−D̂Rπr + δOS

)
πa

]}
,

from which we obtain the Langevin equation

π′′k + (2H + γ) aπ′k + c2
sk2πk = a2δOS(k, η). (5.28)

23This modifies the influence functional by adding terms that depend on δO2
S These terms will tell us about

the statistics of the noise δOS .

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The new variable δOS behaves as a Gaussian field, getting a non-vanishing two-point function
under the path integral

⟨δOS(k, η)δOS(k′, η′)⟩ = 2β1
a4(η)δ(η − η′)(2π)3δ(k + k′). (5.29)

A matching of β1 with the microphysical parameters of [55] is possible by comparing the
Gaussian statistics of the noise, from which we recover the above result. Alternatively,
the linear Langevin equation eqs. (5.28) contains all the information needed to derive the
power spectrum. One can indeed solve the Langevin equation using the convolution of
the retarded Green function with the noise and then compute the power spectrum from
the inhomogeneous solution

πk(η) =
∫ η

−∞
dη′a4(η′)GR(k; η, η′)δOS(k, η′). (5.30)

The two-point function of the noise given in eq. (5.29) sources the two-point function of
the fluctuations through

⟨πk1(η1)πk2(η2)⟩ =
∫ η1

−∞
dη′1

∫ η2

−∞
dη′2a4(η′1)a4(η′2)

GR(k1, η1, η′1)GR(k2, η2, η′2)⟨δOS(k1, η′1)δOS(k2, η′2)⟩, (5.31)

which obviously reproduces the above result

⟨πk1(η1)πk2(η2)⟩ = (2π)3δ(k1 + k2)
[
β1

∫ η1

−∞

dη

H4η4 GR(k1, η1, η)GR(k2, η2, η) + (η1 ↔ η2)
]

.

(5.32)

As shown above, the matching between the results from section 3 and [55] is obtained for

β1 = νO
2ρf

m4

f2 . (5.33)

5.2.2 Interacting theory

The Langevin equation formalism can be extended to include the interaction terms of the
influence functional (2.3). We divide our analysis into three different types of interactions,
one for each power of πa.

Linear terms in πa. The terms of eq. (5.9) linear in πa are

Seff ⊃
∫

d4x

[(8γ2 − γ)
2f2

π

a2π′2r πa + γ

2f2
π

a2 (∂iπr)2 πa

]
. (5.34)

These terms being already linear in πa, their inclusion in the Langevin equation is straight-
forward, leading to24

π′′ + (2H + γ) aπ′ − ∂2
i π = a2δOS + (8γ2 − γ)

2f2
π

π′2 + γ

2f2
π

(∂iπ)2 . (5.35)

We can then compare with eq. (5.4), which leaves

ρf = f2
π , and 8γ2 = (1− 2πξ) γ. (5.36)

24If these terms were proportional to derivatives of πa, we would need to integrate by parts, which is further
discussed in appendix E.

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Quadratic terms in πa. There exist terms that may source a bispectrum signal that are
quadratic in πa. They can be included into the Langevin equation as a coupling between
the noise and the system’s variable π. To achieve this task, we first need to rewrite the
interactions as being proportional to one of the quadratic terms in πa in eq. (3.1). We
then modify the Hubbard-Stratonovich trick [97, 98], allowing for a small deviation in the
coupling δOSπa in eq. (5.26). Then, assuming the quadratic noise to be much stronger than
the noise-system coupling, we can work in perturbation theory. This is equivalent to the
tree-level approximation at the level of the path integral over the noise δOS . Therefore, the
Wilsonian coefficients of the open effective functional (2.3) and the coefficients appearing in
the Langevin equation eq. (5.4) are only equal at first order perturbatively.

The term we need to consider is

Seff ⊃ −2i
β5
f2

π

∫
d4xa3π′rπ2

a, (5.37)

which couples to the quadratic noise iβ1π2
a. The modified Hubbard-Stratonovich trick reads

exp
[
−
∫

d4xa4
(

β1 − 2β5
f2

π

π′r
a

)
π2

a

]
= N (π′r)

∫
D[δOS ] (5.38)

× exp
{∫

d4xa4
[
−δO2

S

4β1
+ iλ(π′r)δOSπa

]}

with
λ2(πr) = 1− 2 β5

f2
πβ1

π′r
a

. (5.39)

Note that the full Hubbard-Stratonovich trick implies that the normalisation of the path
integral over δOS depends on π′r. Consequently, to follow this procedure we have to take
the tree-level approximation where we drop the dependence on π′r from the normalisation of
the integral and expand the square root in powers of λ(π′r), leading to

exp
[
−
∫

d4xa4
(

β1−2β5
f2

π

π′r
a

)
π2

a

]
=N

∫
D[δOS ] (5.40)

×exp
[∫

d4xa4
(
−δO2

S

4β1
+iδOSπa−i

β5
f2

πβ1

π′r
a

δOSπa

)]
.

This approach generates a perturbativity condition that is similar to demanding that the
three-point function is smaller than the corresponding two-point signal, that is

2β5π′rπ2
a

f2
πaβ1π2

a

≪ 1. (5.41)

At last, we can include the noise-system coupling obtained on the right-hand side of the
Langevin equation, leading to

π′′ + (2H + γ) aπ′ − ∂2
i π = a2δOS − β5

f2
πβ1

aπ′δOS . (5.42)

Comparing with eq. (5.4), we recover

β5 = 2πρfξβ1. (5.43)

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Cubic terms in πa. Finally, the terms that are cubic in πa can also be included at leading
order into the Langevin equation by modifying the Hubbard-Stratonovich trick (5.26). These
terms lead to quadratic corrections in the noise appearing on the right-hand side of the
Langevin equation. As we will see below, these corrections mimic non-Gaussian statistics
of the noise found for instance in [55]. In particular, we need to modify the coefficient
controlling δO2

S in eq. (5.26) by allowing it to depend on πa. In order to recover the Langevin
equation, we again rely on perturbation theory, assuming the quadratic noise to be much
stronger than the cubic operator in πa. Therefore, the Wilsonian coefficients in the open
effective functional (2.3) and the coefficients in the Langevin equation are again only equal
at first order in perturbation theory.

