aainstitutetext: Department of Physics, Faculty of Natural Sciences, Norwegian
University of Science and Technology, Høgskoleringen 5, N-7491 Trondheim,
Norwaybbinstitutetext: Faculty of Science and Technology, University of
Stavanger, 4036 Stavanger, Norway

# Stochastic inflation from quantum field theory and the parametric dependence
of the effective noise amplitude

Jens O. Andersen a,b Magdalena Eriksson b Anders Tranberg
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The non-linear dynamics of long-wavelength cosmological fluctuations may be
phrased in terms of an effective classical, but stochastic evolution equation.
The stochastic noise represents short-wavelength modes that continually
redshift into the long-wavelength domain. The effective evolution may be
derived from first principles quantum field theory in an expanding background,
through a sequence of approximations calling for additional scrutiny. We
perform such an analysis, putting particular emphasis on the amplitude of the
stochastic noise, which ultimately determines the cosmological correlations
and provides a non-perturbative IR regulator to the dynamics.

###### Keywords:

Cosmology of Theories beyond the SM, Effective Field Theories, Nonperturbative
Effects

##  1 Introduction

The primordial density perturbations in the Universe are expected to have
originated as vacuum fluctuations of a light, weakly-interacting scalar field
$\phi$, amplified during a period of accelerating cosmological expansion known
as inflation (see for instance Riotto:2002yw ; LythLiddle ). However, it has
long been known that when computing cosmological correlations in perturbative
quantum field theory (QFT), the (near-)massless propagator leads to unphysical
divergent and secular IR behaviour, which call for regularisation
Tsamis:2005hd ; vanderMeulen:2007ah .

One way to achieve this is through resumming infinite sets of perturbative
diagrams, using resummation techniques well-known from Minkowski space
computations, adapted to de Sitter space (dS). These include the large-$N$
expansion Serreau:2011fu ; Serreau:2013psa , truncations of the two-particle-
irreducible (2PI) effective action formalism Arai:2011dd ; LopezNacir:2013alw
, the non-perturbative renormalisation group (RG) Serreau:2013eoa , and
dressing of the Euclidean zero mode by summing an infinite class of diagrams
Rajaraman:2010xd ; Beneke:2012kn ; LopezNacir:2016gzi . These all reveal that
a dynamical mass is generated by non-linear interactions, regularising the
correlation functions, even in the case of minimally coupled massless scalar
fields.

An alternative to the diagrammatic QFT analysis is stochastic inflation
Starobinsky:1986fx ; Starobinsky:1994bd , where a separation of scales is
introduced between the (far) super-horizon (IR) modes and the near- and sub-
horizon (UV) modes. By integrating out the UV modes, an effective classical,
but stochastic, IR theory arises, which retains much of the nonlinear,
nonperturbative dynamics of the exact theory. The IR dynamics again generate a
dynamical mass. This is reminiscent of similar approaches to quantum thermal
field theories of gauge and fermion fields bodeker , where UV degrees of
freedom source classical IR dynamics in the form of a stochastic noise. One
important difference is that in stochastic inflation, the noise is the result
of a sliding scale separation turning UV modes into IR modes. It therefore
appears even in the non-interacting theory, and is hence unsuppressed by
powers of the couplings. It also follows that the stochastic approach is
specific to near-de Sitter geometries in that the Hubble horizon is used as a
factorization scale between the long and short wavelength modes, with modes
leaving the horizon only for an accelerating expansion.

In the simplest overdamped (slow-roll) limit, $\ddot{\phi}\ll 3H\dot{\phi}$,
and in addition neglecting spatial gradients, the effective IR dynamics can be
described by a Langevin equation, and field correlations be obtained via the
corresponding Fokker-Planck equation. This is the framework most often applied
to models of inflation, although the problem is still numerically tractable
even when relaxing some of these assumptions Cable:2020dke .

The stochastic approach has been shown to reproduce the correct IR behaviour
of the full QFT statistical propagator to leading order in the coupling
Tsamis:2005hd ; Finelli:2008zg ; Garbrecht:2013coa ; Garbrecht:2014dca ;
Onemli:2015pma  and was recently favourably compared at two-loop order in
perturbation theory Kamenshchik:2021tjh . Other authors have studied the
evolution of the density matrix for the IR modes to recover a Fokker-Planck
equation Burgess:2015ajz ; Collins:2017haz , and Ref. Baumgart:2019clc  argued
the emergence of the stochastic formalism from a leading-log diagrammatic
analysis at coincident points. The conceptual and computational tractability
offered by the stochastic approach has made it a popular method, and in recent
years there has been an upsurge in activity aiming to extend it e.g. to beyond
leading order in the coupling Collins:2017haz ; Tokuda:2017fdh ; Cohen:2021fzf
, the slow-roll approximation Pattison:2019hef ; Cable:2020dke , or to include
derivative interactions Kitamoto:2018dek . However, compared to its extensive
use, studies of the embedding and range of validity of the stochastic approach
in QFT remain scarce (see however Moss:2016uix ), which has motivated the
present work.

In this paper we will revisit the first-principles derivation at the level of
the path integral, paying attention to the sequence of approximations required
to reach the standard form of the effective dynamics. These include the coarse
graining procedure, the scale separation specified by a so-called window
function and the parametrisation of the free UV vacuum sourcing the horizon-
crossing modes. In particular, the stochastic noise is often taken to have a
particular, universal amplitude, which we will see only arises in some very
specific parametric limits.

The article is organised as follows. In the following subsection 1.1, we
present the effective stochastic theory, and how it is applied to compute IR
field correlators given an inflation model. Having described the objective of
our discussion, we in section 2 introduce the real-time QFT formalism for the
problem at hand, and show how an effective IR theory arises (sections 2.2 and
2.4) and how to interpret it as a stochastic theory (section 2.5). We
reconnect to the standard form of the stochastic noise resulting from the
choice of a sharp window function applied to the field variables in momentum
space Starobinsky:1994bd  (section 2.6). This allows us to identify the set of
assumptions and approximations going into the derivation, and their validity.
We investigate an alternative mode separation procedure at the level of the
path integral in section 3, and proceed to again derive the target theory, in
the process identifying the approximations required. The procedure is
illustrated and approximations checked through an example (Gaussian) window
function (sections 3.1–3.4). Section 4 represents the bulk of our analysis,
where we for a selection of window functions compute and compare the amplitude
of the stochastic noise, considering both massless and massive UV modes, dS
and slow-roll spacetimes, and test the validity of the central assumptions.
Conclusions are gathered in section 5.

###  1.1 Application of stochastic dynamics to inflationary perturbations

We first briefly review the stochastic inflationary formalism, and how a
dynamical mass is generated. The dynamics of the coarse-grained IR field
$\phi_{\text{\tiny IR}}$ on super-Hubble scales is described by a Langevin
equation

$\displaystyle\dot{\phi}_{\text{\tiny
IR}}(t)+\frac{V^{\prime}(\phi_{\text{\tiny IR}})}{3H}=\xi_{\phi}(t),$ (1)

where $H\equiv\dot{a}(t)/a(t)$ is the Hubble rate,

$\displaystyle V(\phi)=\tfrac{1}{2}m^{2}\phi^{2}+V_{\text{I}}(\phi),$ (2)

is the potential and $\xi_{\phi}$ is a Gaussian stochastic noise with the
correlation

$\displaystyle\langle\xi_{\phi}(t)\xi_{\phi}(t^{\prime})\rangle=\frac{H^{3}}{4\pi^{2}}f(\epsilon_{H},\epsilon_{M},\dotso)\delta(t-t^{\prime}),$
(3)

corresponding to white, Markovian, noise statistics. Most commonly, the
function $f$ is taken to be trivial, $f=1$, but we will allow for the noise
amplitude to a priori depend on several variables, including the slow-roll and
mass parameters $\epsilon_{H}$ and $\epsilon_{M}$ defined as

$\displaystyle\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}},\qquad\epsilon_{M}\equiv\frac{m^{2}}{3H^{2}}.$
(4)

The Langevin equation gives rise to a Fokker-Planck equation for the one-point
probability distribution of the IR field

$\displaystyle\partial_{t}P(t,\phi_{\text{\tiny
IR}})=\left(\frac{V^{\prime\prime}(\phi_{\text{\tiny
IR}})}{3H}+\frac{V^{\prime}(\phi_{\text{\tiny
IR}})}{3H}\partial_{\phi_{\text{\tiny
IR}}}+\frac{H^{3}f}{8\pi^{2}}\partial^{2}_{\phi_{\text{\tiny
IR}}}\right)P(t,\phi_{\text{\tiny IR}}).$ (5)

Given an initial distribution at the beginning of inflation, the distribution
at any later time then follows, and observables may be computed as

$\displaystyle\langle\mathcal{O}(\phi_{\text{\tiny
IR}}(t))\rangle=\int\differential\phi_{\text{\tiny IR}}P(t,\phi_{\text{\tiny
IR}})\mathcal{O}(\phi_{\text{\tiny IR}}).$ (6)

Often, the late-time equilibrium solution is assumed to have been
reached111Although this may not always be the case Enqvist:2012 .

$\displaystyle P_{\text{eq}}(\phi_{\text{\tiny
IR}})\sim\,\mathrm{exp}\left[-\frac{8\pi^{2}V(\phi_{\text{\tiny
IR}})}{3fH^{4}}\right],$ (7)

and for instance for a free, massive theory,
$V(\phi)=\tfrac{1}{2}m^{2}\phi^{2}$, the variance is given by

$\displaystyle\langle\phi^{2}_{\text{\tiny
IR}}\rangle=\frac{3fH^{4}}{8\pi^{2}m^{2}}.$ (8)

For $f=1$, this is indeed the finite part of the field correlator in the
Bunch-Davies vacuum at leading order in the limit $m^{2}\ll H^{2}$. This can
be seen directly from expanding the free semi-dS propagator in the same limit
Prokopec:2003tm ; Janssen:2007ht ,

$\displaystyle G_{F}(y)=\frac{H^{2}}{4\pi^{2}}\left(\frac{1}{y}-\frac{1}{2}\ln
y+\frac{1}{2(\epsilon_{M}-\epsilon_{H})}-1+\ln
2+\mathcal{O}(\epsilon_{M},\epsilon_{H})\right),$ (9)

where $y$ is the dS invariant length scale,

$\displaystyle
y(x,x^{\prime})=4\sin^{2}\left(\tfrac{1}{2}l(x,x^{\prime})H\right),$ (10)

with $l(x,x^{\prime})$ denoting the geodesic distance. Setting
$\epsilon_{H}=0$ and inserting (4) into the leading finite term, we recover
(8).

However, when applying the stochastic formalism to a massless self-interacting
field with $V_{\text{I}}(\phi)=\frac{\lambda}{4}\phi^{4}$, the late-time
Fokker-Planck distribution straightforwardly leads to Starobinsky:1994bd

$\displaystyle\lim_{t\rightarrow\infty}\langle\phi^{2}_{\text{\tiny
IR}}(t)\rangle=\sqrt{\frac{3f}{2\pi^{2}}}\frac{\Gamma\left[\tfrac{3}{4}\right]}{\Gamma\left[\tfrac{1}{4}\right]}\frac{H^{2}}{\sqrt{\lambda}}.$
(11)

One may then infer that a mass has been generated dynamically by comparing the
massive dS result (8) with (11), i.e.,

$\displaystyle
m^{2}_{\text{dyn}}=\frac{\sqrt{6}}{\sqrt{f}8\pi}\frac{\Gamma\left[\tfrac{1}{4}\right]}{\Gamma\left[\tfrac{3}{4}\right]}\sqrt{\lambda}H^{2}\simeq
0.288\frac{\sqrt{\lambda}}{\sqrt{f}}H^{2}.$ (12)

The effective mass squared is proportional to $H^{2}$ and importantly to
$\sqrt{\lambda}$, rather than an integer power of the coupling. This non-
analytic dependence suggests that the stochastic prescription amounts to a
resummation of Feynman diagrams from all orders of perturbation theory,
providing an IR regulator in an expanding background ($H\neq 0$). We see that
the amplitude of the noise, including the function $f(\dotso)$, sets the scale
for correlators and the dynamical mass.

The original stochastic formalism was restricted to the slow-roll limit where
$\ddot{\phi}_{\text{\tiny IR}}$ is neglected, and where also
$V^{\prime\prime}/H^{2}\ll 1$. However, one may generalise to include second-
order derivatives in time by introducing a ”momentum noise” $\xi_{\pi}$ to
accompany the new degree of freedom. The Langevin dynamics can then be written
as two coupled equations for the IR field and its time derivative:

$\displaystyle\dot{\phi}_{\text{\tiny IR}}=\pi_{\text{\tiny
IR}}-\xi_{\phi},\qquad\dot{\pi}_{\text{\tiny IR}}+3H\pi_{\text{\tiny
IR}}+V^{\prime}(\phi_{\text{\tiny IR}})=\xi_{\pi},$ (13)

alternatively,

$\displaystyle\ddot{\phi}_{\text{\tiny IR}}+3H\dot{\phi}_{\text{\tiny
IR}}+V^{\prime}(\phi_{\text{\tiny
IR}})=3H\xi_{\phi}+\dot{\xi}_{\phi}+\xi_{\pi}\equiv\xi.$ (14)

For later reference, we note that in the particular case, when
$3H\xi_{\phi}\gg\dot{\xi}_{\phi},\xi_{\pi}$ (in a sense to be discussed
below), the normalisation (3) implies

$\displaystyle\langle\xi(t)\xi(t^{\prime})\rangle\rightarrow
9H^{2}\langle\xi_{\phi}(t)\xi_{\phi}(t^{\prime})\rangle=\frac{9H^{5}}{4\pi^{2}}f(\epsilon_{H},\epsilon_{M},\dotso)\delta(t-t^{\prime}).$
(15)

In the following we will see that these results emerge also (in certain
limits) in QFT. Starting from the closed-time-path (CTP) action, we will
derive the Langevin equation using a suitable set of approximations. In
particular we keep track of the normalisation of the noise, parametrised by a
function $f(\dotso)$, which as we have seen appears prominently in the late-
time Fokker-Planck distribution and hence the computed observables. The
variance for the non-interacting dS vacuum corresponds to $f=1$ to leading
order in the mass, and provides a natural benchmark for comparison.

##  2 Stochastic dynamics from quantum field theory

We first revisit the derivation of the Langevin evolution from a field
theoretic perspective, paying special attention to the scale separation
between the long- and short-wavelength degrees of freedom Starobinsky:1994bd ;
Moss:2016uix ; Morikawa1990 .

