CERN-TH-2022-206

Large Deviations in the Early Universe

Timothy Cohen${}^{\hskip 0.8pt1,2,3}$, Daniel Green${}^{\hskip 0.8pt4}$, and
Akhil Premkumar${}^{\hskip 0.8pt5}$

1Theoretical Physics Department, CERN, 1211 Geneva, Switzerland 2Theoretical
Particle Physics Laboratory, EPFL, 1015 Lausanne, Switzerland 3Institute for
Fundamental Science, University of Oregon, Eugene, OR 97403, USA 4Department
of Physics, University of California at San Diego, La Jolla, CA 92093, USA
5Kavli Institute for Cosmological Physics, University of Chicago, IL 60637,
USA

Abstract

Fluctuations play a critical role in cosmology. They are relevant across a
range of phenomena from the dynamics of inflation to the formation of
structure. In many cases, these fluctuations are coarse grained and follow a
Gaussian distribution as a consequence of the Central Limit Theorem. Yet, some
classes of observables are dominated by rare fluctuations and are sensitive to
the details of the underlying microphysics. In this paper, we argue that the
Large Deviation Principle can be used to diagnose when one must to appeal to
the fundamental description. Concretely, we investigate the regime of validity
for the Fokker-Planck equation that governs Stochastic Inflation. For typical
fluctuations, this framework leads to the central limit-type behavior expected
of a random walk. However, fluctuations in the regime of the Large Deviation
Principle are determined by instanton-like saddle points accompanied by a new
energy scale. When this energy scale is above the UV cutoff of the EFT, the
tail is only calculable in the microscopic description. We explicitly
demonstrate this phenomenon in the context of determining the phase transition
to eternal inflation, the distribution of scalar field fluctuations in de
Sitter, and the production of primordial black holes.

###### Contents

  1. 1 Introduction
  2. 2 Random Walks and the Renormalization Group
    1. 2.1 Central Limit Theorem
    2. 2.2 Large Deviation Principle
  3. 3 Eternal Inflation
    1. 3.1 Review of Stochastic Inflation
    2. 3.2 Central Limit Theorem as a Resummation
    3. 3.3 Large Deviations and the EFT of Inflation
  4. 4 Random Walks with External Forces
    1. 4.1 Equilibrium Distributions
    2. 4.2 Markovian Evolution
  5. 5 Light Scalar Fields in de Sitter
    1. 5.1 Effective Potentials and Markovian Dynamics
    2. 5.2 Light Scalars in de Sitter with Derivative Interactions
  6. 6 Models of Primordial Black Hole Generation
    1. 6.1 Non-Gaussian Tails
    2. 6.2 Relation to Factorial Enhancement
  7. 7 Conclusions

##  1 Introduction

The cosmological evolution of our Universe was shaped by fluctuations. The
formation of dark matter halos, and hence galaxies and galaxy clusters, was
the result of large density fluctuations, which can be modeled using Gaussian
random fields. Rare fluctuations, which are determined by the tail of the
probability distribution, may also be important for cosmology. For example,
determining if some (or all) of the abundance of dark matter is due to the
presence of primordial black holes requires a precise knowledge of the tail of
the distribution. Such rare fluctuations could also have played a critical
role in the dynamics of (eternal) inflation at very early times.

Despite the broad interest in this subject (_e.g._ for inflation see [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and for eternal inflation see [14, 15, 16,
17, 18, 19]), characterizing the regime of validity of perturbation theory for
the tail of the distribution is an under developed subject. The probability
distribution of a field in a de Sitter background can be calculated using a
Fokker-Planck equation, known as the framework of Stochastic Inflation [20,
21] (see also [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37]). Recently, the introduction of the Soft de Sitter Effective Theory
(SdSET) [38, 39] (see [40] for review) has facilitated the systematic
computation of corrections to the evolution equations of Stochastic Inflation
(for alternative approaches, see [41, 42, 35, 43, 44, 36]). In the presence of
primordial non-Gaussianity (non-trivial interactions during inflation),
perturbation theory for the probability distribution can be naturally
organized as an Edgeworth series, such that the coefficients of the expansion
are determined by the cumulants of the distribution. However, such an
expansion is expected to break down when computing observables that are
sensitive to very rare configurations.