To recover the non-Gaussian statistics of the noise found in [55], we consider

Seff = δ1
f2

π

∫
d4xa4π3

a. (5.44)

The modification of the Hubbard-Stratonovich trick leads to

exp
[
−
∫

d4xa4
(

β1 + i
δ1
f2

π

πa

)
π2

a

]
= N (πa)

∫
D[δOS ] (5.45)

× exp

∫ d4xa4

− δO2
S

4β1 + 4i δ1
f2

π
πa

+ iδOSπa

 .

We expand at leading order in πa both the denominator and the normalisation constant,
which leads to a term of the form δO2

Sπa, that is

exp
[
−
∫

d4xa4
(

β1 + i
δ1
f2

π

πa

)
π2

a

]
≈ N

∫
D[δOS ] (5.46)

× exp
[∫

d4xa4
(
−δO2

S

4β1
+ iδOSπa + i

δ1
4f2

πβ2
1

δO2
Sπa

)]
.

Here, we again find a perturbativity condition

δ1π3
a

f2
πβ1π2

a

≪ 1, (5.47)

that can be related to the heuristic estimate made in eq. (2.79). The δO2
Sπa term enters

the Langevin equation as

π′′ + (2H + γ) aπ′ − ∂2
i π = a2δOS + δ1

4f2
πβ2

1
a2δO2

S (5.48)

There is no direct matching with eq. (5.4) yet, the connection can be made manifest if one
introduces a field redefinition. Indeed, one can map the Gaussian noise δOS to a noise with
a non-Gaussian statistics through

δOng
S = δOS + δ1

4f2
πβ2

1
δO2

S , (5.49)

or equivalently in Fourier space

δOng
S (k, η) = δOS(k, η) + δ1

4f2
πβ2

1

∫
q

δOS(q, η)δOS(k − q, η). (5.50)

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Under this field redefinition, the non-Gaussian noise adquires a three-point function of the form

⟨δOng
S (k1, (η1)δOng

S (k2, η2)δOng
S (k3, η3)⟩′ = δ(η1 − η2)δ(η2 − η3)

24
a4(η1)a4(η2)

δ1
f2

π

, (5.51)

where we used the notation ⟨·⟩ = (2π)3δ(k1 + k2 + k3)⟨·⟩′. Under this construction, we
obtain the matching of the last parameter

6δ1 = ρf
m6

f3 νO3 , (5.52)

which completes the reconstruction of the full non-linear Langevin equation of [55] from
the path integral language of the open EFToI.

6 Discussions

General summary. When energy is not conserved, imprints of new physics on observable
sectors might not follow the rules of unitary EFTs. In this work, we constructed a bottom-up
open EFT for the Nambu-Goldstone boson of the spontaneous breaking of time-translation
symmetry during inflation in the presence of an unknown environment. The theory recovers
the usual EFToI [3] in the unitary limit, and extends it to account for dissipation and
noise generated by a surrounding environment, which is generally out-of-equilibrium. New
scales enter the problem such as the dissipation scale or the amplitude of noise fluctua-
tions. Symmetries ensure the existence of a nearly scale invariant power spectrum for the
curvature perturbations, which can be compared to cosmological observations [99–101]. Non-
Gaussianities are generated that peak in the equilateral configuration for large dissipation
and in the folded configurations for small dissipation. The latter possibility constitutes
a distinctive feature of this class of model, which departs from non-Bunch Davies initial
states [40–46] in that it is finite for all physical kinematics. Consistency relations [47–54]
still hold due to the symmetry content of the theory. In short, the open EFToI provides an
embedding for local dissipative models of inflation such as [55].

New results. Some results recovered in this article were already known in the past literature,
mostly established in [18, 22, 55]. Still, our work makes progress compared to the previous
literature in many directions. The benefit of the current construction is the import of
techniques from non-equilibrium EFTs [21], which provides a path-integral formulation for
dissipative hydrodynamics [19, 34]. Our approach clarifies the comparison between the
particle physics approach of unitary in-out evolution [24] and non-unitary in-in dynamics
that emerges from the coarse-graining of UV degrees of freedom in cosmology [96, 102].
As found in [19, 21, 34], imposing unitarity of the UV evolution tightly constrains the IR
dynamics through a set of non-perturbative non-equilibrium constraints. Further enforcing
symmetries through the in-in coset construction [35], the most generic generating functional
can be constructed in a systematic manner. The application of these techniques to time-
translation symmetry breaking in [22] has been an important milestone in the development of
non-equilibrium EFTs for cosmology [103] at the origin of this work. While in the case of a
closed system both π+ and π− transform non-linearly under time-translation and boosts, the

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open dynamics breaks the symmetry group G+ × G− to its diagonal subgroup Gr such that
only πr transforms non-linearly. Through this property, the open EFToI still relates operators
at different orders, not only in the unitary direction (such as the well-studied example of the
speed of sound [3]) but also in the dissipative [18] and diffusive directions. At last, locality
ensures the existence of a radiatively-stable power counting scheme that can be truncated
to a finite number of interactions for any desired precision.

Further investigation. One might look to further constrain the EFT parameters. When the
environment reaches thermal equilibrium, the fluctuation-dissipation relations [104] impose
non-trivial constraints among the operators of the EFT [22, 103] through the Kubo-Martin-
Schwinger symmetry [19, 20, 34]. One may eventually desire to enforce this symmetry as
in [103] to recover warm inflation results.25 Yet, many situations of cosmological interest
evade the thermal assumption [77] and require the complete treatment of this work. In
the opposite direction, one might also relax some of the underlying assumptions such as
the locality of our EFT. Indeed, locality is not necessary per se because the environment
can mediate interactions at distance and this induces non-locality in time (and/or non-
Markovianity [83]). Nevertheless, we restricted ourselves to EFTs that are local in time and
space as implied by an appropriate underlying separation of scales. This is an important
assumption leading to a major simplification, a power counting scheme that can be truncated
to a finite number of operators. The radiative stability of the power counting scheme under
loop corrections and the renormalization procedure should be further studied along the line
of [105, 106]. The non-equilibrium constraints eqs. (2.8), (2.9) and (2.10) should provide
non-perturbative structure ensuring the stability at any order, yet it would be reassuring
to control this in a specific example.