###  2.1 The closed-time-path formalism for inflation

We consider the real-time evolution of an interacting scalar field in an FRLW
spacetime, whose line element is given by

$\displaystyle\differential s^{2}=-\differential
t^{2}+a^{2}(t)\differential\mathbf{x}^{2},$ (16)

where $a(t)$ is the scale factor and the slow-roll parameter $\epsilon_{H}$
parameterises the deviation from a constant expansion rate, corresponding to
dS space where $a(t)\propto e^{Ht}$. Expectation values for operators are
computed using the CTP formalism222Also known as the in-in or Schwinger-
Keldysh formalism. of QFT Chou84 ; Calzetta:1986ey . In the CTP formalism, the
time coordinate runs on a closed time path from $t_{\text{in}}$ to $t$ (the
’+’ branch) and back again (the ’$-$’ branch). The field $\phi$ and source $J$
are split up into path-ordered constituents $\phi^{\pm}$ and $J^{\pm}$, where
for equal time $\phi^{+}(t)=\phi^{-}(t)$. The quantum correlators evaluated at
time $t$ can be obtained by functional differentiation of from the generating
functional

$\displaystyle
Z[\mathbbm{J}]=\int\mathcal{D}\Phi\,\mathrm{exp}\left[\frac{i}{2}\int_{x}\left(\Phi^{T}\mathbbm{G}^{-1}\Phi-2V_{\text{I}}(\Phi)+\mathbbm{J}^{T}\Phi\right)\right],$
(17)

where $\int_{x}\equiv\int\differential^{4}x\sqrt{-g(x)}$ for brevity,
$V_{\text{I}}(\Phi)=V_{\text{I}}(\phi^{+})-V_{\text{I}}(\phi^{-})$ is the
interacting potential and we define

$\displaystyle\Phi(x)\equiv\begin{bmatrix}\phi^{+}(x)\\\
\phi^{-}(x)\end{bmatrix},\quad\mathbbm{G}^{-1}(x,x^{\prime})\equiv\begin{bmatrix}G^{-1}(x)&0\\\
0&-G^{-1}(x)\end{bmatrix}\delta(x,x^{\prime}),\quad\mathbbm{J}(x)\equiv\begin{bmatrix}J^{+}(x)\\\
-J^{-}(x)\end{bmatrix}.$ (18)

The free inverse propagator is $G^{-1}(x)=\Box_{x}-m^{2}$ with
$\Box_{x}\equiv\partial_{\mu}(\sqrt{-g(x)}g^{\mu\nu}\partial_{\nu})/\sqrt{-g(x)}$.
The matrix $\mathbbm{G}(x,x^{\prime})$ consists of the CTP propagators with
all four possible time orderings:

$\displaystyle\mathbbm{G}(x,x^{\prime})=\begin{bmatrix}G^{++}(x,x^{\prime})&G^{+-}(x,x^{\prime})\\\
G^{-+}(x,x^{\prime})&G^{--}(x,x^{\prime})\end{bmatrix}=-i\begin{bmatrix}\langle
T\phi^{+}(x)\phi^{+}(x^{\prime})\rangle&\langle\phi(x)^{+}\phi^{-}(x^{\prime})\rangle\\\
\langle\phi^{-}(x^{\prime})\phi^{+}(x)\rangle&\langle\bar{T}\phi^{-}(x)\phi^{-}(x^{\prime})\rangle\end{bmatrix},$
(19)

where $T$ ($\bar{T}$) denote (anti-)time ordering, satisfying

$\displaystyle\langle T\phi(x)\phi(x^{\prime})\rangle$
$\displaystyle=\Theta(t-t^{\prime})\langle\phi(x)\phi(x^{\prime})\rangle+\Theta(t^{\prime}-t)\langle\phi(x^{\prime})\phi(x)\rangle,$
(20) $\displaystyle\langle\bar{T}\phi(x)\phi(x^{\prime})\rangle$
$\displaystyle=\Theta(t-t^{\prime})\langle\phi(x^{\prime})\phi(x)\rangle+\Theta(t^{\prime}-t)\langle\phi(x)\phi(x^{\prime})\rangle.$
(21)

The two-point functions are related via

$\displaystyle
G^{++}(x,x^{\prime})+G^{--}(x,x^{\prime})=G^{+-}(x,x^{\prime})+G^{-+}(x,x^{\prime}).$
(22)

For classical considerations it is useful to work in the Keldysh basis, in
which the Schwinger basis fields $\phi^{\pm}$ are transformed into ”classical”
and ”quantum” fields $\phi^{\text{c}}$, $\phi^{\text{q}}$ via a transformation
matrix $U$ as

$\displaystyle\Phi(x)\rightarrow
U\Phi(x)=\begin{bmatrix}\tfrac{1}{2}[\phi^{+}(x)+\phi^{-}(x)]\\\
\phi^{+}(x)-\phi^{-}(x)\end{bmatrix}\equiv\begin{bmatrix}\phi^{\text{c}}(x)\\\
\phi^{\text{q}}(x)\end{bmatrix},\qquad
U=\begin{bmatrix}\tfrac{1}{2}&\tfrac{1}{2}\\\ 1&-1\end{bmatrix}.$ (23)

This notation can be understood heuristically in the sense that
$\phi^{\text{q}}$ expresses the amplitude of the quantum fluctuations around
the mean field value $\phi^{\text{c}}$. The kinetic operator in the Keldysh
basis is

$\displaystyle\mathbbm{G}^{-1}(x,x^{\prime})\rightarrow
U\mathbbm{G}^{-1}(x,x^{\prime})U^{T}=\begin{bmatrix}0&G^{-1}(x)\\\
G^{-1}(x)&0\end{bmatrix}\delta(x,x^{\prime}),$ (24)

and the propagators are

$\displaystyle\mathbbm{G}(x,x^{\prime})=\begin{bmatrix}-iG_{F}(x,x^{\prime})&G_{R}(x,x^{\prime})\\\
G_{A}(x,x^{\prime})&0\end{bmatrix},$ (25)

with

$\displaystyle-iG_{F}(x,x^{\prime})$
$\displaystyle=\tfrac{1}{2}\left[G^{+-}(x,x^{\prime})+G^{-+}(x,x^{\prime})\right]=-\tfrac{i}{2}\langle\\{\phi(x),\phi(x^{\prime})\\}\rangle,$
(26) $\displaystyle G_{R}(x,x^{\prime})$
$\displaystyle=G^{++}(x,x^{\prime})-G^{+-}(x,x^{\prime})=-i\Theta(t-t^{\prime})\langle[\phi(x),\phi(x^{\prime})]\rangle,$
$\displaystyle G_{A}(x,x^{\prime})$
$\displaystyle=G^{++}(x,x^{\prime})-G^{-+}(x,x^{\prime})=-i\Theta(t^{\prime}-t)\langle[\phi(x^{\prime}),\phi(x)]\rangle.$

We recognise the statistical Feynman propagator $G_{F}$ and spectral
retarded/advanced propagators $G_{R}(x,x^{\prime})=G_{A}(x^{\prime},x)$. The
free propagators satisfy the equations of motion

$\displaystyle G^{-1}(x)G_{F}(x,x^{\prime})=0,\qquad
G^{-1}(x)G_{R,A}(x,x^{\prime})=\frac{\delta(x-x^{\prime})}{\sqrt{-g(x)}}.$
(27)

###  2.2 Coarse graining and window functions

In order to derive the effective IR dynamics of the inflaton field $\phi$, one
may proceed to split it into UV and IR parts as $\phi=\phi_{\text{\tiny
UV}}+\phi_{\text{\tiny IR}}$. The field split has been addressed at the level
of the effective action in earlier work Morikawa1990  (see also
Matarrese:2003ye ; PerreaultLevasseur:2013kfq ), where an effective IR
equation of motion is obtained by integrating out the UV modes.

The IR/UV field split is a coarse-graining procedure, in the sense that it
amounts to averaging in position space using a smoothing window function
$\overline{W}$ dependent on a characteristic smoothing scale $L$ as
Winitzki:1999ve

$\displaystyle\phi_{\text{\tiny
IR}}(x)=\int\differential^{3}\mathbf{x^{\prime}}\phi(x^{\prime})\overline{W}(\mathbf{x}-\mathbf{x^{\prime}},L).$
(28)

The IR field value at a given point in space is then given by the value of the
full field $\phi$ averaged over a region of size $L$. The convolution of the
window function in position space then corresponds to multiplication in
momentum space. Implicit in (28) is that $\overline{W}$ is taken to be local
in time, through a form

$\displaystyle\overline{W}(x,x^{\prime})=\delta(t-t^{\prime})\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}\overline{W}(k,t)e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})}.$
(29)

Note that $k$ is the co-moving momentum, with $k/a(t)$ being the physical
momentum. We will use this form in the following as well. From the field split
$\phi=\phi_{\text{\tiny UV}}+\phi_{\text{\tiny IR}}$ with $\phi_{\text{\tiny
IR}}$ defined as in (28), it follows that we can define the UV part of the
field in terms of a completing function $W$,

$\displaystyle
W(x,x^{\prime})+\overline{W}(x,x^{\prime})=\delta(x-x^{\prime}),$ (30)

such that

$\displaystyle\phi_{\text{\tiny
UV}}(x)=\int_{x^{\prime}}W(x,x^{\prime})\phi(x^{\prime}),\qquad\phi_{\text{\tiny
IR}}(x)=\int_{x^{\prime}}\overline{W}(x,x^{\prime})\phi(x^{\prime}).$ (31)

In order for the window function to truly split the field into an IR and UV
part, we require them to satisfy $W(k,t)=1$ for field modes with $k\gg aH$ and
$W(k,t)=0$ for $k\ll aH$, and vice versa for $\overline{W}(k,t)$. In
particular, if $W(k,t)$ has precisely the value 1 or 0 for a given mode, that
entire mode belongs to either the UV or IR field. For a smooth window function
interpolating between 1 and 0, modes of all $k$ will contribute to both the IR
and the UV field. In that case, the field split is into a ”mostly-UV” and
”mostly-IR” part.

We will perform the split in terms of physical momentum $k/a(t)$, rather than
the co-moving momentum defining the mode functions. In practice, this
introduces a time-dependence of the window function, as the UV modes are
continuously red-shifted and join the coarse-grained classical IR modes. For
the IR modes the effect of these quantum fluctuations crossing the Hubble
horizon is what amounts to a continuous noise source.

###  2.3 The UV vacuum mode functions

Expanding the propagator in terms of quantised field momentum modes

$\displaystyle\phi(x)=\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}\big{[}a_{\mathbf{k}}\phi(k,t)e^{i\mathbf{k}\cdot\mathbf{x}}+\text{h.c.}\big{]},$
(32)

with
$[a_{\mathbf{k}},a_{\mathbf{k^{\prime}}}^{\dagger}]=(2\pi)^{3}\delta^{3}(\mathbf{k}+\mathbf{k^{\prime}})$
and $a_{\mathbf{k}}$ annihilating the Bunch-Davies vacuum, the free equation
of motion (27) has the standard solution

$\displaystyle\phi(k,t)=\frac{\sqrt{\pi}}{2\sqrt{a^{3}H(1-\epsilon_{H})}}H_{\nu}^{(1)}\left(\frac{k}{aH(1-\epsilon_{H})}\right),$
(33)

where $\nu^{2}=\frac{9}{4}-3\epsilon_{M}+3\epsilon_{H}$ and $H_{\nu}^{(1)}(x)$
are the Hankel functions of the first kind grad . In the massless dS case,
$\nu=\frac{3}{2}$ and the mode solution reduces to

$\displaystyle\phi(k,t)=-\frac{H}{\sqrt{2k^{3}}}\bigg{(}i+\frac{k}{aH}\bigg{)}e^{ik/aH},\qquad\epsilon_{M}=0,\quad\epsilon_{H}=0.$
(34)

In the long-wavelength limit, the Hankel function can be approximated333For
small $x$ and $\nu>0$ the leading terms are
$H_{\nu}^{(1)}(x)\simeq-i\frac{\Gamma[\nu]}{\pi}\left(\frac{x}{2}\right)^{-\nu}\left[1+\frac{1}{\nu-1}\left(\frac{x}{2}\right)^{2}+\dotso\right].$
such that the leading term in $k/aH$ of the mode function (33) is given by

$\displaystyle\phi(k,t)\simeq-i\frac{H(1-\epsilon_{H})}{\sqrt{2k^{3}}}\left(\frac{k}{aH(1-\epsilon_{H})}\right)^{\epsilon_{M}-\epsilon_{H}},\qquad\epsilon_{M}\ll
1,\quad k/aH\ll 1.$ (35)

In the following, whenever the quantum UV modes enter through their
correlators, these will be represented by the free vacuum state through the
above mode functions (33)–(35). The long-wavelength approximate solution (35)
applies to our UV modes, since as we will see below, the modes responsible for
the stochastic noise are in fact super-horizon, in the sense required by (35).
They are modes transitioning from the UV to the IR.

#### Interacting UV mode functions

In order to solve explicitly for the UV mode functions, it is usually assumed
that interactions may be neglected. It is however worth noting, that since the
mass only enters through $\epsilon_{M}$, the expressions generalise trivially
to Gaussian interacting UV modes including an effective mass
$M^{2}=m^{2}+\dotso$. In the complete theory, where also the UV modes
experience an effective mass $\propto\sqrt{\lambda}H^{2}$, modes are never
truly massless, and for $m^{2}=0$, the small-mass criterion $\epsilon_{M}\ll
1$ becomes a constraint on the coupling, $\sqrt{\lambda}\ll 1$.

###  2.4 IR effective theory

In the language of open quantum systems, the quantum UV modes can be viewed as
a bath affecting the system of classical IR modes. The generating functional
(17) can then be written in terms of an influence functional $\mathcal{F}$ as

$\displaystyle Z[\mathbbm{J}]=\int\mathcal{D}\Phi_{\text{\tiny
IR}}e^{i(S[\Phi_{\text{\tiny IR}}]-\mathbbm{J}^{T}\Phi_{\text{\tiny
IR}})}\mathcal{F}[\Phi_{\text{\tiny IR}},\mathbb{J}],$ (36)

where

$\displaystyle\mathcal{F}[\Phi_{\text{\tiny
IR}},\mathbb{J}]=\int\mathcal{D}\Phi_{\text{\tiny
UV}}\,\mathrm{exp}\left[i\int_{x}\left(\tfrac{1}{2}\Phi_{\text{\tiny
UV}}^{T}\mathbbm{G}^{-1}\Phi_{\text{\tiny UV}}+\Phi_{\text{\tiny
IR}}^{T}\mathbbm{G}^{-1}\Phi_{\text{\tiny UV}}-V_{\text{I}}(\Phi_{\text{\tiny
IR}},\Phi_{\text{\tiny UV}})+\mathbbm{J}^{T}\Phi_{\text{\tiny
UV}}\right)\right],$ (37)

contains all the instances of $\Phi_{\text{\tiny UV}}$ and is to become an
effective contribution the IR dynamics upon integrating out the UV part of the
field. Here we have defined $V_{\text{I}}(\Phi_{\text{\tiny
IR}},\Phi_{\text{\tiny UV}})\equiv V_{\text{I}}(\Phi_{\text{\tiny
IR}}+\Phi_{\text{\tiny UV}})-V_{\text{I}}(\Phi_{\text{\tiny IR}})$, which
typically includes non-linear self-interactions among the UV modes, and
between the UV and IR modes. We are interested in the stochastic noise that
arises solely from the time-dependent field split of the free theory, and we
will therefore simply neglect this non-linear interaction in our calculation,
and as in the preceding section consider only free UV fields.444Stochastic
contributions from the non-linear IR-UV interactions involve powers of a
coupling which for inflation tend to be small. The self-interaction of the IR
field with itself is retained in (36) Setting also $\mathbb{J}=0$ in (37), we
may complete the square and perform the path integral over $\Phi_{\text{\tiny
UV}}$, to obtain