One natural hope is that this tail of the probability distribution can be
captured by some kind resummation of the perturbative series. However,
implementing such a resummation simply yields the Fokker-Planck equation for
the distribution of cosmological fluctuations, with small corrections that are
equivalent to performing a ‘local’ Kramers–Moyal expansion of the underlying
master equation [39] (for an alternative approach, see [45]). To assess the
range of validity for these methods, a first step is to compute the
corrections to these equations systematically, which allows one to explore the
properties of the resulting probability distributions. To this end, we
computed the corrections to Stochastic Inflation from the leading effects of
primordial non-Gaussianity in [46]. By studying the phase transition to
eternal inflation (an observable that is exponentially sensitive to the tails
of the probability distribution), we observed that the resulting probability
distribution was not under perturbative control. While one might anticipate
the tail is sensitive to non-perturbative effects, the precise origin and
location of this breakdown should be calculable from the Effective Field
Theory (EFT) point of view.

This paper will explain how the tails of these distributions are determined by
a new instanton-like111The saddles we will discuss are not instantons in the
conventional sense. However, they are occasionally referred to as instantons
in the broader literature.  saddle point. These new saddles have their own
associated energy scale, which can invalidate the naive EFT expansion, where
the IR scale is set by the Hubble scale $H$. As we will quantify below, there
are circumstances where these saddles are under control within the EFT
description, and they simply reproduce the behavior of Stochastic Inflation.
However, when computing the probability for observables that are sensitive to
sufficiently large deviations, the saddle lies beyond the EFT description and
must be calculated by appealing to the full theory.

We will provide a framework for anticipating such a breakdown by recasting
these questions in terms of random walks. In fact, the appearance of the
Fokker-Planck equation in inflationary cosmology suggests that the
phenomenology of fluctuations in dS spacetime can be mapped onto the behavior
of random walks. Specifically, we will show that random walks with independent
and identically distributed (i.i.d.) steps give rise to the same behavior as
cosmological systems. Concretely, consider an i.i.d. walk with zero mean that
traverses a distance $X$ in a number of steps $N$. For typical fluctuations
($X\propto\sqrt{N}$) the Central Limit Theorem (CLT) tells us that $X$ is
Gaussian distributed, even if the individual steps are not. As we will review
below, one can view the CLT as the result of a renormalization group flow to a
Gaussian fixed point, with all non-Gaussianity being irrelevant – typical
fluctuations are insensitive to the microscopic details of the walk. On the
other hand, large deviations ($X\propto N$) defy this general behavior and do
depend on the microscopic details. The fact that large deviations are not
determined by universal long-distance behavior will have a precise analogy in
cosmology, and will explain the breakdown of EFT for large fluctuations.

Figure 1:  Large deviations in a random walk. The walker makes one large jump
of size ${\cal L}$ in the left figure, on top of many small positive and
negative steps. In contrast, the figure on the right depicts the walker
covering the same distance ${\cal L}$ in a series of smaller steps that are
mostly in the same direction. Although both of these are extremely unlikely,
the one on the right is much more probable than the one on the left. If we
assume that the individual steps are sampled from a Gaussian distribution, the
probability of making one large jump is $\sim e^{-{\cal L}^{2}}$, whereas the
walker can get there using a series of smaller aligned steps with probability
$\sim e^{-{\cal L}}$. Therefore, the latter dominates the tail of the coarse
grained probability distribution. This shows that the most probable paths are
described by small fluctuations around a single classical deterministic
trajectory, which is associated with a novel saddle point solution.

A deeper understanding of such fluctuations can be gleaned from the Large
Deviation Principle (LDP) [47, 48]. Stated simply, the LDP is a scaling law of
the form $P_{N}\simeq e^{-NI}$, where $P_{N}$ is a probability distribution
parameterized by some large number $N$, and $I$ is a positive number called
the rate function. In the context of random walks $I\sim O(1)$ for large
fluctuations, which means $P_{N}\sim e^{-N}$ at the tail. As illustrated in
Fig. 1, the dominant contributions to this tail come from walks that resemble
a classical trajectory, which looks quite different from the usual zero mean
walks. This is the new saddle that the CLT (or EFT) can fail to capture. The
central goal of this work is to develop a precise map between the LDP
framework and the physics of Stochastic Inflation.222Although to our
knowledge, our paper is the first to make the connection between the LDP and
the physics of inflation, the LDP has been previously applied to questions in
observational cosmology [49, 50, 51, 52, 53, 54, 55, 56, 57].