Caveats. The most manifest blind spot of the current analysis is the focus on observables that
are not dominated by the mixing with gravity. Working in the decoupling limit of [3] along the
lines of [18], here the physics of the Goldstone π was studied neglecting metric fluctuations.
It would be desirable to go beyond the decoupling limit, which would require an improved
understanding of diffeomorphism invariance in the Schwinger-Keldysh formalism [35].26 More
generally, a better control over global and gauge symmetries in the in-in formalism in the
presence of entangling dynamics is desirable. It would help us to improve expectations for
dissipative open dynamics in flat [112, 113] and de Sitter [114] spacetimes. In light of the
no-go theorems [115, 116], it would also be interesting to identify parity-violating operators
which might be triggered when integrating out spin-1 particles [17, 117]. In addition to
symmetries, causality might also play a role in restricting the range of available dynamics.
In the flat space case, it imposes γ > 0 at the level of the retarded Green function to
avoid divergences [75]. A more systematic understanding of the causality structure of the
theory might help us to further constrain the open EFToI, for example constraining loop
contributions as studied recently in [118].

25Another direction would consists in using the full out-of-equilibrium treatment to investigate the equili-
bration properties through the emergence of the KMS symmetry in cosmological settings.

26Exploring corrections beyond the leading slow-roll contribution might also be relevant [107], as well as
resonant setups [108–111].

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Interesting regimes. Two limits of interest of the current construction are the unitary and
semiclassical limits. The former allows us to recover the usual EFToI construction [3]. If
the limit is conceptually clear at the level of the open effective functional, the complexity of
the de Sitter propagators obscures the comparison whenever γ ̸= 0. Better understanding
γ → 0 in a self consistent manner (which would certainly require a double-scaling with
β1 → 0) would be valuable to better control systematic expansions around the unitary
result. Notice that a possible direction to improve the computability in the presence of
dissipation would consist in working at the level the transport equations [119–123], which
may greatly simplify the computations of correlators. In this language, the unitary limit
might also be more accessible. The second limit of interest is the semiclassical ℏ → 0 limit.
The Schwinger-Keldysh formalism is well-known for its ability to organise a systematic ℏ
expansion in powers of πa ∝ O(ℏ) [90]. Yet, this expansion is valid only around states with
large occupation numbers [76] such that the equivalence should not hold for the flat-space
vacuum. In primordial cosmology, the Bunch-Davies vacuum being dynamically populated, a
deeper investigation of the semiclassical expansion in this context would be valuable, with
eventual connections to hybrid quantum-classical dynamics studied in [74, 124–129].

Connections with the literature. Systematic bottom-up approaches to open quantum
dynamics aims at rendering visible general features following physical principles (e.g. sta-
bility, locality, perturbativity, etc.) [32]. It might find applications in the study of parti-
cle production in cosmology [89, 130–133], either when the environment thermalizes such
as in warm inflation [14, 15, 36, 37, 134–140] or in the larger framework of dissipative
models of inflation [17, 55, 117, 136, 141–147]. This construction might also help in fos-
tering formal connections between stochastic inflation [28, 29, 148–162] and open EFT
techniques [26, 27, 32, 73, 77, 81–84, 96, 102, 125, 163–196], with an eventual window on
quantum diffusion during inflation and its known phenomenology of primordial black holes
and associated gravitational waves [197–214]. In the top-down context, both stochastic
inflation [29–31, 202, 207, 215–221] and open EFT techniques [26, 27, 77, 167, 177, 222, 223]
have demonstrated ability to go beyond standard perturbation theory by implementing
non-perturbative resummations. It would be interesting to further explore to which extend
these resummation techniques can be used in the bottom-up context, in connection with
the development of non-perturbative methods for cosmology [68, 69].

Future works. In this context, the construction of bottom-up open Effective Field Theories
for Dark Energy [224–228] and the Large Scale Structure [229, 230] can be a direct extension
of the current work. Applications to the late-time expansion of the universe would explore the
remaining parameter space [231–233] after GW170817 [234, 235] and DESI 2024 [236] results
beyond unitary constructions. The relation between modified gravity and out-of-equilibrium
thermodynamics [237, 238], entropic forces [239–241] and stochastic diffusion [242, 243] is
an ever increasing field for which open EFT might provide a systematic framework. The
path integral formulation of viscous fluids [19, 21] whose application to cosmology is at
its dawn [103] is a promising avenue, which should reduce in its simplest version to the
original formulation of the EFTofLSS [229, 230]. At last, our formalism is a starting point
to study quantum information aspects of inflation (entanglement growth [244–255], purity

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and decoherence [27, 83, 84, 189, 256], etc.). Connections between positivity bounds and
entanglement entropy were recently unravelled [257, 258], rising the hope of overcoming
obstructions related to the lack of well-defined amplitudes in primordial cosmology (despite
the efforts into this direction [24, 259, 260]). Accounting for the singularity of cosmology,
such as the lack of IR unitary description, might by the route forward to the extension of
particle physics techniques to quantum field theory in curved spacetime.

Acknowledgments

We thank C. P. Burgess, S. Cespedes, P. Creminelli, J. Grain, R. Holman, G. Kaplanek,
S. Melville, A. Nicolis, R. Penco, L. Santoni, B. Salehian, D. Stefanyszyn, V. Vennin and
G. Villa for the insightful discussions. T.C. warmly thanks D. Comelli, F. Piazza, A. Tolley
and F. Vernizzi for inspiring this research direction. This work has been supported by
STFC consolidated grant ST/X001113/1, ST/T000694/1, ST/X000664/1 and EP/V048422/1.
S.A.S. is supported by a Harding Distinguished Postgraduate Scholarship.

A Derivation of Seff constraints

The derivation follows from [34]. Let us consider a UV evolution operator Û(t, t0) under
which the UV state evolves according to

ρ̂(t) = Û(t, t0) |Ω⟩ ⟨Ω| Û†(t, t0), (A.1)

starting from an initial vacuum state |Ω⟩. The reduced density matrix is obtained by tracing
out the environmental degrees of freedom denoted σ such that

ρ̂red(t) = Trσ

[
Û(t, t0) |Ω⟩ ⟨Ω| Û†(t, t0)

]
, (A.2)

=
∫

dσ ⟨σ| Û(t, t0) |Ω⟩ ⟨Ω| Û†(t, t0) |σ⟩ . (A.3)

Let us consider the field-basis matrix element of reduced density matrix

ρππ′(t) ≡ ⟨π| ρ̂red(t) |π′⟩ (A.4)

=
∫

dσ ⟨π| ⊗ ⟨σ| Û(t, t0) |Ω⟩ ⟨Ω| Û†(t, t0) |π′⟩ ⊗ |σ⟩ (A.5)