$\displaystyle\mathcal{F}_{0}[\Phi_{\text{\tiny
IR}}]=N\,\mathrm{exp}\left[-\frac{i}{2}\int_{x}\int_{x^{\prime}}\Phi_{\text{\tiny
IR}}^{T}(x)\mathbbm{G}^{-1}(x)\mathbbm{G}_{\text{\tiny
UV}}(x,x^{\prime})\mathbbm{G}^{-1}(x^{\prime})\Phi_{\text{\tiny
IR}}(x^{\prime})\right],$ (38)

where $N\propto(\det\mathbbm{G}_{\text{\tiny UV}}^{-1})^{-1/2}$ is a
normalisation factor and UV propagator is defined by (25) with (31) as

$\displaystyle\mathbbm{G}_{\text{\tiny
UV}}(x,x^{\prime})=W(x)\mathbbm{G}(x,x^{\prime})W(x^{\prime}),$ (39)

Here $G^{-1}(x^{\prime})$ can be integrated by parts to act on
${G}_{\text{\tiny UV}}(x,x^{\prime})$ rather than the field $\Phi_{\text{\tiny
IR}}(x^{\prime})$. Later we will also consider departures from strict dS space
($\epsilon_{H}\neq 0$), but for the moment we continue with simply

$\displaystyle
G^{-1}(x)=-\partial_{t}^{2}-3H\partial_{t}+\frac{\nabla^{2}_{\mathbf{x}}}{a^{2}(t)}-m^{2}.$
(40)

Writing out the kernel in (38) in the Keldysh c-q matrix form, we explicitly
have

$\displaystyle\mathcal{F}_{0}[\Phi_{\text{\tiny IR}}]$
$\displaystyle=N\mathrm{exp}\bigg{[}-\frac{i}{2}\int_{x}\int_{x^{\prime}}\bigg{(}-i[\phi_{\text{\tiny
IR}}^{\text{c}}(x),\phi_{\text{\tiny
IR}}^{\text{q}}(x)]\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})}$
(41)
$\displaystyle\quad\times\left[\begin{smallmatrix}0&[{\mathcal{Q}}_{t}+W_{t}G^{-1}_{t}][{\mathcal{Q}}_{t^{\prime}}+W_{t^{\prime}}G^{-1}_{t^{\prime}}]G_{A}(t,t^{\prime})\\\
[{\mathcal{Q}}_{t}+W_{t}G^{-1}_{t}][{\mathcal{Q}}_{t^{\prime}}+W_{t^{\prime}}G^{-1}_{t^{\prime}}]G_{R}(t,t^{\prime})&-i{\mathcal{Q}}_{t}{\mathcal{Q}}_{t^{\prime}}G_{F}(t,t^{\prime})\end{smallmatrix}\right]\bigg{[}\,\begin{matrix}\phi_{\text{\tiny
IR}}^{\text{c}}(x^{\prime})\\\ \phi_{\text{\tiny
IR}}^{\text{q}}(x^{\prime})\end{matrix}\,\bigg{]}\bigg{)}\bigg{]},$

where the UV propagator components $G_{F}$, $G_{R/A}$, $G^{-1}$ and the window
function $W$ enter through their momentum space counterparts, and we have
defined the operator

$\displaystyle{\mathcal{Q}}_{x}=\int\frac{\differential^{3}{\bf
k}}{(2\pi)^{3}}{\mathcal{Q}}_{t}(k)e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})},\quad{\mathcal{Q}}_{t}(k)\equiv-\ddot{W}(k,t)-3H\dot{W}(k,t)-2\dot{W}(k,t)\partial_{t}.$
(42)

As we will argue below, the off-diagonal c-q terms in (41) may under some
circumstances be neglected compared to the diagonal q-q component. Since the
UV fields are taken to be non-interacting, we can make use of (27), and after
partially integrating with respect to time, one may write for the off-diagonal
components

$\displaystyle\int_{x}\int_{x^{\prime}}\phi_{\text{\tiny
IR}}^{\text{q}}(x)\phi_{\text{\tiny
IR}}^{\text{c}}(x^{\prime})[{\mathcal{Q}}_{x}+W_{x}G^{-1}_{x}][{\mathcal{Q}}_{x^{\prime}}+W_{x^{\prime}}G^{-1}_{x^{\prime}}]G_{R}(x,x^{\prime})$
(43) $\displaystyle=\int_{x}\int_{x^{\prime}}\phi_{\text{\tiny
IR}}^{\text{q}}(x)\left[\Upsilon(x,x^{\prime})+\frac{\delta(t-t^{\prime})}{a^{3}(t)}\left(W_{x^{\prime}}G^{-1}_{x^{\prime}}+2\mathcal{Q}_{x^{\prime}}W_{x^{\prime}}+2W_{x^{\prime}}\dot{W}_{x^{\prime}}\partial_{t^{\prime}}\right)\right]\phi_{\text{\tiny
IR}}^{\text{c}}(x^{\prime}),$

where we have defined the quantity

$\displaystyle\Upsilon(x,x^{\prime})\equiv\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}{\mathcal{Q}}_{t}{\mathcal{Q}}_{t^{\prime}}G_{R}(k,t,t^{\prime})e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})}.$
(44)

The last three terms in (43) include a window function $W$ acting on an IR-
field. As pointed out in Morikawa1990 , if $W(k,t)$ is a projection operator
(so that $W^{2}\phi=W\phi$, $\overline{W}W\phi=0$), terms for which the IR
field $\phi_{\text{\tiny IR}}=\overline{W}\phi$ is directly convoluted with
$W$ vanish.

Requiring window functions to be projections is a rather strict constraint,
since it requires $W$ to take on only the values 0 and 1, and hence be a
(sequence of) discontinuous step functions. A smooth window function is not a
projection, but one may still expect that terms of the form $W\phi_{\rm IR}$
are suppressed also for e.g. a slightly smoothed-out step function. Discarding
the last three terms of (2.28) then implies that the window function is
assumed to be ”sufficiently step-like”. In the following we will neglect these
terms, but keep in mind this requirement on the window function. Discarding of
the final off-diagonal term $\Upsilon$, connecting $\phi_{\text{\tiny
IR}}^{\text{c}}$ and $\phi_{\text{\tiny IR}}^{\text{q}}$, will be justified
below by an explicit computation, but it is kept for now.

When the simplifications discussed above can be made, the influence functional
can be written as

$\displaystyle\mathcal{F}_{0}[\Phi_{\text{\tiny
IR}}]=N\mathrm{exp}\bigg{[}i\int_{x}\int_{x^{\prime}}\bigg{(}\frac{i}{2}\phi_{\text{\tiny
IR}}^{\text{q}}(x)\real\Pi(x,x^{\prime})\phi_{\text{\tiny
IR}}^{\text{q}}(x^{\prime})-2\Theta(t-t^{\prime})\phi_{\text{\tiny
IR}}^{\text{q}}(x)\imaginary\Pi(x,x^{\prime})\phi_{\text{\tiny
IR}}^{\text{c}}(x^{\prime})\bigg{)}\bigg{]},$ (45)

with

$\displaystyle\Pi(x,x^{\prime})=\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}{\mathcal{Q}}_{t}{\mathcal{Q}}_{t^{\prime}}\phi(k,t)\phi^{*}(k,t^{\prime})e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})},$
(46)

having used that (for $t>t^{\prime}$)

$\displaystyle
G_{F}(k,t,t^{\prime})=\real\phi(k,t)\phi^{*}(k,t^{\prime}),\qquad
G_{R}(k,t,t^{\prime})=2\Theta(t-t^{\prime})\imaginary\phi(k,t)\phi^{*}(k,t^{\prime}).$
(47)

In order to compute these quantities explicitly, one is required to choose a
concrete representation for the UV field mode functions $\phi(k,t)$ as well as
a window function $W(k,t)$. To this end we will be using the mode solutions
defined in section 2.3, at different levels of approximations.

###  2.5 Stochastic IR theory

Following a well-known procedure for turning our effective theory into a
stochastic one Stratonovich ; Hubbard , the q-q component of (45) is rendered
a real-valued quantity by introducing an auxiliary field $\xi(x)$ via a
Hubbard-Stratonovich transformation,

$\displaystyle e^{-\frac{1}{2}\int_{x}\mathcal{M}\phi_{\text{\tiny
IR}}^{2}}=\int\mathcal{D}\xi
e^{-\frac{1}{2}\int_{x}\mathcal{M}^{-1}\xi^{2}+i\int_{x}\xi\phi_{\text{\tiny
IR}}},$ (48)

for which we obtain

$\displaystyle\mathcal{F}_{0}[\Phi_{\text{\tiny
IR}}]=N\int\mathcal{D}\xi\,\mathrm{exp}\bigg{[}\int_{x}\int_{x^{\prime}}$
$\displaystyle\Big{(}-\tfrac{1}{2}\xi(x)\real\Pi(x,x^{\prime})^{-1}\xi(x^{\prime})+i\xi(x)\phi_{\text{\tiny
IR}}^{\text{q}}(x^{\prime})$ (49)
$\displaystyle\qquad-i2\Theta(t-t^{\prime})\phi_{\text{\tiny
IR}}^{\text{q}}(x)\imaginary\Pi(x,x^{\prime})\phi_{\text{\tiny
IR}}^{\text{c}}(x^{\prime})\Big{)}\bigg{]}.$

Here the variables $\xi$ are defined to be Gaussian, with

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle=\int\mathcal{D}\xi\mathcal{P}[\xi]\xi(x)\xi(x^{\prime})=\real\Pi(x,x^{\prime}).$
(50)

In this relation $\langle\cdot\rangle$ denotes the ensemble average and the
right-hand-side is evaluated as a quantum expectation value.

An equivalent, but perhaps more familiar form of the stochastic noise
correlator (50) can be obtained by rewriting (46) into555Using the metric
determinant to write
$-a^{3}(t){\mathcal{Q}}_{t}\phi(k,t)=\partial_{t}[a^{3}(t)\dot{W}(k,t)\phi(k,t)]+a^{3}(t)\dot{W}(k,t)\dot{\phi}(k,t)$
and partial integrating the first term in the action.

$\displaystyle\real\int_{x}\int_{x^{\prime}}\phi_{\text{\tiny
IR}}^{\text{q}}(x)\Pi(x,x^{\prime})\phi_{\text{\tiny
IR}}^{\text{q}}(x^{\prime})=\real\int_{x}\int_{x^{\prime}}\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}\left(-\dot{\phi}^{\text{q}}_{{\text{\tiny
IR}}}(t)\phi(t)+\phi^{\text{q}}_{{\text{\tiny
IR}}}(t)\dot{\phi}(t)\right)\dot{W}_{t}$ (51)
$\displaystyle\times\dot{W}_{t^{\prime}}\Big{(}-\dot{\phi}^{\text{q}}_{{\text{\tiny
IR}}}(t^{\prime})\phi^{*}(t^{\prime})+\phi^{\text{q}}_{{\text{\tiny
IR}}}(t^{\prime})\dot{\phi}^{*}(t^{\prime})\Big{)}e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})}$
,

and introducing instead two auxiliary fields $\xi_{\phi}$ and $\xi_{\pi}$.
Writing $\pi=\dot{\phi}$, the generating functional becomes

$\displaystyle\mathcal{F}_{0}[\Phi_{\text{\tiny IR}}]$
$\displaystyle=N\int\mathcal{D}\xi_{\phi}\mathcal{D}\xi_{\pi}\,\mathrm{exp}\bigg{[}\int_{x}\int_{x^{\prime}}\bigg{(}-\tfrac{1}{2}\left[\,\xi_{\phi}(x),\xi_{\pi}(x)\,\right]\mathcal{M}^{-1}(x,x^{\prime})\bigg{[}\,\begin{matrix}\xi_{\phi}(x^{\prime})\\\
\xi_{\pi}(x^{\prime})\end{matrix}\,\bigg{]}$ (52)
$\displaystyle\quad+i\left[\,-\pi_{\text{\tiny
IR}}^{\text{q}}(x),{\phi}_{\text{\tiny
IR}}^{\text{q}}(x)\,\right]\bigg{[}\,\begin{matrix}\xi_{\phi}(x^{\prime})\\\
\xi_{\pi}(x^{\prime})\end{matrix}\,\bigg{]}-i2\Theta(t-t^{\prime})\phi_{\text{\tiny
IR}}^{\text{q}}(x)\imaginary\Pi(x,x^{\prime})\phi_{\text{\tiny
IR}}^{\text{c}}(x^{\prime})\bigg{)}\bigg{]},$

where

$\displaystyle\mathcal{M}(x,x^{\prime})=\real\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})}\dot{W}_{t}\begin{bmatrix}\phi(k,t)\phi^{*}(k,t^{\prime})&\phi(k,t)\pi^{*}(k,t^{\prime})\\\
\pi(k,t)\phi^{*}(k,t^{\prime})&\pi(k,t)\pi^{*}(k,t^{\prime})\end{bmatrix}\dot{W}_{t^{\prime}}.$
(53)

Schematically, we may then write for the stochastic noise

$\displaystyle\xi_{\phi}(x)$
$\displaystyle=\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{\frac{3}{2}}}\dot{W}_{t}\big{[}a_{\bf
k}\phi(k,t)e^{i\mathbf{k}\cdot\mathbf{x}}+\textrm{h.c.}\big{]},$ (54)
$\displaystyle\xi_{\pi}(x)$
$\displaystyle=\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{\frac{3}{2}}}\dot{W}_{t}\big{[}a_{\bf
k}\pi(k,t)e^{i\mathbf{k}\cdot\mathbf{x}}+\textrm{h.c.}\big{]},$

keeping in mind that $\xi(x)$ is a number and the right-hand side is an
operator, and the correspondence is at the level of expectation values.

#### The stochastic equation

Combining the expressions (37) and (49), the influence functional (36) now
reads

$\displaystyle Z[0]=\int\mathcal{D}\Phi_{\text{\tiny
IR}}e^{iS[\Phi_{\text{\tiny IR}}]}\mathcal{F}_{0}[\Phi_{\text{\tiny
IR}}]=\int\mathcal{D}\Phi_{\text{\tiny IR}}\mathcal{D}\xi e^{iS_{\rm
eff}}[\Phi_{\text{\tiny IR}},\xi],$ (55)

with

$\displaystyle S[\Phi_{\text{\tiny
IR}}]=\int_{x}\left(\tfrac{1}{2}\phi_{\text{\tiny
IR}}^{\text{c}}G^{-1}\phi_{\text{\tiny
IR}}^{\text{q}}-V_{\text{I}}[\phi_{\text{\tiny
IR}}^{\text{c}},\phi_{\text{\tiny IR}}^{\text{q}}]\right).$ (56)

By variation of the effective action, we finally arrive at the stochastic
equation of motion for the IR field, which becomes666We note that the
potential term
$V_{\text{I}}[\Phi]=V_{\text{I}}[\phi^{+}]-V_{\text{I}}[\phi^{-}]$ in the
Keldysh basis has the property that $\differential
V_{\text{I}}[\Phi]/\differential\phi^{\text{q}}|_{\phi^{\text{q}}=0}=V_{\text{I}}^{\prime}[\phi^{\text{c}}]$.