There is a vast literature on large deviations that extends well beyond i.i.d.
random walks, including applications to a number of physical problems in
equilibrium and non-equilibrium statistical physics, _e.g._ see the review
article [58]. For our purposes, the language of the LDP will be useful for two
reasons: First, it makes precise the sense in which the physics leading to
large deviations requires a departure from the usual long distance description
in terms of the CLT and implies that one must appeal to the microscopic nature
of the walk. Second, the rate functions $I$ are often calculable in terms of a
novel saddle point approximation. Combining these two insights will allow us
to characterize the regime of validity of Stochastic Inflation, highlighting
the situations where it improves perturbation theory and why it ultimately
breaks down. We will demonstrate this concretely in the context of eternal
inflation and $\lambda\phi^{4}$ in a fixed de Sitter background. The LDP will
explain the behavior of the probability distribution of scalar fluctuations
calculated used the stochastic framework. We then extend this understanding to
models that generate primordial black holes.

This paper is organized as follows. We begin with a discussion of models that
have a vanishing potential. We first explore this scenario using random walks
in Sec. 2. This allows us to show how the CLT emerges from a coarse graining
procedure, and to both explain the LDP and apply it in a simple context. Using
this formalism, we can then understand the probability distribution for the
fluctuations of the inflaton, which is the topic of Sec. 3. We then turn on a
non-trivial potential for a random walk in Sec. 4, and explore the role of the
LDP when computing the probability distribution for this example. This is
exactly the framework we need to understand the behavior of massless scalar
field theory in a dS background in Sec. 5. We then apply the same techniques
to explore models whose goal is to generate primordial black holes in Sec. 6.
Finally, Sec. 7 provides our conclusions and a discussion of future
directions.

##  2 Random Walks and the Renormalization Group

Random walks offer a simple setting within which we can understand the
conceptual details of the present work. We will review the Central Limit
Theorem (CLT) and demonstrate how the long wavelength behavior of a wide class
of walks fall into the universality class modeled by a Gaussian distribution
[59]. The failure of the CLT for large fluctuations is exactly analogous to
the breakdown of EFT we discuss later in the paper. We can understand this
regime better with the Large Deviation Principle (LDP), as illustrated with
some simple examples worked out in detail below. This framework will
eventually allow us to gain insight into the evolution of the scalar
fluctuations of the inflaton from a new perspective.

###  2.1 Central Limit Theorem

Consider a one dimensional i.i.d. random walk. Starting from the origin, the
walker takes one step per discrete time interval, each with a displacement $x$
chosen independently from some ‘microscopic’ distribution333We use the word
‘distribution’ to mean the probability density function for a continuous
random variable, as well as the probability distribution for a discrete random
variable. In some texts this word applies exclusively to the latter, but we
make no such distinction in this work. $p(x)$ with finite moments. To
facilitate the discussion below, we will consider two specific examples:

$\displaystyle p_{f}(x)$
$\displaystyle=\frac{1}{2}\delta(|x|-1)\qquad\text{(Fixed step length
distribution)}\ ,$ (2.1a) $\displaystyle p_{g}(x)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}\qquad\text{(Gaussian
distribution)}\ .$ (2.1b)

Both of these distributions have mean $\langle x\rangle=0$ and variance
$\langle x^{2}\rangle-\langle x\rangle^{2}=1$. A walker governed by $p_{f}(x)$
can take a step with either $x=1$ or $x=-1$ at each turn. The Gaussian
walker’s probability for each step is governed by $p_{g}(x)$, so it is able to
take a step of any size, with the most probable values being $|x|\lesssim 1$.
Examples of a typical random walk generated by $p_{f}(x)$ and $p_{g}(x)$ are
given in Fig. 2. If we zoom in on each walk we can make out the difference: it
is evident that the red trajectory is borne of fixed size steps whereas the
blue one is not. However, at the macroscopic level the two walks look like
they could have been generated by the same $p(x)$, suggesting that the long
wavelength behaviors of both walkers have something in common.

Figure 2:  Random walks generated by a fixed step length distribution and a
Gaussian distribution. If we zoom in, the difference in step size is evident
from the shape of the trajectories, but at the macroscopic level it is
difficult to tell which $p(x)$ generated each walk.