=
∫

dπidπ′i

∫
dσ

∫
dσidσ′i ⟨π| ⊗ ⟨σ| Û(t, t0) |πi⟩ ⊗ |σi⟩

ρ
(0)
πiπ′

i
ρ

(0)
σiσ′

i
⟨π′i| ⊗ ⟨σ′i| Û†(t, t0) |π′⟩ ⊗ |σ⟩ , (A.6)

where |π⟩, |σ⟩ are eigenstates of the position operators π̂, σ̂ and we used four representations
of the identity, two on each side of the vacuum density matrix. The initial matrix elements are

ρ
(0)
πiπ′

i
≡ ⟨πi|Ωπ⟩ ⟨Ωπ|π′i⟩ , (A.7)

ρ
(0)
σiσ′

i
≡ ⟨σi|Ωσ⟩ ⟨Ωσ|σ′i⟩ , (A.8)

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where we consider |Ω⟩ = |Ωπ⟩ ⊗ |Ωσ⟩. The path integral representation of the evolution
operator is

⟨π| ⊗ ⟨σ| Û(t, t0) |πi⟩ ⊗ |σi⟩ =
∫ π

πi

Dπ+

∫ σ

σi

Dσ+eiS0[π+,σ+], (A.9)

⟨π′i| ⊗ ⟨σ′i| Û†(t, t0) |π′⟩ ⊗ |σ⟩ =
∫ π′

π′
i

Dπ−

∫ σ

σ′
i

Dσ−e−iS0[π−,σ−], (A.10)

such that we obtain

ρππ′(t) =
∫

dπidπ′i

∫ π

πi

Dπ+

∫ π′

π′
i

Dπ−eiSeff [π+,π−]ρ
(0)
πiπ′

i
, (A.11)

with the influence functional

eiSeff [π+,π−] =
∫

dσ

∫
dσidσ′i

∫ σ

σi

Dσ+

∫ σ

σ′
i

Dσ−eiS0[π+,σ+]−iS0[π−,σ−]ρ
(0)
σiσ′

i
. (A.12)

The central step of the proof is to consider σ evolves as if π is a background (external
source) see appendix A of [34]. In this case, we can consider the sourced evolution∫ σ

σi

Dσ+eiS0[π+,σ+] = ⟨σ| Û(t, t0; {π+}) |σi⟩ , (A.13)∫ σ

σ′
i

Dσ−e−iS0[π−,σ−] =
[
⟨σ| Û(t, t0; {π−}) |σ′i⟩

]†
, (A.14)

where Û(t, t0; {π+}) is the sourced evolution acting on Hσ only. One can the reconsider
the influence functional as being

eiSeff [π+,π−] =
∫

dσ ⟨σ| Û(t, t0; {π+})

I︷ ︸︸ ︷[∫
dσi |σi⟩ ⟨σi|

]
|Ωσ⟩ (A.15)

⟨Ωσ|
[∫

dσ′i |σ′i⟩ ⟨σ′i|
]

︸ ︷︷ ︸
I

Û†(t, t0; {π−}) |σ⟩

which finally reduces to

eiSeff [π+,π−] = ⟨Ωσ| Û†(t, t0; {π−})

I︷ ︸︸ ︷[∫
dσ |σ⟩ ⟨σ|

]
Û(t, t0; {π+}) |Ωσ⟩ , (A.16)

that is

Influence functional as a transition probability

eiSeff [π+,π−]| = ⟨Ω{π−}
σ (t)|Ω{π+}

σ (t)⟩ (A.17)

where we defined the sourced-evolved states

|Ω{π+}
σ (t)⟩ = Û(t, t0; {π+}) |Ωσ⟩ , (A.18)

⟨Ω{π−}
σ (t)| = ⟨Ωσ| Û†(t, t0; {π−}). (A.19)

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In this sense, the influence functional can be interpreted as a probability of obtaining
a configuration (π+, π−) given the unitary evolution of the UV theory and taking into
consideration the lack of knowledge about the environment. The transition rate is physical if

|| ⟨Ω{π−}
σ (t)|Ω{π+}

σ (t)⟩ ||2 ≤ 1 (A.20)

such that

|eiSeff [π+,π−]| ≤ 1 ⇒ ℑmSeff [π+, π−] ≥ 0. (A.21)

Then, from eq. (A.17), one can easily see that

e−iS∗
eff [π+,π−] = ⟨Ω{π+}

σ (t)|Ω{π−}
σ (t)⟩ = eiSeff [π−,π+] (A.22)

from which we deduce

Seff [π+, π−] = −S∗eff [π−, π+] . (A.23)

Lastly, the causality structure which is obvious in the unitary theory, S0[π+]− S0[π−] = 0 if
π+ = π−, is much less straightforward in the non-unitary case yet still holds as

eiSeff [π+,π+] = ⟨Ω{π+}
σ (t)|Ω{π+}

σ (t)⟩ = 1 ⇒ Seff [π+, π+] = 0. (A.24)

B de Sitter Keldysh functions

This appendix gathers the outcome of the lenghty integrals appearing in the main text in
the expression of eq. (3.42). Indeed, to analytically access the Keldysh function given in
eq. (3.42), we have three integrals to compute

A(1)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ J2

νγ
(z′), (B.1)

B(1)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Jνγ (z′)Yνγ (z′), (B.2)

C(1)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Y 2

νγ
(z′), (B.3)

that lead for the first contribution to

A(1)
νγ

(z) = 1
4

[
Γ(νγ − 1)

Γ
(
νγ − 1

2

)
Γ
(
2νγ − 1

2

)
− z3Γ

(
νγ + 1

2

)
2F̃3

(3
2 , νγ + 1

2;
5
2 , νγ + 1, 2νγ + 1;−z2

)]
, (B.4)

for the second contribution to

B(1)
νγ

(z) =
z3−2νγ 2F3

(
1
2 , 3

2 − νγ ; 1− νγ , 5
2 − νγ , νγ + 1;−z2

)
3πνγ − 2πνγ

2

+
z3Γ(−νγ) 2F3

(
3
2 , νγ + 1

2 ;
5
2 , νγ + 1, 2νγ + 1;−z2

)
3
√

πΓ
(

1
2 − νγ

)
Γ(2νγ + 1)

−
Γ
(

3
2 − νγ

)
4Γ(2− νγ)Γ

(
2νγ − 1

2

) ,

(B.5)