$\displaystyle 0=\frac{\delta S_{\rm eff}}{\delta\phi_{\text{\tiny
IR}}^{\text{q}}(x)}\bigg{|}_{\phi_{\text{\tiny IR}}^{\text{q}}=0}$
$\displaystyle=\ddot{\phi}_{\text{\tiny
IR}}^{\text{c}}(x)+3H\dot{\phi}_{\text{\tiny
IR}}^{\text{c}}(x)+\frac{\nabla^{2}_{\mathbf{x}}}{a^{2}(t)}\phi_{\text{\tiny
IR}}^{\text{c}}(x)+V^{\prime}(\phi_{\text{\tiny IR}}^{\text{c}}(x))-\xi(x)$
(57)
$\displaystyle\qquad+\int_{x^{\prime}}2\Theta(t-t^{\prime})\imaginary\Pi(x,x^{\prime})\phi_{\text{\tiny
IR}}^{\text{c}}(x^{\prime}).$

One may note that if the IR field is sufficiently super-horizon, the gradient
term is negligible, since by construction

$\displaystyle\frac{k^{2}}{a^{2}(t)H^{2}}\phi_{\text{\tiny
IR}}^{\text{c}}\ll\phi_{\text{\tiny IR}}^{\text{c}}.$ (58)

This must then be compared to the remaining terms. To the extent that
$\Upsilon(x,x^{\prime})=2\Theta(t-t^{\prime})\imaginary\Pi(x,x^{\prime})$ may
be neglected, we recover the stochastic equation advertised in (14);

$\displaystyle\ddot{\phi}_{\text{\tiny IR}}+3H\dot{\phi}_{\text{\tiny
IR}}+V^{\prime}(\phi_{\text{\tiny IR}})=\xi.$ (59)

One may equivalently vary instead (52), to obtain the identification

$\displaystyle 3H\xi_{\phi}(x)+\dot{\xi}_{\phi}(x)+\xi_{\pi}(x)\equiv\xi(x),$
(60)

as discussed in section 1.1. In the following sections we will show explicitly
that in certain limits, $3H\xi_{\phi}$ is indeed the dominant contribution to
the noise. For the moment, we will simply note that for $k\ll aH$ the mode
derivative is $\dot{\phi}(k,t)\simeq\epsilon_{M}H\phi(k,t)$, as can be seen
from e.g. (54) and (81). It follows that for small $\epsilon_{M}$, the
remaining correlators in (53) are $\mathcal{O}(\epsilon_{M})$.

Under the above assumptions, and additionally in the slow-roll regime where
$\ddot{\phi}\ll 3H\dot{\phi}$, the original stochastic inflation form (1) is
recovered,

$\displaystyle\dot{\phi}_{\text{\tiny
IR}}(t)+\frac{V^{\prime}(\phi_{\text{\tiny IR}}(t))}{3H}=\xi_{\phi}(t).$ (61)

Let us summarise the procedure so far. Before applying standard slow-roll and
large wavelength assumptions common to inflationary dynamics, we have ignored
UV-UV and UV-IR self-interactions, made use of $W$ being close to a projection
onto UV modes, neglected $\Upsilon$ and neglected noise contributions other
than $\xi_{\phi}$. The IR fields have also implicitly been assumed classical,
so that the variation of the path integral with respect to $\phi_{\text{\tiny
IR}}^{\text{q}}$ gives the complete dynamics. Only then do we recover the
standard form of the stochastic equation. A number of these assumptions will
be checked explicitly below.

Before we proceed, we emphasise that expressing the path integral in terms of
a stochastic process does not imply any physical reality to each individual
stochastic trajectory. Only the ensemble average over initial conditions and
realisations of the noise have meaning, and so conceptually, the corresponding
Fokker-Planck equation and distribution is perhaps the preferred object to
work with.

###  2.6 An example: the step window function

The simplest example of a window function is the step function

$\displaystyle W(k,t)={\Theta(k/aH-\mu)},$ (62)

which projects out modes with $k<\mu aH$, and was the original choice made in
Starobinsky:1986fx ; Starobinsky:1994bd . It is a true projection, and so some
of the simplifications discussed above go through straightforwardly. The
dimensionless parameter $\mu<1$ is taken to be small but non-vanishing, and
the coarse-graining scale is then $(\mu aH)^{-1}$.

The noise correlations can be computed from (53), where from (35) we obtain to
first order in mass and slow-roll parameters,

$\displaystyle\begin{bmatrix}\langle\xi_{\phi}(x)\xi_{\phi}(x^{\prime})\rangle\\\
\langle\xi_{\phi}(x)\xi_{\pi}(x^{\prime})\rangle\\\
\langle\xi_{\pi}(x)\xi_{\pi}(x^{\prime})\rangle\end{bmatrix}$
$\displaystyle=\frac{H^{3}}{4\pi^{2}}(1-\epsilon_{H})^{3}\frac{\sin\mu
aHr}{\mu
aHr}\delta(t-t^{\prime})\bigg{(}\frac{\mu}{1-\epsilon_{H}}\bigg{)}^{2\epsilon_{M}-2\epsilon_{H}}$
(63) $\displaystyle\qquad\times\begin{bmatrix}1+\mu^{2}(1+2\epsilon_{M})\\\
\left(-\epsilon_{M}+\mu^{2}(1+\tfrac{3}{2}\epsilon_{M}-\epsilon_{H})\right)H\\\
\left(\epsilon_{M}^{2}+2\mu^{2}\epsilon_{M}\right)H^{2}\end{bmatrix}+\mathcal{O}(\mu^{4}).$

In the limit $\epsilon_{M},\epsilon_{H}\rightarrow 0$ the first noise
correlator in (63) reduces to the original result Starobinsky:1994bd  to
leading order in $\mu$. It remains to take the limits $\mu\ll 1$ and $r<aH$ to
recover the correlation,

$\displaystyle\langle\xi_{\phi}(x)\xi_{\phi}(x^{\prime})\rangle=\frac{H^{3}}{4\pi^{2}}\delta(t-t^{\prime}).$
(64)

This result is elegant in its simplicity, but as we have seen, it relies on a
number of approximations and assumptions.

The expression (63) is our first encounter of the correction function
$f(\epsilon_{M},\epsilon_{H},\mu,...)$ advertised in section 1.1. We see that
the limits $\epsilon_{M},\epsilon_{H}\rightarrow 0$ do not commute with
$\mu\rightarrow 0$. Indeed, as soon at $\mu$ is taken to be small, the
standard noise normalisation (64) only follows in the strict massless and dS
limit. From (63) it also becomes apparent that the contribution of
$\xi_{\phi}$ dominates for small $\mu$.

It was pointed out in Winitzki:1999ve  that some window functions fail to
reproduce the correct long-distance behaviour of the full (non-coarse grained)
correlators, and that some result in coloured noise Casini:1998wr ;
Winitzki:1999ve .777For instance, the resulting correlations
$\langle\dot{\phi}_{\text{\tiny IR}}(x)\dot{\phi}_{\text{\tiny
IR}}(x^{\prime})\rangle$ from the Langevin equation should at least match the
behaviour of $\langle\dot{\phi}(x)\dot{\phi}(x^{\prime})\rangle\sim r^{-4}$ at
large distances. This can be achieved by demanding sufficient smoothness of
the window function, such that the noise correlation
$\displaystyle\langle\xi_{\phi}(x)\xi_{\phi}(x^{\prime})\rangle\sim\int\differential
kh(k)\sin
kr=\frac{h(0)}{r}-\frac{h^{\prime\prime}(0)}{r^{3}}+\frac{h^{\prime\prime\prime}(0)}{r^{5}}-\dotso,$
(65) has at least $h^{\prime\prime\prime}(0)$ finite. The authors of
Winitzki:1999ve  provided a sufficient (but not necessary or exhaustive)
condition for a window function to satisfy this. The step function is one such
example, and in the following, we will consider smooth window functions.

##  3 Momentum decomposition of the propagator

An alternative method for the scale separation between IR and UV modes was
applied in Moss:2016uix , where the propagator rather than the field itself is
subject to a window function in momentum space. The propagator is decomposed
into two constituents that are dominant in the IR and UV respectively:

$\displaystyle\mathbbm{G}(x,x^{\prime})=\mathbbm{G}_{\text{\tiny
IR}}(x,x^{\prime})+\mathbbm{G}_{\text{\tiny UV}}(x,x^{\prime}),$ (66)

where the IR propagator is weighted according to

$\displaystyle\mathbbm{G}_{\text{\tiny
IR}}(x,x^{\prime})\equiv\int_{y}\int_{y^{\prime}}\overline{W}(x,y)\mathbbm{G}(y,y^{\prime})\overline{W}(y^{\prime},x^{\prime}),$
(67)

and $\overline{W}$ is as earlier used to project the field onto long
wavelengths. The UV propagator is constructed from (66) as (c.f. (39))

$\displaystyle\mathbbm{G}_{\text{\tiny
UV}}(x,x^{\prime})=\int_{y^{\prime}}\int_{y}\left[\mathbbm{G}(x,y)W(y,x^{\prime})+W(x,y)\mathbbm{G}(y,x^{\prime})-W(x,y)\mathbbm{G}(y,y^{\prime})W(y^{\prime},x^{\prime})\right].$
(68)

An additional step is to make the formal rewriting of the field into two
constituents,

$\displaystyle\Phi(x)=\Phi_{1}(x)+\Phi_{2}(x).$ (69)

Unlike when decomposing at the level of the fields, these field constituents
are not under any constraints: they are simply two new field variables defined
over all of momentum space. The decomposition (69) is used as a tool together
with the propagator constituents (66) to make use of a Gaussian identity888
For some function $f(x)$ depending only on the combination $x=y+z$ the
relation  $\displaystyle\sqrt{\frac{|a+b|}{2\pi ab}}\int\differential
y\int\differential
z\,\mathrm{exp}\left[-\left(\frac{y^{2}}{2a}+\frac{z^{2}}{2b}+f(y+z)\right)\right]=\int\differential
x\,\mathrm{exp}\left[-\left(\frac{x^{2}}{2(a+b)}+f(x)\right)\right],$ (70) can
be verified e.g. by a change of variables $x=y+z$ and integration over $u=by-
az$.  in order to rewrite the generating functional (17) as

$\displaystyle
Z[\mathbbm{J}]=\int\mathcal{D}\Phi_{1}\mathcal{D}\Phi_{2}\,\mathrm{exp}\left[\frac{i}{2}\int_{x}\left(\Phi_{1}^{T}\mathbbm{G}^{-1}_{\text{\tiny
IR}}\Phi_{1}+\Phi_{2}^{T}\mathbbm{G}^{-1}_{\text{\tiny
UV}}\Phi_{2}-2V_{\text{I}}(\Phi_{1}+\Phi_{2})+\mathbbm{J}^{T}(\Phi_{1}+\Phi_{2})\right)\right].$
(71)

We have in mind a smooth window function $W$($\overline{W}$) that suppresses
the IR(UV). From the construction of the propagators, the UV components of the
$\Phi_{1}$ field will have suppressed contributions to the path integral, and
analogously the IR components of the $\Phi_{2}$ field. Note that when
decomposing at the level of the propagator, we no longer have bilinear kinetic
terms mixing the UV and IR physics in the generating functional. To obtain an
expression for the IR kinetic operator in (71), we write (formally,
suppressing integration labels)

$\displaystyle\mathbbm{G}_{\text{\tiny
IR}}^{-1}=\frac{1}{1-\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}}\mathbbm{G}^{-1}.$ (72)

For the large-wavelength dynamics governed by $\mathbbm{G}_{\text{\tiny
IR}}^{-1}$, the propagator $\mathbbm{G}_{\text{\tiny UV}}$ is by construction
acting on a subspace of the field configuration in which it is suppressed,
hence allowing the expansion

$\displaystyle\mathbbm{G}_{\text{\tiny IR}}^{-1}$
$\displaystyle=\mathbbm{G}^{-1}+\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}+\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny UV}}\mathbbm{G}^{-1}+\dotso.$
(73)

To clarify, the ”small quantity” in the expansion is the degree to which the
”mostly-UV” propagator $\mathbbm{G}_{\text{\tiny UV}}$ is small when acting on
the ”mostly-IR” field $\Phi_{1}$.999For a strict step function such as (62),
this procedure is ill-defined, since the denominator of (72) vanishes for UV
modes. For a given window function, it may be prudent to confirm convergence a
posteriori, and we will do this below.

The first term in (73) replaces $G_{\text{\tiny IR}}^{-1}$ by $G^{-1}$ in the
quadratic part of the action (71), such that

$\displaystyle Z[0]$
$\displaystyle=\int\mathcal{D}\Phi_{1}\,\mathrm{exp}\left[\frac{i}{2}\int_{x}\left(\Phi_{1}^{T}\mathbbm{G}^{-1}\Phi_{1}-2V_{\text{I}}(\Phi_{1})\right)\right]$
(74)
$\displaystyle\quad\times\,\mathrm{exp}\left[\frac{i}{2}\int_{x}\Phi_{1}^{T}\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}\Phi_{1}+\Phi_{1}^{T}\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}\Phi_{1}+\dotso\right]$
$\displaystyle\quad\times\int\mathcal{D}\Phi_{2}\,\mathrm{exp}\left[\frac{i}{2}\int_{x}\left(\Phi_{2}^{T}\mathbbm{G}^{-1}_{\text{\tiny
UV}}\Phi_{2}-2V_{\text{I}}(\Phi_{1},\Phi_{2})\right)\right].$

where similarly to (37), we have defined
$V_{\text{I}}(\Phi_{1},\Phi_{2})\equiv
V_{\text{I}}(\Phi_{1}+\Phi_{2})-V_{\text{I}}(\Phi_{1})$.

By the same logic as when the UV-IR and UV-UV self-interactions were neglected
in the previous section, we will treat the $\Phi_{2}$ field as non-
interacting, in which case the integral over $\Phi_{2}$ is Gaussian and only
provides a multiplicative constant. The second line will now play the role of
the influence functional, while the first line is the free action
$S[\Phi_{1}]$.

Our new influence functional has as its leading term (corresponding to the
next-to-leading order (NLO) in the inverse propagator expansion (73))

$\displaystyle\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}=W\mathbbm{G}^{-1}+\mathbbm{G}^{-1}W-\mathbbm{G}^{-1}W\mathbbm{G}W\mathbbm{G}^{-1},$
(75)

which includes the integrand in (38). The derivation of the Langevin equation
then goes through in the same manner as previously described. The additional
terms $\mathbbm{G}^{-1}W$ and $W\mathbbm{G}^{-1}$ become directly convoluted
with $\Phi_{1}$. By construction, $\Phi_{1}$ has a suppressed short-wavelength
component, and similarly to the discussion above, we expect the contribution
of these terms to be negligible only for a sharp enough window function.

Unlike in section 2.4, the inverse propagator expansion in (73) gives further
contributions at higher order. From (76) one obtains at next-to-next-to-
leading order (NNLO)

$\displaystyle\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny UV}}\mathbbm{G}^{-1}$
$\displaystyle=\mathbbm{G}^{-1}{W}^{2}+{W}^{2}\mathbbm{G}^{-1}+{W}\mathbbm{G}^{-1}{W}-\mathbbm{G}^{-1}{W}^{2}\mathbbm{G}{W}\mathbbm{D}^{-1}$
(76)
$\displaystyle\quad-\mathbbm{G}^{-1}{W}\mathbbm{G}{W}^{2}\mathbbm{G}^{-1}-{W}\mathbbm{G}^{-1}{W}\mathbbm{G}{W}\mathbbm{G}^{-1}-\mathbbm{G}^{-1}{W}\mathbbm{G}{W}\mathbbm{G}^{-1}{W}$
$\displaystyle\quad+\mathbbm{G}^{-1}{W}\mathbbm{G}{W}\mathbbm{G}^{-1}+\mathbbm{G}^{-1}{W}\mathbbm{G}{W}\mathbbm{G}^{-1}{W}\mathbbm{G}{W}\mathbbm{G}^{-1},$

where we note the reappearance of the NLO term on the last line with the
opposite sign. It is not manifest that the above is subleading compared to
(75), and we shall return to this point in the next section when considering a
concrete example.