We can make this intuition precise by studying the net displacement of a
random walker after $N$ steps, which we denote as

$X=\sum_{i=1}^{N}x_{i}\ .$ (2.2)

If each step is sampled from $p(x)$, the probability of finding the walker at
a distance $X$ from the origin after $N$ steps is

$P(X)=\int\prod_{i=1}^{N}{\rm
d}x_{i}\,p(x_{i})\,\delta\left(X-\sum_{i=1}^{N}x_{i}\right)\ .$ (2.3)

$P(X)$ can be evaluated by Fourier transforming the delta function,

$\displaystyle\delta\bigg{(}X-\sum_{i=1}^{N}x_{i}\bigg{)}=\int\frac{{\rm
d}k}{2\pi}e^{-ik\left(X-\sum_{i=1}^{N}x_{i}\right)}\ ,$ (2.4)

to obtain

$\displaystyle P(X)$ $\displaystyle=\int\frac{{\rm
d}k}{2\pi}\,e^{-ikX}\int\prod_{i=1}^{N}{\rm d}x_{i}\,p(x_{i})\hskip
0.8pte^{ikx_{i}}$ $\displaystyle=\int\frac{{\rm
d}k}{2\pi}\,e^{-ikX}\left\langle e^{ikx}\right\rangle^{N}\ .$ (2.5)

The second equality holds because we are assuming that the $x_{i}$ are i.i.d.
random variables, which means their expectation values are independent. The
quantity

$\left\langle e^{ikx}\right\rangle=\int{\rm d}x\,p(x)\hskip 0.8pte^{ikx}\ ,$
(2.6)

is called the characteristic function of $p(x)$. For example $\left\langle
e^{ikx}\right\rangle_{g}=e^{-k^{2}/2}$, if the steps are sampled from the
Gaussian $p_{g}(x)$ in Eq. 2.1b. Plugging this into Eq. 2.5 yields

$P_{g}(X)=(2\pi N)^{-1/2}e^{-\frac{X^{2}}{2N}}\ ,$ (2.7)

which is an exact result valid for all $X$.

We can repeat this exercise with $p_{f}(x)$; after $N$ steps, the walker will
be at $X$ with a probability

$P_{f}(X)=\frac{N!}{\left(\frac{N-X}{2}\right)!\left(\frac{N+X}{2}\right)!}\frac{1}{2^{N}}\
.$ (2.8)

In the large $N$ limit, we can use Stirling’s approximation
$n!\xrightarrow{n\to\infty}n^{n}e^{-n}$ to write

$P_{f}(X)\simeq e^{-NI_{f}(X/N)},$ (2.9)

with

$I_{f}(X/N)=\frac{1}{2}\left[\left(1-\frac{X}{N}\right)\ln\left(1-\frac{X}{N}\right)+\left(1+\frac{X}{N}\right)\ln\left(1+\frac{X}{N}\right)\right]\
.$ (2.10)

If $X\ll N$, we can Taylor expand $I_{f}(X)$ to find

$P_{f}(X)\simeq e^{-\frac{X^{2}}{2N}-\frac{X^{4}}{12N^{3}}+\cdots}\ .$ (2.11)

So long as we restrict ourselves to $X\lesssim\sqrt{N}$, the first term in the
exponent dominates, and we can approximate the distribution in Eq. 2.8 as a
Gaussian. In particular, $P_{f}(X)$ and $P_{g}(X)$ have the same behavior up
to $X\sim\sqrt{N}$. A plot of $X$ over time can be obtained by downsampling
the trajectories in Fig. 2 by a factor of $N$. From this perspective it is
clear that $P(X)$ captures the long wavelength dynamics of the walk, which is
the same at leading order for the fixed step length and the Gaussian walkers.

We can generalize these conclusions to a wide class of $p(x)$. Going back to
Eq. 2.6, the Taylor expansion in $k$ of the logarithm of the characteristic
function yields the cumulant expansion444The expression Eq. 2.12 may be more
familiar to some readers as the generating function for connected correlation
functions, which are $\langle x^{m}\rangle_{C}$ here. In that language, the
discussion above can be re-contextualized as 0+1 dimensional field theory.