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and for the third contribution to

C(1)
νγ

(z) =
2 cot(πνγ)z3−2νγ 2F3

(
1
2 , 3

2 − νγ ; 1− νγ , 5
2 − νγ , νγ + 1;−z2

)
3πνγ − 2πνγ

2

+
4νγΓ(νγ)2z3−4νγ 2F3

(
3
2 − 2νγ , 1

2 − νγ ; 1− 2νγ , 5
2 − 2νγ , 1− νγ ;−z2

)
π2(4νγ − 3)

−
4−νγ z3 cot2(πνγ) 2F3

(
3
2 , νγ + 1

2 ;
5
2 , νγ + 1, 2νγ + 1;−z2

)
3Γ(νγ + 1)2

+ Γ(νγ − 1)
4Γ
(
νγ − 1

2

)
Γ
(
2νγ − 1

2

) − 43νγ−2 sin(2πνγ)Γ(3− 4νγ)
Γ(1− νγ)Γ(2− νγ)

. (B.6)

C de Sitter power spectrum with derivative noises

In this appendix, we compute the inflationary power spectrum in the presence of derivatives
noises π′2a and (∂iπa)2. It amounts to explicitly evaluate eq. (3.43) and eq. (3.44) in the
coincident time limit. We proceed as we did in the main text in section 3.3 for the π2

a noise.

Time-derivative noise. Injecting eq. (3.39) in eq. (3.43), we obtain the Keldysh propagator

GK
2 (k; η1, η2) = i

π2H2

4k3 (β4 − β2)(z1z2)νγ

[
Yνγ (z1)Yνγ (z2)F (2)

νγ
(z2) + Jνγ (z1)Jνγ (z2)H(2)

νγ
(z2)

+ Jνγ (z1)Yνγ (z2)G(2)
νγ

(z2) + Jνγ (z2)Yνγ (z1)G̃(2)
νγ

(z2)
]
+ (1 ↔ 2) (C.1)

where

F (2)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Jνγ (z′)

{[
z′2 − 2(νγ − 1)νγ

]
Jνγ (z′)

+ (νγ − 1)z′Jνγ−1(z′)− (νγ − 2)z′Jνγ+1(z′)
}

(C.2)

G(2)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Yνγ (z′)

{
−
[
z′2 − 2(νγ − 1)νγ

]
Jνγ (z′)

− (νγ − 1)z′Jνγ−1(z′) + (νγ − 2)z′Jνγ+1(z′)
}

(C.3)

G̃(2)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Jνγ (z′)

{
−
[
z′2 − 2(νγ − 1)νγ

]
Yνγ (z′)

+ (1− νγ)z′Yνγ−1(z′) + (νγ − 2)z′Yνγ+1(z′)
}

(C.4)

H(2)
νγ

(z) ≡
∫ ∞

z
dz′z′2−2νγ Yνγ (z′)

{[
z′2 − 2(νγ − 1)νγ

]
Yνγ (z′)

− (1− νγ)z′Yνγ−1(z′)− (νγ − 2)z′Yνγ+1(z′)
}

(C.5)

are complicated functions easily obtained from Mathematica (the integrals being convergent)
yet too lengthy to have any practical interest being explicitly spelled here. Just as above,

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the power spectrum is obtained in the coincident time limit

Pζ(k) =
π2

8k3 (β4 − β2)
H4

f4
π

z2νγ

[
Yνγ (z)Yνγ (z)F (2)

νγ
(z) + Jνγ (z)Jνγ (z)H(2)

νγ
(z)

+ Jνγ (z)Yνγ (z)G(2)
νγ

(z) + Jνγ (z)Yνγ (z)G̃(2)
νγ

(z)
]

(C.6)

In the super-Hubble regime z ≪ 1,27 the power spectrum again freezes and we obtain

∆2
ζ(k) =

15
32(β4 − β2)

H4

f4
π

22νγ
Γ (νγ − 2) Γ (νγ)2

Γ
(
νγ − 3

2

)
Γ
(
2νγ − 1

2

) . (C.8)

One can expand this result in the large dissipation regime, leading to

∆2
ζ(k) ∝ (β4 − β2)

H4

f4
π

√
H

γ

[
1 +O

(
H

γ

)]
, (C.9)

which also easily fulfils the observational constraint ∆2
ζ = 10−9 under imposing some hierarchy

between the various scales. The small dissipation regime, on the contrary, leads to a positivity
violation with an unphysical negative power spectrum for β4−β2 > 0, which is consistent with
the sub-Hubble behaviour discussed below eq. (C.7). It appears π′2a generates an unphysical
regime whenever νγ < 2 (or equivalently γ < H).

Spatial-derivative noise. Injecting eq. (3.39) in eq. (3.44), we obtain the Keldysh prop-
agator

GK
3 (k; η1, η2) = i

π2β2H2

4k3 (z1z2)νγ

{
Yνγ (z1)Yνγ (z2)F (3)

νγ
(z2) + Jνγ (z1)Jνγ (z2)H(3)

νγ
(z2)

−
[
Jνγ (z1)Yνγ (z2) + Jνγ (z2)Yνγ (z1)

]
G(3)

νγ
(z2)

}
+ (1 ↔ 2) (C.10)

where

F (3)
νγ

(z) ≡
∫ ∞

z
dz′z′4−2νγ J2

νγ
(z′) (C.11)

G(3)
νγ

(z) ≡
∫ ∞

z
dz′z′4−2νγ Jνγ (z′)Yνγ (z′) (C.12)

H(3)
νγ

(z) ≡
∫ ∞

z
dz′z′4−2νγ Y 2

νγ
(z′) (C.13)

are complicated functions which are explicitly given for the first contribution by

F (3)
νγ

(z)= 3
8

 Γ(νγ−2)
Γ
(
νγ− 3

2

)
Γ
(
2νγ− 3

2

)−z5Γ
(

νγ+
1
2

)
2F̃3

(5
2 ,νγ+

1
2;

7
2 ,νγ+1,2νγ+1;−z2

) ,

(C.14)
27In the sub-Hubble regime z ≫ 1, the power spectrum reads

Pζ(k; η) = 1
2k3 (β4 − β2)H4

f4
π

z3

νγ − 2 (C.7)

which turns negative for νγ < 2 (or equivalently γ < H), indicating a positivity violation in small dissipation
regime. Moreover, the sub-Hubble power spectrum scale as z3 which might violate Hadamard condition in the
asymptotic past.