###  3.1 An example: the Gaussian window function

A common example of a smooth window function is the Gaussian,

$\displaystyle W(k,t)=1-e^{-\frac{1}{2}\left(\frac{k}{\sigma
a_{t}H}\right)^{2}}.$ (77)

The Gaussian window function fulfills the criteria set out in Ref.
Winitzki:1999ve , and is parameterised by a dimensionless parameter $\sigma$,
controlling both the location and steepness of the transition from IR to UV.
This is in contrast to a smoothed step function (which we will consider
below), where the location is fixed (by the parameter $\mu$ in (62)) and a
second parameter that controls the steepness. In other words, the step
function is not a limit of the Gaussian, and as $\sigma\rightarrow 0$, the
cutoff becomes steep, but also moves to $k\rightarrow 0$.

The noise correlation (50) can be simplified slightly using that the time
derivative of the full mode solution can be factorised as

$\displaystyle\dot{\phi}(k,t)=q_{\nu}(k,t)H\phi(k,t),$ (78)

such that

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle=\real\int\frac{k\differential
k}{2\pi^{2}r}\sin
kr\tilde{\mathcal{Q}}(k,t)\tilde{\mathcal{Q}}^{*}(k,t^{\prime})\phi(k,t)\phi^{*}(k,t^{\prime}).$
(79)

Here we have defined

$\displaystyle\tilde{\mathcal{Q}}(k,t)\equiv-\ddot{W}(k,t)-[3+2q_{\nu}(k,t)]H\dot{W}(k,t).$
(80)

where $q_{\nu}$ may be expressed as

$\displaystyle
q_{\nu}(k,t)=-\epsilon_{M}-\frac{k}{a_{t}H}\frac{H_{\nu-1}^{(1)}\big{(}\frac{k}{a_{t}H(1-\epsilon_{H})}\big{)}}{H_{\nu}^{(1)}\big{(}\frac{k}{a_{t}H(1-\epsilon_{H})}\big{)}}\;.$
(81)

to first order in $\epsilon_{M},\epsilon_{H}$. We will proceed with
$\epsilon_{H}=0$ and generalise to $\epsilon_{H}\neq 0$ later.

To begin with, using the approximate mode solution (35) for which
$q_{\nu}(k,t)\simeq-\epsilon_{M}$, the noise correlator is found to be

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle$
$\displaystyle=\frac{H^{6}}{8\pi^{2}}\sigma^{2\epsilon_{M}}(\sech
H\tau)^{2+\epsilon_{M}}\Big{[}\sech^{2}H\tau\Gamma[4+\epsilon_{M}]{}_{1}F_{1}\Big{(}4+\epsilon_{M};\frac{3}{2};-\frac{r^{2}}{4\alpha}\Big{)}$
(82)
$\displaystyle\qquad+2(1-2\epsilon_{M})\Gamma[3+\epsilon_{M}]{}_{1}F_{1}\Big{(}3+\epsilon_{M};\frac{3}{2};-\frac{r^{2}}{4\alpha}\Big{)}$
$\displaystyle\qquad+(1-2\epsilon_{M})^{2}\Gamma[2+\epsilon_{M}]{}_{1}F_{1}\Big{(}2+\epsilon_{M};\frac{3}{2};-\frac{r^{2}}{4\alpha}\Big{)}\Big{]},$

where $\tau\equiv t-t^{\prime}$ denotes the time separation,
$r\equiv|\mathbf{x}-\mathbf{x^{\prime}}|$ the spatial separation,
${}_{1}F_{1}(a,b,z)$ is the confluent hypergeometric function grad , and

$\displaystyle\alpha\equiv\frac{1}{2\sigma^{2}H^{2}}\Big{(}\frac{1}{a_{t}^{2}}+\frac{1}{a_{t^{\prime}}^{2}}\Big{)}=\frac{R^{2}}{\sigma^{2}\sech
H\tau}\qquad\text{with}\qquad R^{2}\equiv\frac{1}{a(t)a(t^{\prime})H^{2}}\;.$
(83)

$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$5.5$$6$$6.5$$7$$7.5$$8$$0$$2$$4$$H\tau$$\frac{4\pi^{2}}{H^{6}}\langle\xi(x)\xi(x^{\prime})\rangle$$\sigma=1.0$$\sigma=0.5$$\sigma=0.3$

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$2$$4$$r/R_{H}$$\frac{4\pi^{2}}{H^{6}}\langle\xi(x)\xi(x^{\prime})\rangle$$\sigma=1.0$$\sigma=0.5$$\sigma=0.3$

Figure 1: Noise correlation (82) with $\epsilon_{M}=0.001$ as a function of
spatial separation $r/R_{H}$ with Hubble scale $R_{H}\equiv R(t=t^{\prime})$
(left panel) and as a function of time separation $\tau\equiv t-t^{\prime}$
for fixed spatial separation $r/R=1$ (right panel).

Because a number of similar expressions will appear later, it is worth pausing
at this point. For equal time $t=t^{\prime}$, we see that $R$ is the Hubble
scale in time, and the spatial separation $r$ then enters the correlation in
units of the Gaussian width $(R/\sigma)^{2}$. Away from the equal-time limit,
$r^{2}/4\alpha$ grows exponentially with $\tau$, resulting in the whole
expression decaying exponentially (see left panel of Fig. 1). For large
spatial separations $r\gg R$, the noise correlation decays exponentially (see
right panel of Fig. 1), signifying that the noise evolution can be considered
local for patches of length scales $r>\sigma R$. It becomes apparent that for
smaller $\sigma$, the decay is slower. On the other hand, since $R$ decreases
very fast in time, the localisation in space is not very sensitive to the
choice of $\sigma$.

In addition, due to the overall factor of $\sigma^{2\epsilon_{M}}$, the limit
$\sigma\rightarrow 0$ does not commute with the massless limit. This factor
can be replaced by 1 if we choose $\sigma$ in such a way that
$\ln\sigma\ll\epsilon_{M}^{-1}/2$, but one may in practise tune $\sigma$ and
$\epsilon_{M}$ to give any noise amplitude.

To get a better understanding of the contributions to the noise correlation
(82), one may consider the decomposed noise terms $\xi_{\phi}$ and $\xi_{\pi}$
of (60), where schematically

$\displaystyle\dot{\xi}_{\phi}(x)=\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{\frac{3}{2}}}\big{(}\ddot{W}_{t}+q_{\nu}(k,t)\dot{W}_{t}\big{)}\big{[}a_{\mathbf{k}}\phi(k,t)e^{i\mathbf{k}\cdot\mathbf{x}}+\text{h.c.}\big{]}.$
(84)

In particular, we obtain

$\displaystyle\langle\xi_{\phi}(x)\xi_{\phi}(x^{\prime})\rangle$
$\displaystyle=\frac{H^{4}}{8\pi^{2}}\sigma^{2\epsilon_{M}}\Gamma[2+\epsilon_{M}](\sech
H\tau)^{2+\epsilon_{M}}{}_{1}F_{1}\Big{(}2+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}$
(85)
$\displaystyle\simeq\frac{H^{4}}{8\pi^{2}}\sech^{2}H\tau+\mathcal{O}(\epsilon_{M}),$

where the last equality applies for $r=0$. The long-wavelength approximate
result (85) reproduces the result at leading order in $\sigma$ obtained using
the exact massless mode solution (34) in earlier works Winitzki:1999ve ;
Matarrese:2003ye .

For the correlation of (84) one finds

$\displaystyle\langle\dot{\xi}_{\phi}(x)\dot{\xi}_{\phi}(x^{\prime})\rangle$
$\displaystyle=\frac{H^{6}}{8\pi^{2}}\sigma^{2\epsilon_{M}}(\sech
H\tau)^{2+\epsilon_{M}}\Big{[}\Gamma[4+\epsilon_{M}]\sech^{2}H\tau{}_{1}F_{1}\Big{(}4+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}$
(86)
$\displaystyle\quad-4\Gamma[3+\epsilon]{}_{1}F_{1}\Big{(}3+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}+4\Gamma[2+\epsilon]{}_{1}F_{1}\Big{(}2+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}\Big{]}$
$\displaystyle\simeq\frac{H^{6}}{8\pi^{2}}\sech^{2}H\tau(6\sech^{2}H\tau-4)+\mathcal{O}(\epsilon_{M}).$

with the last equality again in the limit $r\rightarrow 0$. For the cross
terms, we find

$\displaystyle\langle\dot{\xi}_{\phi}(x){\xi}_{\phi}(x^{\prime})\rangle+\langle{\xi}_{\phi}(x)\dot{\xi}_{\phi}(x^{\prime})\rangle$
$\displaystyle=\frac{H^{5}}{4\pi^{2}}\sigma^{2\epsilon_{M}}(\sech
H\tau)^{2+\epsilon_{M}}\Big{[}\Gamma[3+\epsilon_{M}]{}_{1}F_{1}\Big{(}3+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}$
(87)
$\displaystyle\qquad-2\Gamma[2+\epsilon]{}_{1}F_{1}\Big{(}2+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}\Big{]}$
$\displaystyle\simeq 0+\mathcal{O}(\epsilon_{M}).$

Correlations involving $\xi_{\pi}\propto q_{\nu}\simeq-\epsilon_{M}$ are
similarly suppressed. In Fig. 2, we show all the contributions to the noise
correlators at $r=0$ as a function of time separation $H\tau$. We see that
$\xi_{\phi}$ indeed dominates, while $\dot{\xi}_{\phi}$ has both positive and
negative contributions.

$-4$$-2$$0$$2$$4$$1$$3$$5$$H\tau$$\frac{4\pi^{2}}{H^{6}}$$9H^{2}\langle\xi_{\phi}\xi_{\phi}\rangle$$\langle\dot{\xi}_{\phi}\dot{\xi}_{\phi}\rangle$$\langle\xi_{\pi}\xi_{\pi}\rangle$$3H\sum\langle\xi_{\phi}\dot{\xi}_{\phi}\rangle$$3H\sum\langle\xi_{\phi}\xi_{\pi}\rangle$$\sum\langle\dot{\xi}_{\phi}\xi_{\pi}\rangle$
Figure 2: Decomposed noise correlations at coincident spatial points for the
Gaussian window with $\sigma=0.1$ and $\epsilon_{M}=0.005$. Here we denote
$\sum\langle\xi_{\phi}\dot{\xi}_{\phi}\rangle\equiv\langle\xi_{\phi}(x)\dot{\xi}_{\phi}(x^{\prime})\rangle+\langle\dot{\xi}_{\phi}(x)\xi_{\phi}(x^{\prime})\rangle$,
etc. The combined noise $\langle\xi(x)\xi(x^{\prime})\rangle$ of (82) is
displayed in black for comparison.

The noise correlator is localised around $H\tau=0$, but has more features than
simply a delta-function. Nevertheless, a final approximation to the delta
function case can be made, even for finite $\sigma$, through the replacement
(for this example, in the massless limit $\epsilon_{M}=0$)

$\displaystyle\sech^{2}H\tau\rightarrow
2\delta(H\tau)=\frac{2}{H}\delta(t-t^{\prime}),$ (88)

where the factor of 2 is chosen so that

$\displaystyle\int\differential(H\tau)\sech^{2}H\tau=\frac{2}{H}\int\differential\tau\delta(t-t^{\prime}).$
(89)

In other words, we replace a moderately localised noise distribution by a
fully localised one with the same integrated power, i.e., such that

$\displaystyle\langle\xi_{\phi}(x)\xi_{\phi}(x^{\prime})\rangle\simeq\frac{H^{4}}{8\pi^{2}}\sech^{2}H\tau\simeq\frac{H^{3}}{4\pi^{2}}\delta(t-t^{\prime}).$
(90)

With this procedure, it can be seen that the integral of the
$\langle\dot{\xi}_{\phi}\dot{\xi}_{\phi}\rangle$ correlator vanishes
identically to leading order, and so non-local time correlations (positive and
negative) are ignored in this procedure. Only the
$\langle\xi_{\phi}\xi_{\phi}\rangle$ contribution remains.

In section 4, we will further investigate how the noise distribution depends
on the choice of window function and its parameters, as well as the choice of
approximation for the UV mode functions (33)–(35). The dependence on the mass
parameter $\epsilon_{M}$ and the generalisation beyond leading order in slow-
roll, $\epsilon_{H}\neq 0$, will also be considered. Ultimately, we will
compute the amplitude of the localised noise, making use of the prescription
(88). The prefactor of the delta-function will be referred to as the noise
amplitude.

Before doing so, however, we will use the Gaussian window function to
explicitly check some of the assumptions made with the derivations in the
previous section. The conclusions will hold also for the other window
functions considered below.

###  3.2 Off-diagonal (q-c) contributions to the IR field evolution

In our derivation of the stochastic evolution equation for the IR modes (57),
we made the claim that the off-diagonal c-q contributions, including the term
$\Upsilon$, can be neglected. In the following we will check this explicitly
for the terms that do not contain any operators acting directly on the IR
field $\phi_{\text{\tiny IR}}$.

With the approximate solution (35) for the dS UV mode functions, we first find
for the retarded propagator

$\displaystyle
G_{R}(k,t,t^{\prime})\simeq-\frac{H^{2}}{3}\Theta(t-t^{\prime})\left[(a_{t}H)^{-3+\epsilon_{M}}(a_{t^{\prime}}H)^{-\epsilon_{M}}-(a_{t^{\prime}}H)^{-3+\epsilon_{M}}(a_{t}H)^{-\epsilon_{M}}\right],$
(91)

which for $\epsilon_{M}\rightarrow 0$ goes as $a^{-3}(t)$. We can then
straightforwardly compute

$\displaystyle\Upsilon(x,x^{\prime})$
$\displaystyle=\frac{H^{7}}{4\pi^{2}}\sigma^{3}(a_{t}a_{t^{\prime}})^{\frac{3}{2}}\sech^{\frac{7}{2}}H\tau\Big{[}\Gamma\left[\tfrac{11}{2}\right]\sech^{2}H\tau{}_{1}F_{1}\Big{(}\frac{11}{2},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}$
(92)
$\displaystyle\qquad+2\Gamma\left[\tfrac{9}{2}\right]{}_{1}F_{1}\Big{(}\frac{9}{2},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}\Big{(}1+\sech
H\tau\Big{(}\frac{a_{t^{\prime}}}{a_{t}}\frac{\partial_{t^{\prime}}}{H}+\frac{a_{t}}{a_{t^{\prime}}}\frac{\partial_{t}}{H}\Big{)}\Big{)}$
$\displaystyle\qquad+(-5+12\epsilon_{M})\Gamma\left[\tfrac{7}{2}\right]{}_{1}F_{1}\Big{(}\frac{7}{2},\frac{3}{2},-\frac{r^{2}}{4\alpha}\Big{)}\Big{]}G_{R}(x,x^{\prime})$
$\displaystyle\qquad-\frac{H^{6}}{3\pi^{2}}(3-2\epsilon_{M})\sigma^{3}\Gamma\left[\tfrac{7}{2}\right]{}_{1}F_{1}\Big{(}\frac{7}{2},\frac{3}{2},-\frac{1}{4}(r\sigma
a_{t}H)^{2}\Big{)}\delta(H\tau),$

where we have used

$\displaystyle(\partial_{t}+\partial_{t^{\prime}})G_{R}=-3HG_{R},\quad\partial_{t}\partial_{t^{\prime}}G_{R}=(3-\epsilon_{M})\epsilon_{M}H^{2}G_{R}+\frac{\delta(t-t^{\prime})}{\Theta(t-t^{\prime})}\partial_{t^{\prime}}G_{R}.$
(93)

As an example, in the $r=0$, $\epsilon_{M}=0$ limit this reduces to

$\displaystyle\Upsilon(x,x^{\prime})$
$\displaystyle=\frac{H^{6}}{\pi^{2}}\sigma^{3}\Gamma\left[\tfrac{7}{2}\right]\Big{[}\sech^{\tfrac{7}{2}}H\tau\Big{(}\tfrac{21}{8}\sinh\tfrac{3}{2}H\tau\sech^{2}H\tau$
(94) $\displaystyle\qquad-\tfrac{7}{2}\sinh\tfrac{1}{2}H\tau\sech
H\tau+\tfrac{1}{3}\sinh\tfrac{3}{2}H\tau\Big{)}\Theta(\tau)-\delta(H\tau)\Big{]}.$

The important result is a suppression by an overall factor of $\sigma^{3}$,
but not by e.g. powers of the scale factor. This is still much faster than for
the diagonal correlator (82), which has a leading $\sigma^{2\epsilon_{M}}$.