$\log\left\langle
e^{ikx}\right\rangle=\sum_{m=1}^{\infty}\frac{(ik)^{m}}{m!}\langle
x^{m}\rangle_{C}\ .$ (2.12)

For instance, $\langle x\rangle_{C}\equiv\langle x\rangle$ and $\langle
x^{2}\rangle_{C}\equiv\langle x^{2}\rangle-\langle
x\rangle^{2}=\sigma_{0}^{2}$ is the variance of $p(x)$. We can exponentiate
the above expression to write Eq. 2.5 as

$P(X)=\int\frac{{\rm d}k}{2\pi}e^{-ikX}\exp\left(ikN\langle
x\rangle-\frac{1}{2}Nk^{2}\sigma_{0}^{2}+\frac{i^{3}}{3!}Nk^{3}\langle
x^{3}\rangle_{C}+O(k^{4})\right)\ .$ (2.13)

For observables where the Gaussian contribution dominates, the integrand has
support for $k\lesssim\frac{1}{\sqrt{N}\sigma_{0}}$. Therefore, the $m^{\rm
th}$ term in the cumulant expansion contributes to the exponent as

$Nk^{m}\langle x^{m}\rangle_{C}\sim N^{1-m/2}\langle
x^{m}\rangle_{C}\xrightarrow{N\to\infty}0\quad\text{for }m>2\ ,$ (2.14)

where $\langle x^{m}\rangle_{C}$ are independent of $N$ since the moments of
the distribution $p(x)$ are finite constants. Therefore, the first and second
terms of the exponent in Eq. 2.13 scale as $O(\sqrt{N})$ and $O(1)$
respectively and the remaining terms shrink as $O(1/\sqrt{N})$ or faster. This
behavior can be made manifest by rescaling $k\to k/\sqrt{N}$ and
$X\to\sqrt{N}X$ in Eq. 2.13 so that

$P(X)\to\frac{1}{\sqrt{N}}\int\frac{{\rm
d}k}{2\pi}\exp\left(ik(\sqrt{N}\langle
x\rangle-X)-\frac{1}{2}k^{2}\sigma_{0}^{2}+O\left(\frac{1}{\sqrt{N}}\right)\right)\simeq
e^{\frac{-(X-\sqrt{N}\langle x\rangle)^{2}}{2\sigma_{0}^{2}}},$ (2.15)

which is a Gaussian probability distribution centered at $\sqrt{N}\langle
x\rangle$ up to corrections that vanish for large $N$. This result is known as
the Central Limit Theorem.555The CLT is more general; it is not necessary for
the steps to be independent[60]. In that case Eq. 2.15 is still valid so long
as the $m^{\rm th}$ cumulant in Eq. 2.13 satisfies
$\sum_{i_{1},\cdots,i_{m}}^{N}\langle x_{i_{1}}\cdots x_{i_{m}}\rangle_{C}\ll
O(N^{m/2})$.

It is very useful to interpret this result in the language of Renormalization
Group (RG) evolution as applied to EFTs. In this context, one identifies a
power counting parameter which facilitates the use of dimensional analysis. In
the classic case of integrating out a heavy particle of mass $M$, the power
counting is determined by the small dimensionless number $E/M$, where $E\ll M$
is the typical energy associated with the process of interest. This allows one
to organize the local operator expansion of the EFT into terms which are
relevant (grow larger polynomially at lower energies), marginal (only evolve
at most logarithmically), and irrelevant (grow smaller polynomially at lower
energies).

We can see the same principles in action by viewing our random walk examples
through the lens of RG. If we organize the terms that appear in the exponent
of Eq. 2.15 by how they scale with $N$, then we see that the mean is a
relevant parameter, the variance is marginal, and the cumulants $\langle
x^{m>2}\rangle_{C}$ are irrelevant. Therefore, as $N\to\infty$, the
distribution is localized about the mean, and the Gaussian distribution
emerges as a universal fixed point of the RG evolution. The interpretation is
that coarse graining the distribution by zooming out (equivalently taking a
large number of steps) erases any detailed memory of the microscopic
distribution $p(x)$, beyond the gross features captured by its mean and
standard deviation. This is in exact analogy with EFTs, where only a small
number of parameters contribute significantly to low energy observables,
regardless of the detailed UV completion.

###  2.2 Large Deviation Principle

However, this is not the whole story. Returning to the examples introduced in
Eq. 2.1, let us consider the probability $P(X>N)$ of finding a random walker
at a distance farther than $N$ from the origin, after $N$ steps. The walker
taking fixed (unit) size steps has no hope of going beyond $N$ even if they
were to take all $N$ steps in the same direction, and therefore
$P_{f}(X>N)=0$. However, the Gaussian walker, with the same mean and standard
deviation, can have $P_{g}(X>N)\neq 0$. Evidently, some information about the
microscopic distribution $p(x)$ is encoded in the region $X\gtrsim N$, which
we refer to as the _tail_ of $P(X)$.