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for the second contribution by

G(3)
νγ

(z)=
z5−2νγ 2F3

(
1
2 , 5

2−νγ ;1−νγ , 7
2−νγ ,νγ+1;−z2

)
5πνγ−2πν2

γ

+
z5Γ(−νγ)2F3

(
5
2 ,νγ+ 1

2 ;
7
2 ,νγ+1,2νγ+1;−z2

)
5
√

πΓ
(

1
2−νγ

)
Γ(2νγ+1)

−
3Γ
(

5
2−νγ

)
8Γ(3−νγ)Γ

(
2νγ− 3

2

) , (C.15)

and for the third contribution by

H(3)
νγ

(z)=
2cot(πνγ)z5−2νγ 2F3

(
1
2 , 5

2−νγ ;1−νγ , 7
2−νγ ,νγ+1;−z2

)
5πνγ−2πν2

γ

+
4νγΓ(νγ)2z5−4νγ 2F3

(
5
2−2νγ , 1

2−νγ ;1−2νγ , 7
2−2νγ ,1−νγ ;−z2

)
π2(4νγ−5)

−
4−νγ z5 cot2(πνγ)2F3

(
5
2 ,νγ+ 1

2 ;
7
2 ,νγ+1,2νγ+1;−z2

)
5Γ(νγ+1)2 + 3Γ(νγ−2)

8Γ
(
νγ− 3

2

)
Γ
(
2νγ− 3

2

)
+
3sin(2πνγ)Γ

(
5
2−νγ

)
Γ
(

5
2−2νγ

)
4πΓ(3−νγ)

. (C.16)

Just as above, the power spectrum is obtained in the coincident time limit

Pζ(k) =
π2

8k3 β2
H4

f4
π

z2νγ

[
Y 2

νγ
(z)F (3)

νγ
(z) + J2

νγ
(z)H(3)

νγ
(z)− 2Jνγ (z)Yνγ (z)G(3)

νγ
(z)
]

(C.17)

In the super-Hubble regime z ≪ 1,28 the power spectrum again freezes and we obtain

∆2
ζ(k) =

3
16β2

H4

f4
π

22νγ
Γ (νγ − 2) Γ (νγ)2

Γ
(
νγ − 3

2

)
Γ
(
2νγ − 3

2

) . (C.19)

One can expand this result in the large dissipation regime, leading to

∆2
ζ(k) ∝ β2

H4

f4
π

√
γ

H

[
1 +O

(
H

γ

)]
, (C.20)

which is enhanced due to the
√

γ/H yet may still fulfil the observational constraint ∆2
ζ = 10−9

under imposing some hierarchy between the various scales. The small dissipation regime, on
the contrary, again leads to a positivity violation, consistent with the sub-Hubble behaviour
discussed below eq. (C.18). It appears that (∂iπa)2, just as π′2a , generates an unphysical
regime whenever νγ < 2 (or equivalently γ < H).

28In the sub-Hubble regime z ≫ 1, the power spectrum takes the same form as the one obtained from π′2
a ,

that is

Pζ(k; η) = 1
2k3 β2

H4

f4
π

z3

νγ − 2 (C.18)

which again turns negative for νγ < 2 (or equivalently γ < H), indicating a positivity violation in small
dissipation regime.

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D Unitary results in Keldysh basis

In this appendix, we derive in flat space some tree level unitary results useful to apprehend
to formalism and build physical intuition on use of the Keldysh basis.

Propagators. For a free bosonic theory, the quadratic open effective functional reads [90]

S
(2)
eff [πr, πa] =

∫
d4x

(
πr(x) πa(x)

)( 0 (∂0 + ϵ)2 − ∂2
i + m2

(∂0 − ϵ)2 − ∂2
i + m2 if(Ek)ϵ

)(
πr(x)
πa(x)

)
(D.1)

where the ϵ are here for convergence purpose and f(Ek) is a function of Ek ≡
√

k2 + m2

encoding the occupation of the state [90]. It generates the equations of motion for the
propagators [

(∂0 ± ϵ)2 + k2 + m2
]

GR/A(k; t1, t2) = δ(t1 − t2) (D.2)

and

GK(k; t1, t2) = −if(Ek)ϵ
∫

dt′GR(k; t, t′)GA(k; t′, t). (D.3)

One can benefit from the flat space frequency decomposition to solve eq. (D.2) in temporal
Fourier space

GR/A(k;ω) = − 1
(ω ∓ iϵ)2 − E2

k

= − 1
2k

[ 1
ω − (Ek ± iϵ) −

1
ω − (−Ek ± iϵ)

]
(D.4)

The real-time Green functions are given by

GR/A(k; τ) =
∫ dω

2π
eiωτ GR/A(k;ω) = ±sin [Ekτ ]

Ek
θ (±τ) (D.5)

where τ ≡ t1 − t2. We can then compute the Keldysh function either in Fourier or real
space. For instance, in real space, it writes

GK(k; t1, t2) = −i
f(Ek)

E2
k

ϵ

∫
dt′
{
sin[Ek(t1 − t′)]eϵ(t1−t′)θ(t′ − t1)

}
(D.6)

×
{
sin[Ek(t′ − t2)]e−ϵ(t′−t2)θ(t′ − t2)

}
which leads to

GK(k; τ) = i
f(Ek)
4E2

k

cos [Ekτ ] . (D.7)

The f(Ek) factor characterises the occupation of the system’s state. The vacuum result is
recovered for f(Ek) = 2Ek which we assume to be the case below. Note that this textbook
result relies on a subtle balance between fluctuation and dissipation controlled by the ϵ

prescription. Indeed, ϵ appearing in D̂R ensures the convergence of the integrals. Taking
ϵ → 0 makes the field interacts for longer and longer in the asymptotic past. Then, the
amplitude of the fluctuations controlled by D̂K must be rescaled by ϵ accordingly such that
the contributions equilibrate despite longer interactions and a finite result is reached.

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Interactions. Let us now recover some known results from the Feynman rules presented in
figure 3. For this purpose, we work with a vacuum state such that f(Ek) = 2Ek below.