Similarly, in (43), one of the terms may by partial integration be made not to
act explicitly on the $\phi_{\text{\tiny IR}}$. We find for this term,

$\displaystyle\frac{2}{a_{t^{\prime}}^{3}}$
$\displaystyle\delta(t-t^{\prime})\int\frac{\differential^{3}\mathbf{k}}{(2\pi)^{3}}e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x^{\prime}})}\tilde{Q}_{t^{\prime}}{W}_{t^{\prime}}$
(95)
$\displaystyle=\delta(H\tau)\frac{H^{6}\sigma^{3}}{2\pi^{2}}\Big{[}3\Gamma[\tfrac{7}{2}]\left({}_{1}F_{1}(\tfrac{7}{2},\tfrac{3}{2},-\tfrac{1}{4}(r\sigma
aH)^{2})-2^{\frac{7}{2}}{}_{1}F_{1}(\tfrac{7}{2},\tfrac{3}{2},-\tfrac{1}{2}(r\sigma
aH)^{2})\right)$
$\displaystyle\qquad+\Gamma\left[\tfrac{5}{2}\right]\left({}_{1}F_{1}(\tfrac{5}{2},\tfrac{3}{2},-\tfrac{1}{4}(r\sigma
aH)^{2})-2^{\frac{5}{2}}{}_{1}F_{1}(\tfrac{5}{2},\tfrac{3}{2},-\tfrac{1}{2}(r\sigma
aH)^{2})\right)\Big{]}$
$\displaystyle\simeq\frac{H^{6}}{\pi^{2}}\sigma^{3}\Gamma\left[\tfrac{7}{2}\right]\bigg{(}\frac{17-128\sqrt{2}}{10}\bigg{)}\delta(H\tau),$

where again the limits $r=0$, $\epsilon_{M}=0$ are taken in the last line. It
is clear that also this quantity is suppressed by $\sigma^{3}$.

In Ref. Morikawa1990 , $\Upsilon$ was also reported to decay as $\mu^{3}$ for
a step window function (62). This is consistent, since for the Gaussian window
function, $\sigma$ also parametrises the cutoff $\mu$. We conclude that
neglecting the off-diagonal contributions in the influence functional is
consistent in the small-$\sigma$ (or $\mu$) regime, for
$\sigma^{3}\ll\sigma^{2\epsilon_{M}}$.

###  3.3 Inverse IR propagator expansion

$0.2$$0.4$$0.6$$0.8$$0.7$$0.8$$0.9$$1$$\sigma$$\mathcal{N}_{\text{NNLO}}/\mathcal{N}_{\text{NLO}}$$r/R=1.0$$r/R=0.1$
Figure 3: The ratio of the noise amplitude from the NNLO contributions (97) to
the NLO noise (82) (with the Gaussian window function) as a function of
$\sigma$ with $\epsilon_{M},\epsilon_{H}=0$.

The expansion of the IR propagator relies on the smallness of the overlap
between the IR field and the UV propagator. To at least get an idea of the
convergence, we proceed to consider the stochastic noise at NNLO in (76),
given by the q-q component

$\displaystyle\begin{aligned} i(\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1})_{\text{qq}}&=G^{-1}WG_{F}WG^{-1}-WG^{-1}WG_{F}WG^{-1}-G^{-1}WG_{F}WG^{-1}W\\\
&\quad-G^{-1}W^{2}G_{F}WG^{-1}-G^{-1}WG_{F}W^{2}G^{-1}\\\
&\quad+G^{-1}WG_{F}WG^{-1}WG_{A}WG^{-1}\\\
&\quad+G^{-1}WG_{R}WG^{-1}WG_{F}WG^{-1}.\end{aligned}$ (96)

Assuming that it is valid to discard terms where the window function $W$ is
directly convoluted with the IR field constituents, and in addition that the
contribution involving retarded/advanced propagators is negligible, (96)
reduces to

$\displaystyle i(\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1}\mathbbm{G}_{\text{\tiny
UV}}\mathbbm{G}^{-1})_{\text{qq}}\simeq
G^{-1}WG_{F}WG^{-1}-G^{-1}W^{2}G_{F}WG^{-1}-G^{-1}WG_{F}W^{2}G^{-1}.$ (97)

Here we note that if the window function is a projection so that $W^{2}=W$,
all the terms add up to become equal to the NLO contribution considered above
(75). Clearly, the step function is a pathological case, for which this
expansion fails.

For a general window function, one can compute the three contributions in (97)
explicitly. Using again the Gaussian window function as a test case, we obtain

$\displaystyle\begin{aligned}
G^{-1}W^{2}G_{F}WG^{-1}&=2G^{-1}WG_{F}WG^{-1}-\frac{H^{6}}{4\pi^{2}}\sigma^{2\epsilon_{M}}\Big{(}\frac{2}{2e^{-H\tau}+e^{H\tau}}\Big{)}^{2+\epsilon_{M}}\\\
&\quad\times\Big{[}2\Gamma[4+\epsilon_{M}]\Big{(}\frac{2}{2e^{-H\tau}+e^{H\tau}}\Big{)}^{2}{}_{1}F_{1}\left(4+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\tilde{\alpha}_{t}}\right)\\\
&\qquad+2(1-2\epsilon_{M})\Gamma[3+\epsilon_{M}]{}_{1}F_{1}\Big{(}3+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\tilde{\alpha}_{t}}\Big{)}\\\
&\qquad+(1-2\epsilon_{M})^{2}\Gamma[2+\epsilon_{M}]{}_{1}F_{1}\Big{(}2+\epsilon_{M},\frac{3}{2},-\frac{r^{2}}{4\tilde{\alpha}_{t}}\Big{)}\Big{]},\end{aligned}$
(98)

with

$\displaystyle\tilde{\alpha}_{t}$
$\displaystyle\equiv\frac{2e^{-H\tau}+e^{H\tau}}{2\sigma^{2}a_{t}a_{t^{\prime}}H^{2}},$
(99)

and similarly for $G^{-1}WGW^{2}G^{-1}$ with
$2e^{-H\tau}+e^{H\tau}\leftrightarrow e^{-H\tau}+2e^{H\tau}$. We notice the
reappearance of the NLO noise kernel $G^{-1}WG_{F}WG^{-1}$ as the first term
on the right-hand-side of (98).

To quantify the relation between the two orders of the expansion, we perform
the localisation procedure of (89), i.e., replacing the noise by a delta
function and matching the normalisation. The ratio of the noise amplitude for
the NNLO noise terms (97) and the NLO result (82) is plotted in Fig. 3.
Judging only by the subset of terms computed, there is no suppression of the
NNLO contributions compared to that of NLO. Hence, we cannot confirm that
truncating the inverse propagator expansion at NLO is a controlled
approximation.

###  3.4 Exact or long-wavelength UV mode functions

$0.2$$0.4$$0.6$$0.8$$1$$9$$11$$13$$15$$\sigma$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$ExactLong-
wavelength appr. Figure 4: Noise amplitude using the Gaussian window function
in massless dS using the exact (34) and long-wavelength approximate propagator
(35) as a function of $\sigma$ with sub-horizon spatial separation $r/R=0.01$.

Another approximation worthy of examination is the use of the IR-approximate
mode solutions (35) instead of the exact mode solutions (33). The latter are
complicated to treat analytically and so we compare the two numerically. As a
first example, we continue our analysis of the Gaussian window function by
computing the noise amplitude $\mathcal{N}$ with both the approximate and the
exact mode functions, shown in Fig. 4. For very small $\sigma$, the two can be
seen to agree asymptotically, but diverge for larger values. Apparently, the
$\sigma\rightarrow 0$ is a crucial ingredient of the standard result for the
noise amplitude.

##  4 The noise amplitude and its parameter dependence

As we illustrated for the Gaussian window function, for spatial separations
much smaller than the horizon, the noise correlations are localised near
$t=t^{\prime}$. For many smooth window functions, this is a general property,
although the noise is not white in a strict sense. Making the interpretation
of the distribution being a smoothed delta function, we can attempt to extract
a characteristic noise amplitude as described in (89), suitable for the
stochastic inflationary dynamics. Ultimately, this normalisation is what
enters into observables, as described in section 1.1. To this end we consider
the noise on patches larger than the cutoff scale and define the amplitude
$\mathcal{N}$ as

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle\rightarrow\mathcal{N}\delta(t-t^{\prime})\Theta(1-\mu
raH),$ (100)

where $\mathcal{N}$ is defined as described in section 3.1 through the
matching

$\displaystyle\mathcal{N}\equiv\int\differential\tau\langle\xi\xi\rangle(\tau).$
(101)

As mentioned in section 2.5, when the noise component $\xi_{\phi}$ dominates,
we may write

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle\rightarrow
9H^{2}\delta(t-t^{\prime})\Theta(1-\mu
raH)\int\differential\tau\langle\xi_{\phi}\xi_{\phi}\rangle(\tau).$ (102)

In addition to the Gaussian window function, we will consider three smoothed
step functions, all with the property than in certain limits, they become a
strict step function (in contrast to the Gaussian). In particular we consider
an error (Erfc) function, which turns out to be particularly tractable
analytically. In section 4.2 we consider two other sigmoid functions for
comparison, namely the Tanh and Arctan functions.

For the smoothed step functions, the parameter $\mu$ controls the position of
the cutoff, just as for the strict step function (62). Small $\mu$ means that
the cutoff is far in the super-horizon IR region. The parameter $\sigma$
controls the width of the smoothed step function, and in the limit
$\sigma\rightarrow 0$, we recover the sharp step function. This is illustrated
in Fig. 5.

$0.1$$0.2$$0.3$$0$$0.5$$1$$k/aH$$W(k,t)$GaussianErfcTanhArctan Figure 5:
Window functions with $\mu=0.1$ and $\sigma=0.03$ compared to a step function
(black line). For the Gaussian, $\sigma$ parameterises both the cutoff scale
and the sharpness of the transition.

###  4.1 Erfc window function

A smooth approximate step function can be constructed from the (complementary)
error function as

$\displaystyle{W}(k,t)=\frac{1}{2}\mathrm{erfc}\left[\frac{1}{\sqrt{2}\sigma}\left(\mu-\frac{k}{a_{t}H}\right)\right].$
(103)

Inserting first the long-wavelength approximate mode solution (35) into (79),
the noise correlation is found grad  to be

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle$
$\displaystyle=-\frac{iH^{6}e^{M_{1}-\frac{\mu^{2}}{\sigma^{2}}}}{16\pi^{3}}\frac{R}{r}\bigg{[}\frac{1}{\sigma^{6}}\bigg{(}\mathcal{T}_{-(5+2\epsilon_{M})}-\frac{2\mu}{\sech\frac{H\tau}{2}}\mathcal{T}_{-(4+2\epsilon_{M})}+\mu^{2}\mathcal{T}_{-(3+2\epsilon_{M})}\bigg{)}$
(104)
$\displaystyle\qquad+\frac{4(1-\epsilon_{M})}{\sigma^{4}}\bigg{(}\frac{1}{\sech
H\tau}\mathcal{T}_{-(3+2\epsilon_{M})}-\frac{\mu}{\sech\frac{H\tau}{2}}\mathcal{T}_{-(2+2\epsilon_{M})}\bigg{)}$
$\displaystyle\qquad+\frac{4(1-\epsilon_{M})^{2}}{\sigma^{2}}\mathcal{T}_{-(1+2\epsilon_{M})}\bigg{]}.$

Here we have defined

$\displaystyle\mathcal{T}_{-n}\equiv\Gamma[n]\left(\frac{\sigma^{2}\sech
H\tau}{2}\right)^{\frac{n}{2}}\left[\,\mathrm{exp}\left[M_{2}\right]D_{-n}(M_{3})-\,\mathrm{exp}\left[M_{2}^{*}\right]D_{-n}(M_{3}^{*})\right],$
(105)

where $D_{-n}(x)$ is the parabolic cylinder function. The arguments are

$\displaystyle M_{1}\equiv\frac{\beta^{2}-r^{2}}{8\alpha}\,,\qquad
M_{2}\equiv-\frac{ir\beta}{4\alpha}\,,\qquad M_{3}\equiv\frac{\beta-
ir}{\sqrt{2\alpha}}\,,$ (106)

with $\alpha$ as in (83) and $\beta$ defined as

$\displaystyle
R^{2}\equiv\frac{1}{a(t)a(t^{\prime})H^{2}}\,,\qquad\alpha\equiv\frac{R^{2}}{\sigma^{2}\sech
H\tau}\,,\qquad\beta\equiv-\frac{2\mu R}{\sigma^{2}\sech\frac{H\tau}{2}}\,.$
(107)

The cutoff scale $\mu$ is intended to be finite, in order to retain some modes
in the IR. The sharpness parameter $\sigma$ can however be arbitrarily small.
We are interested in the noise amplitude in the limit of a steep step momentum
transition $\sigma\ll 1$. To study this limit, we proceed by eliminating the
spatial dependence by expanding the noise (104) to leading order in spatial
separation $\mu r/R\ll 1$,

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle$
$\displaystyle\simeq\frac{H^{6}}{8\pi^{3}}e^{\frac{\mu^{2}}{\sigma^{2}}\left(\frac{1}{2}\frac{\sech
H\tau}{\sech^{2}\frac{H\tau}{2}}-1\right)}\bigg{[}\frac{1}{\sigma^{6}}\bigg{(}\tilde{\mathcal{T}}_{-(5+2\epsilon_{M})}-\frac{2\mu}{\sech\frac{H\tau}{2}}\tilde{\mathcal{T}}_{-(4+2\epsilon_{M})}+\mu^{2}\mathcal{T}_{-(3+2\epsilon_{M})}\bigg{)}$
(108)
$\displaystyle\qquad+\frac{4(1-\epsilon_{M})}{\sigma^{4}}\bigg{(}\frac{1}{\sech
H\tau}\tilde{\mathcal{T}}_{-(3+2\epsilon_{M})}-\frac{\mu}{\sech\frac{H\tau}{2}}\tilde{\mathcal{T}}_{-(2+2\epsilon_{M})}\bigg{)}$
$\displaystyle\qquad+\frac{4(1-\epsilon_{M})^{2}}{\sigma^{2}}\tilde{\mathcal{T}}_{-(1+2\epsilon_{M})}\bigg{]},$

where101010We use the recurrence relation
$D_{-n}(x)-\frac{1}{x}D_{-n+1}(x)=-\frac{n}{x}D_{-n-1}(x)$.