For both examples above, it is extremely unlikely that the walker makes it to
$X\sim N$ in $N$ steps, which implies $P(X\sim N)$ is very small. In order to
probe the tail, we need to devise an observable that would be sensitive to
such rare events. To this end, we can compute $\langle e^{\theta X}\rangle$
assuming $P_{g}(X)$ with $\theta>0$. Naively, we might think that $\langle
e^{\theta X}\rangle\sim e^{\theta O(\sqrt{N})}$ since $P_{g}(X)$ is dominated
by $X\lesssim\sqrt{N}$. However, we can perform the following computation:

$\displaystyle\langle e^{\theta X}\rangle=\langle e^{\theta
x}\rangle^{N}=e^{N\theta^{2}/2}\ ,$ (2.16)

which in fact scales as $e^{N}$. The explanation for the breakdown of the
naive intuition is simply due to the fact that $e^{\theta X}$ takes on very
large values with small probabilities, so that contributions from such values
cannot be ignored [61]. In other words, $\langle e^{\theta X}\rangle$ probes
the tail of $P(X)$.

We now introduce a new random variable

${\widetilde{X}}=\frac{1}{N}\sum_{i=1}^{N}x_{i}\equiv\frac{X}{N}\ ,$ (2.17)

called the _sample mean_ of the i.i.d. random variables $x_{i}$. Noting that
the distributions transform under a change of variables as
$P({\widetilde{X}}){\rm d}{\widetilde{X}}=P(X){\rm d}X$, we may rewrite
$P_{g}$ and $P_{f}$ for large $N$ as

$\displaystyle
P\big{(}{\widetilde{X}}\big{)}\simeq\exp\left(-NI({\widetilde{X}})\right)\ ,$
(2.18)

where $I$ is a positive quantity called the rate function that can be read off
from Eq. 2.7 and Eq. 2.9:

$\displaystyle I_{g}({\widetilde{X}})$
$\displaystyle=\frac{{\widetilde{X}}^{2}}{2}\ ,$ (2.19a) $\displaystyle
I_{f}({\widetilde{X}})$
$\displaystyle=\frac{1}{2}\big{[}\big{(}1-{\widetilde{X}}\big{)}\ln\big{(}1-{\widetilde{X}}\big{)}+\big{(}1+{\widetilde{X}}\big{)}\ln\big{(}1+{\widetilde{X}}\big{)}\big{]}\
.$ (2.19b)

A probability distribution $P_{N}$ that satisfies a scaling law of the form
$P_{N}\simeq e^{-NI}$ is said to obey the Large Deviation Principle (LDP).

Figure 3:  Central Limit Theorem vs. Large Deviation Principle: The
probability distributions of the sample mean ${\widetilde{X}}\equiv X/N$ for a
fixed step length and Gaussian random walk for $N=10$, computed using the LDP.
For values of ${\widetilde{X}}$ close to the mean both curves overlap, in
accordance with the CLT. However, they differ for large deviations from the
mean. In particular the tail of the distribution, shown magnified, reveals
that $P_{f}({\widetilde{X}})$ vanishes at ${\widetilde{X}}=1$ whereas
$P_{g}({\widetilde{X}})$ does not. The dashed lines are the rate functions for
each distribution.

The distributions of the sample mean described by Eq. 2.18 and Eqs. (2.19) are
examples of the LDP.

According to Cramér’s theorem, the distribution of a sample mean
${\widetilde{X}}$ of i.i.d. random variables satisfies a LDP with a rate
function $I({\widetilde{X}})$ given by

$I({\widetilde{X}})=\sup_{\theta}\left[\theta{\widetilde{X}}-\lambda(\theta)\right]\
,$ (2.20)

where $\lambda(\theta)$ is the cumulant generating function for each i.i.d.
variable666If the variables are not i.i.d. we should use
$\lambda(\theta)=\lim_{N\to\infty}\frac{1}{N}\ln\left[\langle e^{\theta
X}\rangle\right]$, the scaled cumulant generating function, in Eq. 2.20. This
generalization is called the Gärtner-Ellis theorem.