Polynomial interactions. Let us first consider the unitary interaction

Lint =
λ

3!
(
π3

+ − π3
−

)
= λ

2

(
π2

rπa + 1
12π3

a

)
. (D.8)

We have two diagrams to consider similar to the one presented in figure 4. The first one leads to

Dπ3
1 = −λ

2

∫ t0

−∞(1±iϵ)
dtGK(k1; t0, t)GK(k2; t0, t)GR(k3; t0, t) + perms. (D.9)

Upon injecting eqs. (D.7) and (D.5) in eq. (D.9), we obtain

Dπ3
1 = λ

32E1E2E3

( 1
ET

+ 1
−E1 + E2 + E3

+ 1
E1 − E2 + E3

− 1
E1 + E2 − E3

)
+ perms,

(D.10)

where we defined ET ≡ E1 + E2 + E3. We observe a collection of naked folded singularities.
In [40], these singularities have been argued to correspond to classical exchanges of momenta
through scattering of physical particles (in the case of eq. (D.10), 1 + 2 → 3). The second
diagram is made of the πa components only and reads

Dπ3
2 = − λ

24

∫ t0

−∞(1±iϵ)
dtGR(k1; t0, t)GR(k2; t0, t)GR(k3; t0, t) + perms. (D.11)

Upon injecting eqs. (D.7) and (D.5) in eq. (D.11), we obtain

Dπ3
2 = λ

96E1E2E3

( 1
ET

+ 1
−E1 + E2 + E3

+ 1
E1 − E2 + E3

+ 1
E1 + E2 − E3

)
+ perms.

(D.12)

Upon performing all the six possible permutations and summing eqs. (D.10) and (D.12),
we recover the standard result

Dπ3
1 + Dπ3

2 = λ

4E1E2E3ET
. (D.13)

Derivative interactions. In order to clarify the use of differential operators, we consider
the unitary interactions π̇2π and (∂iπ)2π which can also be described in terms of the diagrams
given in figure 4. Yet, one has to be careful about the use of the differential operators.
Indeed, π̇2π decomposes into

−λ

2
(
π̇2

+π+ − π̇2
−π−

)
= −λ

2

(
π̇2

rπa + 2π̇rπ̇aπr +
1
4 π̇2

aπa

)
(D.14)

where the minus sign in front comes from the Lorentzian signature. It leads to three
contributions (two for the diagram D1 and one for the diagram D2). The first one is

Dπ̇2π
1a = λ

2

∫ t0

−∞(1±iϵ)
dt∂tG

K(k1; t0, t)∂tG
K(k2; t0, t)GR(k3; t0, t) + perms. (D.15)

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Upon injecting eqs. (D.7) and (D.5) in eq. (D.15)

Dπ̇2π
1a = λ

32E3

( 1
ET

− 1
−E1+E2+E3

− 1
E1−E2+E3

− 1
E1+E2−E3

)
+perms. (D.16)

The second contribution is

Dπ̇2π
1b = λ

∫ t0

−∞(1±iϵ)
dt∂tG

K(k1; t0, t)GK(k2; t0, t)∂tG
R(k3; t0, t) + perms. (D.17)

Upon injecting eqs. (D.7) and (D.5) in eq. (D.15), we obtain

Dπ̇2π
1b = λ

16E2

( 1
ET

− 1
−E1+E2+E3

+ 1
E1−E2+E3

+ 1
E1+E2−E3

)
+perms. (D.18)

Lastly, the final contribution is given by

Dπ̇2π
2 = λ

8

∫ t0

−∞(1±iϵ)
dt∂tG

R(k1; t0, t)∂tG
R(k2; t0, t)GR(k3; t0, t) + perms. (D.19)

Upon injecting eqs. (D.7) and (D.5) in eq. (D.19), we obtain

Dπ̇2π
2 = λ

32E3

( 1
ET

+ 1
−E1+E2+E3

+ 1
E1−E2+E3

− 1
E1+E2−E3

)
+perms. (D.20)

Upon performing all the six possible permutations and summing the various contributions,
we recover the expected result

Dπ̇2π
1a + Dπ̇2π

1b + Dπ̇2π
2 = 2λ (E1E2 + E2E3 + E1E3)

8E1E2E3ET
. (D.21)

The spatial derivative case (∂iπ)2π follows accordingly upon decomposing into

λ

2
[
(∂iπ+)2π+ − (∂iπ−)2π−

]
= λ

2

[
(∂iπr)2πa + 2∂iπr∂iπaπr +

1
4(∂iπa)2πa

]
(D.22)

Following the same procedure, we recover the standard result

D
(∂iπ)2π
1a + D

(∂iπ)2π
1b + D

(∂iπ)2π
2 = 2λ (k1.k2 + k2.k3 + k1.k3)

8E1E2E3ET
. (D.23)

Exchange diagram. At last, we aim at recovering the tree level exchange in the presence
of the cubic interaction specified in eq. (D.8). For the two vertices corresponding to the
operators π2

rπa and π3
a, there are three diagrams to compute that are represented in figure 12.

For simplicity, we focus on the s-channel (12) → (34) for which we define ks ≡ |k1 + k2|
and Es ≡

√
|k1 + k2|2 + m2. For convenience, we also define Eij···n ≡ Ei + Ej + · · ·En,

EL ≡ Es + E12 and ER ≡ Es + E34.
The first diagram in figure 12 is given by

A =2
(

i
λ

2

)2
(−i)

∫ t0

−∞(1±iϵ)
dt1

∫ t0

−∞(1±iϵ)
dt2 (D.24)[

GK(k1; t0, t1)GK(k2; t0, t1)GR(kS ; t2, t1)GK(k3; t0, t2)GR(k4; t0, t2) + perms.
]

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2
4
8

Figure 12. The three diagrams to compute given in eqs. (D.24), (D.27) and (D.28).

which is manifestly real (the Keldysh propagator being pure imaginary) and where the
permutations are over each side of the exchange E1 ↔ E2 and E3 ↔ E4, together with the
two partial poles of the exchange (E1, E2) ↔ (E3, E4). There is a symmetry factor 2 in front
coming from the freedom of choosing one out of two legs of the left vertex going into the
exchange on the A-diagram. It leads to A ≡ A1 + A2 with

A1 = λ2

16E1E2E3E4

1
ET ELER

[
1− ET (ET − Es)

(EL − 2Es)(ER − 2Es)

]
(D.25)

and

A2 = − λ2

16E1E2E3E4

{
E34

(E134 − E2)(E234 − E1) [(E1 − E2)2 − E2
s ]

+ E12
(E123 − E4)(E124 − E3) [(E3 − E4)2 − E2

s ]