$\displaystyle\tilde{\mathcal{T}}_{-n}\equiv\Gamma[n+1]\bigg{(}\frac{\sigma^{2}\sech
H\tau}{2}\bigg{)}^{\frac{n+1}{2}}D_{-n-1}\bigg{(}-\sqrt{2}\frac{\mu}{\sigma}\frac{\sqrt{\sech
H\tau}}{\sech\frac{H\tau}{2}}\bigg{)}.$ (109)

For large negative arguments, this may be further approximated as111111The
parabolic cylinder functions have the asymptotic behaviour
$D_{-n-1}(-|x|)\big{|}_{x\rightarrow\infty}\simeq\frac{\sqrt{2\pi}}{\Gamma[n+1]}e^{x^{2}/4}|x|^{n}\left[1+\frac{n(n-1)}{2x^{2}}+\frac{n(n-1)(n-2)(n-3)}{8x^{4}}+\dotso\right].$

$\displaystyle\tilde{\mathcal{T}}_{-n}$
$\displaystyle\simeq\sqrt{\pi}e^{\frac{1}{2}\frac{\mu^{2}}{\sigma^{2}}\frac{\sech
H\tau}{\sech^{2}\frac{H\tau}{2}}}\left(\mu\frac{\sech
H\tau}{\sech\frac{H\tau}{2}}\right)^{n}\sigma\sqrt{\sech
H\tau}\bigg{[}1+\left(\frac{\sigma}{\mu}\right)^{2}\frac{n(n-1)}{4}\frac{\sech^{2}\frac{H\tau}{2}}{\sech
H\tau}$ (110)
$\displaystyle\qquad+\left(\frac{\sigma}{\mu}\right)^{4}\frac{n(n-1)(n-2)(n-3)}{32}\frac{\sech^{4}\frac{H\tau}{2}}{\sech^{2}H\tau}+\dotso\bigg{]},$

which is then valid for large values of $\mu/\sigma$. In the limit
$\epsilon_{M}\rightarrow 0$, the noise correlator becomes

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle$
$\displaystyle\simeq\frac{H^{6}}{4\pi^{2}}\frac{1}{2\sqrt{\pi}}e^{\frac{\mu^{2}}{\sigma^{2}}\left(\frac{\sech
H\tau}{\sech\frac{H\tau}{2}}-1\right)}\bigg{[}-\left(\frac{\mu}{\sigma}\right)^{5}\cosh^{3}\tfrac{H\tau}{2}\sech^{\frac{11}{2}}H\tau\sinh^{2}\tfrac{H\tau}{2}$
(111)
$\displaystyle\qquad+\frac{1}{4}\left(\frac{\mu}{\sigma}\right)^{3}\cosh\tfrac{H\tau}{2}(7-2\cosh
H\tau-3\cosh 2H\tau)\sech^{\frac{9}{2}}H\tau$
$\displaystyle\qquad+\frac{1}{4}\frac{\mu}{\sigma}\left(31-6\cosh
H\tau+16\cosh
2H\tau\right)\cosh\tfrac{H\tau}{2}\sech^{\frac{7}{2}}H\tau\bigg{]}.$

#### Noise localisation and amplitude

The noise distribution (111) is centered around $t=t^{\prime}$, and becomes
more localised for increasing values of the ratio $\mu/\sigma$, as illustrated
in Fig. 6. Therefore we may expand in $H\tau$, to find

$\displaystyle\langle\xi(x)\xi(x^{\prime})\rangle$
$\displaystyle\simeq\frac{H^{6}}{4\pi^{2}}\frac{1}{2\sqrt{\pi}}e^{\left(\frac{\mu}{\sigma}\right)^{2}\left(-\frac{1}{4}(H\tau)^{2}+\frac{5}{48}(H\tau)^{4}-\frac{61}{1440}(H\tau)^{6}+\dotso\right)}\bigg{[}\frac{\sigma}{\mu}\left(\frac{9}{4}+8\right)$
(112)
$\displaystyle\qquad+\left(\frac{\sigma}{\mu}\right)^{3}\left[\frac{1}{2}-\frac{45}{16}(H\tau)^{2}\right]+\left(\frac{\sigma}{\mu}\right)^{5}\left[-\frac{1}{4}(H\tau)^{2}+\frac{55}{96}(H\tau)^{4}\right]+\dotso\bigg{]}.$

The noise amplitude is then obtained by integrating (112) over time
separation. It turns out that some contributions are recovered keeping only
the first-order term in the exponential, i.e., following a Gaussian
distribution,

$\displaystyle\frac{H^{6}}{4\pi^{2}}\frac{1}{2\sqrt{\pi}}\int\differential(H\tau)\bigg{[}\bigg{(}\frac{9}{4}+8\bigg{)}\frac{\mu}{\sigma}-\frac{45}{16}\left(\frac{\sigma}{\mu}\right)^{3}(H\tau)^{2}+\frac{55}{96}\left(\frac{\sigma}{\mu}\right)^{5}(H\tau)^{4}\bigg{]}e^{-\frac{1}{4}\left(\frac{\mu}{\sigma}\right)^{2}(H\tau)^{2}}$
(113) $\displaystyle=\bigg{(}\frac{7}{2}+8\bigg{)}\frac{H^{6}}{4\pi^{2}},$

while two terms receive corrections from the higher-order terms in the
exponential,

$\displaystyle\frac{H^{6}}{4\pi^{2}}\frac{1}{2\sqrt{\pi}}\int\differential(H\tau)\left[-\frac{1}{4}\left(\frac{\sigma}{\mu}\right)^{5}(H\tau)^{4}+\frac{1}{2}\left(\frac{\sigma}{\mu}\right)^{3}\right]e^{\left(\frac{\mu}{\sigma}\right)^{2}\left[-\frac{(H\tau)^{2}}{4}+\frac{5(H\tau)^{4}}{48}-\frac{61(H\tau)^{6}}{1440}\right]}\simeq-\frac{5}{2}\frac{H^{6}}{4\pi^{2}}.$
(114)

The sum of contributions in this limit is the standard result
$\mathcal{N}=9H^{5}/4\pi^{2}$. This is confirmed using the exact mode solution
(33) in (79), the noise amplitude of which is illustrated in Fig. 7. From this
we see that the long-wavelength approximate mode solution (35) used in (104)
is accurate for cutoff scales with $\mu\lesssim 0.1$ for small values of
$\sigma$.

The dependence on $\sigma$ is quite non-trivial. In particular, both positive
and negative correlation regions arise as the ratio $\mu/\sigma$ increases, as
can be seen in Fig. 6. For illustration we have also included the Gaussian
window function result at the same value of $\sigma$. While a Gaussian
distribution for small $\sigma$ may be be said to be well-approximated by a
strict delta-function and a noise amplitude, a substantial amount of
information is lost with such a replacement when the distribution is both
negative and positive.

$-4$$-2$$0$$2$$4$$0$$5$$10$$15$$20$$H\tau$$\frac{4\pi^{2}}{H^{6}}\langle\xi(x)\xi(x^{\prime})\rangle$GaussianErfc
$\mu=0.12$Erfc $\mu=0.10$Erfc $\mu=0.08$Erfc $\mu=0.06$ Figure 6: Noise
correlation as a function of time separation $\tau\equiv t-t^{\prime}$ for the
Erfc and Gaussian window functions with $\sigma=0.03$, $\epsilon_{M}=0$ and
$r/R=0.01$.

$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$8$$9$$10$$11$$12$$13$$\mu$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$ExactLong-
wavelength appr.

$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$3$$5$$7$$9$$\sigma$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$$\mu=0.12$$\mu=0.10$$\mu=0.08$$\mu=0.06$

Figure 7: Left panel: Noise amplitudes for the Erfc window function with exact
(34) and long-wavelength approximate (35) massless dS propagators as a
function of $\mu$, with $\sigma=0.005$ and spatial separation set to
$r/R=0.01$. Right panel: Noise amplitudes for different $\mu$ computed with
the exact propagator (34) as a function of $\sigma$.

#### Splitting up the noise contributions

$-4$$-3.5$$-3$$-2.5$$-2$$-1.5$$-1$$-0.5$$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$-15$$-10$$-5$$0$$5$$10$$15$$20$$25$$30$$35$$H\tau$$\frac{4\pi^{2}}{H^{6}}$$9H^{2}\langle{\xi}_{\phi}{\xi}_{\phi}\rangle$$\langle\dot{\xi}_{\phi}\dot{\xi}_{\phi}\rangle$$\langle\xi_{\pi}\xi_{\pi}\rangle$$3H\sum\langle\dot{\xi}_{\phi}\xi_{\phi}\rangle$$3H\sum\langle{\xi}_{\pi}\xi_{\phi}\rangle$$\sum\langle\dot{\xi}_{\phi}\xi_{\pi}\rangle$

$0.01$$0.02$$0.03$$0.04$$0$$3$$6$$9$$\epsilon_{M}$$\frac{4\pi^{2}}{H^{5}}$$9H^{2}\mathcal{N}_{\xi_{\phi}\xi_{\phi}}$$\mathcal{N}_{\dot{\xi}_{\phi}\dot{\xi}_{\phi}}$$\mathcal{N}_{\xi_{\pi}\xi_{\pi}}$$3H\sum\mathcal{N}_{\dot{\xi}_{\phi}\xi_{\phi}}$$3H\sum\mathcal{N}_{{\xi}_{\pi}\xi_{\phi}}$$\sum\mathcal{N}_{\dot{\xi}_{\phi}\xi_{\pi}}$

Figure 8: Left panel: Noise correlators for the Erfc window function using
$\epsilon_{M}=0.01$, $r/R=0.01$ and values $\sigma=0.005$, $\mu=0.03$. As the
ratio $\mu/\sigma$ increases the visible curves sharpen around $H\tau=0$ and
the visible peaks grow towards $\pm\infty$. Right panel: The corresponding
noise amplitude contributions as a function of the mass parameter
$\epsilon_{M}$.

It is possible to identify the origin of both negative and positive
correlations, and the dominant contribution to the noise amplitude, by
studying the decomposed noise correlations $\xi_{\phi},\xi_{\pi}$ of (60). In
particular, the visible negative correlation regions originate from the
$\langle\dot{\xi}_{\phi}\dot{\xi}_{\phi}\rangle$ correlator with $\xi_{\phi}$
given by (54). In (84) only the first term contributes in the massless limit,
and so

$\displaystyle\langle\dot{\xi}_{\phi}(x)\dot{\xi}_{\phi}(x^{\prime})\rangle$
$\displaystyle\simeq\frac{H^{6}}{8\pi^{3}}e^{\frac{\mu^{2}}{\sigma^{2}}\left(\frac{\sech
H\tau}{\sech^{2}\frac{H\tau}{2}}-1\right)}\bigg{[}\frac{1}{\sigma^{6}}\bigg{(}\tilde{\mathcal{T}}_{-5}-\frac{2\mu}{\sech\frac{H\tau}{2}}\tilde{\mathcal{T}}_{-4}+\mu^{2}\mathcal{T}_{-3}\bigg{)}$
(115) $\displaystyle\qquad-\frac{2}{\sigma^{4}}\bigg{(}\frac{1}{\sech
H\tau}\tilde{\mathcal{T}}_{-3}-\frac{\mu}{\sech\frac{H\tau}{2}}\tilde{\mathcal{T}}_{-2}\bigg{)}+\frac{\tilde{\mathcal{T}}_{-1}}{\sigma^{2}}\bigg{]}.$

Given the parametric choices made above, the integral of this function over
$\tau$ vanishes, and so does not contribute to the noise amplitude in our
definition. Nevertheless, the function itself is both positive and negative.
Similarly, for the cross terms,

$\displaystyle\langle\dot{\xi}_{\phi}(x){\xi}_{\phi}(x^{\prime})\rangle+\langle{\xi}_{\phi}(x)\dot{\xi}_{\phi}(x^{\prime})\rangle$
$\displaystyle\simeq\frac{H^{5}}{8\pi^{3}}e^{\frac{\mu^{2}}{\sigma^{2}}\left(\frac{\sech
H\tau}{\sech^{2}\frac{H\tau}{2}}-1\right)}$ (116)
$\displaystyle\quad\times\bigg{[}\frac{2}{\sigma^{4}}\bigg{(}\frac{1}{\sech
H\tau}\tilde{\mathcal{T}}_{-3}-\frac{\mu}{\sech\frac{H\tau}{2}}\tilde{\mathcal{T}}_{-2}\bigg{)}-\frac{2}{\sigma^{2}}\tilde{\mathcal{T}}_{-1}\bigg{]},$

which vanish by (110). The decomposed noise correlators, including also
$\xi_{\pi}$, are all illustrated in Fig. 8. From this we can conclude that
although the resulting full noise correlator is rather complicated, the main
contribution to the noise amplitude comes from the component

$\displaystyle\langle\xi_{\phi}\xi_{\phi}\rangle$
$\displaystyle=-\frac{iH^{4}}{16\pi^{3}}\frac{e^{M_{1}-\left(\frac{\mu}{\sigma}\right)^{2}}}{\sigma^{2}}\frac{R}{r}\mathcal{T}_{-(1+2\epsilon_{M})},$
(117)

whose behaviour when expanded near $\tau=0$ is well-described by a Gaussian-
like, positive function

$\displaystyle\langle\xi_{\phi}\xi_{\phi}\rangle$
$\displaystyle\simeq\frac{H^{4}}{4\pi^{2}}\frac{\mu^{2\epsilon_{M}}}{2\sqrt{\pi}}{e^{-\frac{1}{4}\left(\frac{\mu}{\sigma}\right)^{2}(H\tau)^{2}}}\frac{\mu}{\sigma}\bigg{[}1+\frac{\epsilon_{M}}{2}\Big{(}\frac{\sigma}{\mu}\Big{)}^{2}$
(118)
$\displaystyle\qquad-\frac{1}{8}\Big{(}5+\frac{3}{2}\Big{(}4+\Big{(}\frac{\sigma}{\mu}\Big{)}^{2}\Big{)}\epsilon_{M}\Big{)}(H\tau)^{2}+\mathcal{O}\Big{(}\Big{(}\frac{\sigma}{\mu}\Big{)}^{4}\Big{)}\Big{]}.$

For small values of $\sigma/\mu$, this is approximated rather well by a delta
function. The noise amplitude in this case is then given by

$\displaystyle\mathcal{N}_{\xi_{\phi}\xi_{\phi}}$
$\displaystyle\simeq\frac{H^{3}}{4\pi^{2}}{\mu^{2\epsilon_{M}}}\Big{[}1-\Big{(}\frac{\sigma}{\mu}\Big{)}^{2}\Big{(}\frac{5}{4}+\epsilon_{M}\Big{)}+\mathcal{O}\Big{(}\Big{(}\frac{\sigma}{\mu}\Big{)}^{4}\Big{)}\Big{]}\equiv\frac{H^{3}}{4\pi^{2}}f(\epsilon_{M},\mu,\sigma).$
(119)

We now have an explicit example of the correction function $f$ advertised in
section 1. It involves the mass through $\epsilon_{M}$ as well as the
parameters of the window function in the combination $\sigma/\mu$ with an
overall factor of $\mu^{2\epsilon_{M}}$.