$\lambda(\theta){:=}\ln\big{\langle}e^{\theta x}\big{\rangle}.$ (2.21)

To see why this is plausible, let us assume that LDP holds for some sample
mean ${\widetilde{X}}$. Then, $P({\widetilde{X}})\simeq
e^{-NI({\widetilde{X}})}$ and

$\big{\langle}e^{\theta
X}\big{\rangle}=\big{\langle}e^{N\theta{\widetilde{X}}}\big{\rangle}\simeq\int{\rm
d}{\widetilde{X}}e^{N\left(\theta{\widetilde{X}}-I({\widetilde{X}})\right)}\simeq
e^{N\sup_{\widetilde{X}}\left[\theta{\widetilde{X}}-I({\widetilde{X}})\right]}.$
(2.22)

In the last step we have used the saddle point approximation to compute the
integral for large $N$. Noting that $\langle e^{\theta X}\rangle=\langle
e^{\theta x}\rangle^{N}\equiv e^{N\lambda(\theta)}$ for i.i.d. random
variables, we have

$\lambda(\theta)=\sup_{\widetilde{X}}\left[\theta{\widetilde{X}}-I({\widetilde{X}})\right].$
(2.23)

Then the rate function given in Eq. 2.20 follows from a Legendre transform of
this result.

As an example, we can work out the rate function $I_{f}({\widetilde{X}})$ for
the sample mean of the fixed step length walk using Crámer’s theorem. Starting
with $\langle e^{\theta x}\rangle=\cosh(\theta)$, we have

$I_{f}({\widetilde{X}})=\sup_{\theta}\left[\theta{\widetilde{X}}-\ln\cosh(\theta)\right].$
(2.24)

The supremum can be obtained using ordinary calculus: taking the first
derivative of the expression in brackets and setting it to zero gives
$\theta_{\text{max}}=\tanh^{-1}({\widetilde{X}})=\frac{1}{2}(\ln(1+{\widetilde{X}})-\ln(1-{\widetilde{X}}))$,
which is where the r.h.s is maximum. Substituting this $\theta_{\text{max}}$
into Eq. 2.24 and simplifying, we recover the rate
function777$I_{f}({\widetilde{X}})$ is only defined for
${\widetilde{X}}\in[-1,1]$ on the real number line, which means the
probability of finding ${\widetilde{X}}$ outside this interval is zero, as
discussed earlier. given in Eq. 2.19b. Finally, notice that the rate function
$I_{f}$ from Eq. 2.19b is just the negative of the entropy for a binary random
variable. We touch upon this fact in Sec. 4.1. A more general discussion is
given in [58].

The CLT arises from the LDP when the rate function $I({\widetilde{X}})$ is
convex and has a single global minimum (at say, ${\widetilde{X}}_{0}$). If
this is the case, we can Taylor expand $I({\widetilde{X}})$ around
${\widetilde{X}}_{0}$ to obtain

$P({\widetilde{X}})\simeq
e^{-\frac{1}{2}NI^{\prime\prime}({\widetilde{X}}_{0})({\widetilde{X}}-{\widetilde{X}}_{0})^{2}}\
.$ (2.25)

For small deviations of ${\widetilde{X}}$ from ${\widetilde{X}}_{0}$, this
quadratic expansion is a good approximation of $I({\widetilde{X}})$, and
therefore the CLT provides the same information as the LDP. On the other hand
large deviations of ${\widetilde{X}}$ are those values at which the rate
function deviates significantly from the quadratic approximation. The CLT does
not correctly describe such large fluctuations, and we need to rely on the LDP
instead.888That a random variable satisfies the CLT does not always imply that
the existence of a rate function with a quadratic minimum, see example 3.4 of
[58].

##  3 Eternal Inflation

Having set up the general ideas of the LDP, we now turn to our first
cosmological application. The connection derives from the fact that scalar
fluctuations during single-field inflation act locally like a 1-dimensional
random walk around a classical trajectory. For a typical path, the end of
inflation is determined by the classical evolution where the field distance
changes linearly in time, $\Delta\phi_{\rm classical}=\dot{\phi}\hskip
0.8ptt$. However, it is possible for quantum fluctuations of the scalar field
to work against the classical motion, giving rise to inflationary periods that
last significantly longer than the classical expectation. In fact, when the
amplitude of fluctuations is large enough, it is known that inflation never
ends everywhere [62, 63] in the universe and instead gives rise to a infinite
reheating volume [64], also known as eternal inflation. Remarkably, the
fluctuations responsible for eternal inflation are necessarily examples of
large deviations, as we will see in this section.