}
(D.26)

The second diagram in figure 12 is given by

B =6
(

i
λ

2

)(
i

λ

24

)
(−i)

∫ t0

−∞(1±iϵ)
dt1

∫ t0

−∞(1±iϵ)
dt2 (D.27)[

GR(k1; t0, t1)GR(k2; t0, t1)GR(kS ; t2, t1)GK(k3; t0, t2)GR(k4; t0, t2)+perms.
]

where the symmetry factor 6 in front comes from the freedom of choosing one out of two
legs of the left vertex and one out of three legs of the right vertex going into the exchange
on the B-diagram. It leads, after performing the exchanges, to B = A1 − A2 so that the A2
contributions cancel out. Lastly, the third diagram in figure 12 is given by

C = 2
(

i
λ

2

)2
(−i)

∫ t0

−∞(1±iϵ)
dt1

∫ t0

−∞(1±iϵ)
dt2 (D.28)[

GR(k1; t0, t1)GK(k2; t0, t1)GK(kS ; t2, t1)GK(k3; t0, t2)GR(k4; t0, t2) + perms.
]

where the symmetry factor 2 in front comes from the freedom of choosing one out of two
legs of the left vertex, which leads, after performing the exchanges, to

C = 2λ2

16E1E2E3E4

1
ELEREs

(EL − Es)(ER − Es)
(EL − 2Es)(ER − 2Es)

. (D.29)

Summing over the three contributions, we recover the expected result, that is

A + B + C = 2λ2

16E1E2E3E4

(ET + Es)
ET ELEREs

. (D.30)

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E Langevin equation with derivative noises

In this appendix, we discuss the inclusion of the derivative noises π′2a and (∂iπa)2 into the
Langevin equation through a modification of the Hubbard-Stratonovich trick (5.26). We
present two approaches that yield the same power spectrum.

Single noise. In order to account for derivative noises, one can introduce a differential
operator D in the coupling πaδOS such that

exp
(

i

∫
d4xa4L(2)

2

)
= N

∫
D[δOS ] exp

[∫
d4xa4

(
−δO2

S

2β1
+ iδOS

→
Dπa

)]
, (E.1)

with
(
→
Dπa)2 = π2

a + β4 − β2
β1

π′2a
a2 + β2

β1

(∂iπa)2

a2 . (E.2)

This way, we can include the derivative noises into the Langevin equation via this differential
operator acting on the Gaussian noise δOS , leading to

D̂R[π(η)] =
[
a4(η)δOS(η)

]←
D

a2(η) . (E.3)

With a non-vanishing right-hand side, we obtain the solution to the differential equation
in terms of the Green function for D̂R, leading to

πk(η) =
∫ 0

−∞
dη′ GR(k; η, η′)

[
a4(η′)δOS(k, η′)

]←
D, (E.4)

where we can then integrate by parts to leave the noise without derivatives29

πk(η) = −
∫ 0

−∞
dη′ a4(η′)δOS(k, η′)

→
D
[
GR(k; η, η′)

]
. (E.5)

We can then compute the power spectrum in the usual way, such that

⟨πk(η)πk′(η)⟩ = 2β1(2π)3δ(k + k′)
∫ 0

−∞
dη′ a4(η′)

→
D
[
GR(k; η, η′)

]→
D
[
GR(k; η, η′)

]
. (E.6)

Using the property (E.2), we simplify the integrand to

2β1
→
D
[
GR(k;η,η′)

]→
D
[
GR(k;η,η′)

]
=2β1

[
GR(k;η,η′)

]2
(E.7)

+2(β4−β2)

[
∂η′GR(k;η,η′)

]2
a2(η) +2β2

[
∂iG

R(k;η,η′)
]2

a2(η) .

Under this form, it is explicit that it matches the result for the power spectrum (3.14)
derived in section 3.

29The boundary terms vanish thanks to the Bunch-Davies vacuum (to the infinite past) and the retarded
boundary condition in the coincident time limit for η′ = η.

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Three noises. Another way to include derivative noises into the Langevin equation consists
in introducing new Gaussian fields δO(i)

S that couple derivatively to πa in the Hubbard-
Stratonovich trick (5.26). For the time derivative noise, the Hubbard-Stratonovich tricks
takes the form

exp
[
−
∫

d4xa2(β4 − β2)π′2a

]
= N (2)

∫
D[δO(2)

S ] (E.8)

× exp
{∫

d4xa4
[
− [δO(2)

S ]2
4β1

+ i

√
β4 − β2

β1

π′a
a

δO(2)
S

]}
.

For the spatial derivative noise, we need to consider the square root of the Laplacian

exp
[
−
∫

d4xa2β2(∂iπa)2
]
= N (3)

∫
D[δO(3)

S ] (E.9)

× exp
{∫

d4xa4
[
− [δO(3)

S ]2
4β1

+ i

√
β2
β1

1
a

δO(3)
S

√
∇2πa

]}
.

We can then modify the Langevin equation which becomes

D̂R[πk(η)] = a2(η)δO(1)
S − 1

a2(η)∂η

[√
β4−β2

β1
a3(η)δO(2)

S

]
−ka(η)

√
β2
β1

δO(3)
S . (E.10)

The statistics of the noises now have a matrix structure with indices labelling the noise
under consideration

⟨δO(a)
S (k, η1)δO(b)

S (k′, η2)⟩ = 2β1H4η4
1(2π)3δ(k + k′)δ(η1 − η2)δab. (E.11)

Using this technique, we fully recover from the Langevin equation the expression of the path
integral Keldysh propagator (3.14) from section 3.

Open Access. This article is distributed under the terms of the Creative Commons Attri-
bution License (CC-BY4.0), which permits any use, distribution and reproduction in any
medium, provided the original author(s) and source are credited.

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	Introduction
	Summary of the main results

	The open effective field theory of inflation
	EFT construction
	Open effective functional
	Setting up expectations

	Power spectrum
	Generating functional
	Flat space intuition
	Dissipative and noise-induced power spectrum

	Bispectrum
	Interactions
	Flat space intuition
	Non-Gaussianities from dissipation and noises

	Matching
	Power spectrum and bispectrum
	Langevin equation

	Discussions
	Derivation of S(eff) constraints
	de Sitter Keldysh functions
	de Sitter power spectrum with derivative noises
	Unitary results in Keldysh basis
	Langevin equation with derivative noises