In the massless case, we get

$\displaystyle
f=1-\frac{5}{4}\Big{(}\frac{\sigma}{\mu}\Big{)}^{2},\quad\epsilon_{M}=0.$
(120)

The standard result is then recovered upon taking the step-function limit
$\sigma\rightarrow 0$. With a massive field, even in that limit we obtain

$\displaystyle f=\mu^{2\epsilon_{M}},\quad\sigma\rightarrow 0,$ (121)

similar to the Gaussian window function, where the parameter $\sigma$ also
plays the role of $\mu$. For small values of $\mu$ and $\sigma$, a nonzero
mass $\epsilon_{M}\neq 0$ has the effect of decreasing the noise amplitude, as
can be seen in Fig. 9. The dependence on mass occurs largely, but not
entirely, through the prefactor $\mu^{2\epsilon_{M}}$ (or
$\sigma^{2\epsilon_{M}}$ in the Gaussian case). Any amplitude smaller than
$9H^{5}/4\pi^{2}$ can be achieved by making a different choice of parameters.

$0.00$$0.01$$0.01$$0.02$$0.02$$0.03$$0.03$$0.04$$0.04$$0.05$$0.05$$5$$6$$7$$8$$9$$\epsilon_{M}$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$GaussianErfc
$\mu=0.025$Erfc $\mu=0.030$Erfc $\mu=0.050$

$0.02$$0.04$$0.06$$0.08$$0.10$$8.4$$8.6$$8.8$$9.0$$\mu$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$$\epsilon_{M}=0$$\epsilon_{M}=0.001$$\epsilon_{M}=0.002$$\epsilon_{M}=0.005$

Figure 9: Left panel: Noise amplitude for the Gaussian and Erfc window
functions using exact mode functions for varying $\epsilon_{M}$ with
$\sigma=0.005$ and $r/R=0.01$. Dashed line is $9\mu^{2\epsilon_{M}}$ with
$\mu=0.05$ and dotted line is $9\sigma^{2\epsilon_{M}}$. Right panel: Cutoff
dependence of the noise amplitude with the Erfc window for different choices
of $\epsilon_{M}$.

###  4.2 Alternative smoothened step window functions

To illustrate that these results are fairly generic, we will also consider two
other sigmoid functions,

$\displaystyle W_{\text{tanh}}(k,t)$
$\displaystyle=\frac{1}{2}\Big{(}1+\tanh[\frac{1}{\sigma}\Big{(}\frac{k}{a_{t}H}-\mu\Big{)}\Big{]}\Big{)},$
(122) $\displaystyle W_{\text{arct}}(k,t)$
$\displaystyle=\frac{1}{2}\Big{(}1+\frac{2}{\pi}\arctan[\frac{\pi}{2\sigma}\Big{(}{\frac{k}{a_{t}H}-\mu}\Big{)}\Big{]}\Big{)},$
(123)

for which

$\displaystyle\tilde{\mathcal{Q}}_{\text{tanh}}(k,t)$
$\displaystyle=-H^{2}\frac{k}{\sigma
a_{t}H}\sech^{2}\Big{[}\frac{1}{\sigma}\Big{(}\frac{k}{a_{t}H}-\mu\Big{)}\Big{]}\Big{(}\frac{k}{\sigma
a_{t}H}\tanh[\frac{1}{\sigma}\Big{(}\frac{k}{a_{t}H}-\mu\Big{)}\Big{]}+1-\epsilon_{M}\Big{)},$
(124)

and

$\displaystyle\tilde{\mathcal{Q}}_{\text{arct}}(k,t)$
$\displaystyle=-H^{2}\frac{\frac{k}{\sigma
a_{t}H}}{1+(\tfrac{\pi}{2\sigma})^{2}(\frac{k}{a_{t}H}-\mu)^{2}}\left[\frac{(\frac{\pi}{2\sigma})^{2}\frac{k}{a_{t}H}(\frac{k}{a_{t}H}-\mu)}{1+(\tfrac{\pi}{2\sigma})^{2}(\frac{k}{a_{t}H}-\mu)^{2}}+1-\epsilon_{M}\right].$
(125)

Restricting ourselves to comparing numerical results, the noise amplitude
$\mathcal{N}$ for all window functions is shown in Fig. 10. We see that in the
limit $\sigma\rightarrow 0$, they all converge to the standard result, but for
finite $\sigma$, they diverge from each other. The convergence for the Arctan
window function to the value $9H^{5}/4\pi^{2}$ is particularly slow, and only
agrees in the limit, whereas the other window functions start converging for
finite values of $\sigma\simeq 0.04$. This qualitative behaviour persists for
other values of $\mu$ and $\epsilon_{M}$ (see Fig. 11, right panel), and is
likely a reflection of the fact that the Arctan window requires smaller values
of $\sigma$ to reach the step function shape compared to the other window
functions. As for the Erfc window function, the noise correlations using the
other sigmoid window functions also develop negative correlation regions for
values $\mu/\sigma\gtrsim 3$, as illustrated in the left panel of Fig 11.

$0.01$$0.02$$0.03$$0.04$$0.05$$0.06$$0.07$$0.08$$0.09$$0.10$$5$$6$$7$$8$$9$$10$$\sigma$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$GaussianErfcTanhArctan

$0.01$$0.02$$0.03$$0.04$$0.05$$0.06$$0.07$$0.08$$0.09$$0.10$$5$$6$$7$$8$$9$$10$$\sigma$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$GaussianErfcTanhArctan

Figure 10: Noise amplitude for different window functions using the exact
propagator (33) as a function of $\sigma$ with masses $\epsilon_{M}=0$ (left
panel) and $\epsilon_{M}=0.002$ (right panel) using $\mu=0.1$ and $r/R=0.01$.

$-5$$-4$$-3$$-2$$-1$$0$$1$$2$$3$$4$$5$$-20$$0$$20$$40$$60$$H\tau$$\frac{4\pi^{2}}{H^{6}}\langle\xi(x)\xi(x^{\prime})\rangle$GaussianErfcTanhArctan

$0.00$$0.01$$0.01$$0.02$$0.02$$0.03$$0.03$$0.04$$0.04$$0.05$$0.05$$5$$6$$7$$8$$9$$\epsilon_{M}$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$GaussianErfcTanhArctan

Figure 11: Left panel: The noise correlations for different window functions
as a function of time separation with $\epsilon_{M}=0.01$, $r/R=0.01$,
$\sigma=0.005$ and in addition $\mu=0.03$ for the sigmoids. Right panel: The
corresponding noise amplitudes as a function of $\epsilon_{M}$. For small
$\sigma$ the Erfc and Tanh windows integrate to very similar values. Smaller
values of $\sigma$ are required for the Arctan window function to recover the
higher noise amplitudes of the other smooth step functions.

###  4.3 Away from dS: leading order in slow-roll

$0.00$$0.01$$0.02$$0.03$$0.04$$0.05$$0.06$$0.07$$0.08$$0.09$$0.10$$9$$10$$11$$\epsilon_{H}$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$$\mu=0.025$$\mu=0.030$$\mu=0.050$$\mu=0.100$

$0.00$$0.01$$0.02$$0.03$$0.04$$0.05$$0.06$$0.07$$0.08$$0.09$$0.10$$9$$11$$13$$15$$\epsilon_{H}$$\frac{4\pi^{2}}{H^{5}}\mathcal{N}$GaussianErfcTanhArctan

Figure 12: Left panel: Noise amplitude with the Erfc window function as a
function of the slow-roll parameter $\epsilon_{H}$ for different values of
$\mu$. Right panel: For all window functions for a cutoff $\mu=0.03$. Here
$\epsilon_{M}=0$, $\sigma=0.005$ and $r/R=0.01$. The Arctan window function
requires even smaller values of $\sigma$ to reach the asymptotic value
$9H^{5}/4\pi^{2}$ in the $\epsilon_{H}\rightarrow 0$ limit.

As a final generalisation, we consider the dependence of the slow-roll
parameter $\epsilon_{H}$. Using that the above window functions depend on $k$
through the combination ${W}(t,k)\equiv{W}(k/\sigma a(t)H(t))$, it is
convenient to make the replacement,

$\displaystyle\dot{{W}}_{t}$
$\displaystyle=-Hz(1-\epsilon_{H})\partial_{z}{W}_{t}(z),\qquad
z\equiv\frac{k}{\sigma a(t)H(t)},$ (126) $\displaystyle\ddot{{W}}_{t}$
$\displaystyle\simeq
H^{2}(1-\epsilon_{H})z\partial_{z}{W}(z)+H^{2}(1-2\epsilon_{H})z^{2}\partial_{z}^{2}{W}(z),$
(127)

where the last equality holds to first order in slow-roll. The function
$\tilde{\mathcal{Q}}_{t}$ in the correlator (79) is then

$\displaystyle\begin{aligned}
-\tilde{\mathcal{Q}}_{t}&=\ddot{{W}}_{t}+(3+2q_{\nu})H(1-\epsilon_{H})\dot{{W}}_{t}\simeq
H^{2}(1-2\epsilon_{H})\left[z^{2}\partial_{z}^{2}-2(1+q_{\nu})z\partial_{z}\right]{W}(z).\end{aligned}$
(128)

$0.00$$0.02$$0.04$$0.06$$0.08$$0.10$$0$$3$$6$$9$$12$$\epsilon_{H}$$\frac{4\pi^{2}}{H^{5}}$$9H^{2}\mathcal{N}_{\xi_{\phi}\xi_{\phi}}$$\mathcal{N}_{\dot{\xi}_{\phi}\dot{\xi}_{\phi}}$$\mathcal{N}_{\xi_{\pi}\xi_{\pi}}$$3H\sum\mathcal{N}_{\dot{\xi}_{\phi}\xi_{\phi}}$$3H\sum\mathcal{N}_{{\xi}_{\pi}\xi_{\phi}}$$\sum\mathcal{N}_{\dot{\xi}_{\phi}\xi_{\pi}}$
Figure 13: Slow-roll dependence of the noise amplitudes of the decomposed
correlators (60) with the Erfc window function, in the massless case, for
$r/R=0.01$, $\sigma=0.005$ and $\mu=0.03$.

The slow-roll parameter $\epsilon_{H}$ appears both in the exact mode
functions through the index $\nu$, and through the derivative operators
$\tilde{\mathcal{Q}}_{t}$. The rest of the computation proceeds in the same
way as before. For instance for the Erfc window with the long-wavelength
approximate solution (35), the normalisation becomes

$\displaystyle\mathcal{N}_{\xi_{\phi}\xi_{\phi}}$
$\displaystyle\simeq\frac{H^{3}}{4\pi^{2}}\frac{(1-\epsilon_{H})^{4}}{1-\frac{1}{2}\epsilon_{H}}\bigg{(}\frac{\mu}{1-\epsilon_{H}}\bigg{)}^{2\epsilon_{M}-2\epsilon_{H}}\bigg{[}1-\Big{(}\frac{\sigma}{\mu}\Big{)}^{2}\Big{(}\frac{5}{4}+\epsilon_{M}-\epsilon_{H}\Big{)}+\mathcal{O}\Big{(}\Big{(}\frac{\sigma}{\mu}\Big{)}^{4}\Big{)}\bigg{]}$
(129)
$\displaystyle\equiv\frac{H^{3}}{4\pi^{2}}f(\epsilon_{M},\epsilon_{H},\mu,\sigma).$

This is a further generalisation of (119), where now also different values of
$\epsilon_{H}$ may alter the noise normalisation.

Inserting the exact mode solutions (33), we evaluate the noise numerically for
the different window functions considered, and the resulting
$\epsilon_{H}$-dependence is shown in Fig. 12. We see that, as for
$\epsilon_{M}$, the dependence on the slow-roll parameter is also substantial,
non-monotonic and dependent on the choice of both window function and the
cutoff parameters. Away from dS, the contributions to the noise amplitude is
spread to the other noise correlators in (60), as illustrated in Fig. 13 for
the Erfc window function. With $\mu,\sigma$ small, the correlator
$\langle\xi_{\phi}\xi_{\phi}\rangle$ remains the leading contributor to the
noise amplitude.

##  5 Conclusions

In this work, we have revisited the derivation of Stochastic Inflation from
first-principles quantum field theory. Inspired by Moss:2016uix , we have
investigated the conditions under which the standard Langevin dynamics for the
super-horizon component of a scalar field may be recovered. The agreement is
subject to an appropriate choice of window function separating the UV and IR
modes, and applies only in certain limits, which we have identified. In
particular, one must pay close attention to the magnitude of the off-diagonal
components of the influence functional kernel (43), and compare them to the
diagonal contributions that parametrise the stochastic noise. Only for a steep
window function with a separation scale in the far super-horizon limit
($\sigma,\mu\ll 1$) is it consistent to neglect them.

The essential ingredient in achieving the standard local Langevin evolution is
to note that noise correlations are exponentially suppressed both outside the
spatial horizon $(aH)^{-1}$, and for large time separations ($H\tau>1$).
Inside the spatial horizon, one may then attempt to represent the correlations
by a completely local noise amplitude, obtained through matching the
integrated power. We showed that this may be a valid approximation in some
cases (e.g. with a Gaussian window function), but that for others, important
information may be lost (e.g. with the Erfc window function).

Our analysis shows that in the limits $\epsilon_{M},\epsilon_{H}=0$, $\mu\ll
1$, $\sigma\rightarrow 0$, the standard noise amplitude $9H^{5}/4\pi^{2}$ is
correct, at least asymptotically. However, as soon as these parameters take on
finite values, the amplitude is modulated by a function $f$, which in
principle can assume any value. We provided a number of examples of what such
a function could look like. We also showed numerically, that the long-
wavelength approximation to the UV mode functions only matches the exact mode-
function results for small $\mu$ and $\sigma$.

Because the stochastic formalism relies on a sequence of approximations and
assumptions, it would be interesting to further map out its region of validity
by comparing to other approaches. These include matching to perturbative and
resummed analytic results Serreau:2011fu ; Serreau:2013psa ; Garbrecht:2013coa
; Kamenshchik:2021tjh , and numerical evaluation within (generalisations of)
the target stochastic model Cable:2020dke . One may also consider a full
numerical implementation of the quantum dynamics, in a semi-dS background,
similar to what was done in Ref. Tranberg:2008ae .

To summarise, Stochastic Inflation as a formalism remains a very elegant and
powerful way of describing the dynamics of cosmological scalar fields at long
wavelengths. However, the detailed properties of the window function, as well
as the field mass and slow-roll parameters are crucial in establishing the
amplitude of the noise, the predicted asymptotic observables and the dynamical
mass generated. This restricts the applicability of the standard formalism to
certain parametric limits, where the overall amplitude of the noise is to be
estimated case-by-case, or calls for a generalised formalism.

## References

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