###  3.1 Review of Stochastic Inflation

The idea that inflation is essentially a random walk has a long history,
starting from nearly the inception of the subject [21, 23, 26]. The intuition
follows from considering the freeze out of modes as they cross the horizon, at
which point the quantum fluctuations of these modes begin to evolve
classically. In any small patch of the universe, the gradients of the field
redshift away and the process is effectively a random walk. (Of course
globally, there are correlations across super horizon scales, which is the one
of the main reasons we invoke inflation in the first place.) Specifically, as
long as the parameters of the inflationary model change slowly in time, the
fluctuations of each mode in any given patch of space follow the same
distribution as a random walk with i.i.d. variables, which is given the
interpretation of _noise_ generated by the Hubble temperature associated with
the horizon. The distribution of fluctuations is also sensitive to the
presence of a potential. In the random walk language, this is the analog of a
classical external force. So the picture is a competition between the noise
and this so-called _drift_. This idea was formalized in the framework of
Stochastic Inflation [20, 21], which is the statement that the probability
distribution for the scalar fluctuations obeys

$\frac{\partial}{\partial
t}P(\phi,t)=\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}P(\phi,t)+\frac{1}{3H}\frac{\partial}{\partial\phi}\big{[}V^{\prime}(\phi)P(\phi,t)\big{]}\
.$ (3.1)

Despite its intuitive appeal, the derivation of Stochastic Inflation from
quantum field theory, a full understanding of its domain of applicability, and
a framework for computing corrections to the formalism had long been elusive.

These puzzles have recently been solved by interpreting Stochastic Inflation
as arising from RG flow (or resumming logs) in quantum field theory [41, 42,
35, 43, 38, 44, 36, 39]. Concretely, by taking moments of Eq. 3.1, one can
relate mixing of operators under time-evolution to the stochastic equation. In
single-field inflation, the fluctuations of $\phi$ can be rewritten in terms
of the adiabatic metric fluctuation $\zeta$. However, $\zeta$ must respect the
single-field consistency conditions [65, 66, 67, 68], which are the
nonlinearly realized SO(4,1) symmetries that act on the metric leaving the
gauge fixed. For example, under the dilatation transformation in this group,
$\zeta$ transforms as
$\delta\zeta=-1-\vec{x}\cdot\vec{\partial}_{\vec{x}}\zeta$. The evolution of
operators under (dynamical) RG must respect these symmetries and restricts the
form of mixing to

$\frac{\partial}{\partial\texttt{t}}\zeta^{N}({\vec{x}},\texttt{t})=\sum^{N}_{n\geq
2}\gamma_{n}\left(\\!\\!\begin{array}[]{c}N\\\
n\end{array}\\!\\!\right)\zeta^{N-n}({\vec{x}},\texttt{t})\ ,$ (3.2)

where $\texttt{t}=H\hskip 0.8ptt$, and $\gamma_{n}$ are the “anomalous
dimensions” which govern the composite operator mixing; the $\gamma_{n}$ are
time-independent for scale invariant correlators. This implies the most
general form of single field Stochastic Inflation is999In order to go from the
dynamical RG for correlators to a Fokker-Planck equation for a probability
distribution $P(\zeta,t)$, one simply identifies
$\langle\zeta^{n}\rangle=\int{\rm d}\zeta P(\zeta,t)\zeta^{n}$.

$\frac{\partial}{\partial\texttt{t}}P(\zeta,\texttt{t})=\sum_{n\geq
2}(-1)^{n}\frac{\gamma_{n}}{n!}\frac{\partial^{n}}{\partial\zeta^{n}}P(\zeta,\texttt{t})\
.$ (3.3)

As discussed in [39], we can view this as the expansion of a general Markovian
process with transition amplitudes $W(\zeta|\zeta^{\prime})$, such that

$\displaystyle\frac{\partial}{\partial\texttt{t}}P(\zeta,\texttt{t})$
$\displaystyle=\int{\rm
d}\zeta^{\prime}\Big{[}P(\zeta^{\prime},\texttt{t})\hskip
0.8ptW(\zeta|\zeta^{\prime})-P(\zeta,\texttt{t})\hskip
0.8ptW(\zeta^{\prime}|\zeta)\Big{]}$ $\displaystyle=\int{\rm
d}\Delta\zeta\Big{[}P(\zeta-\Delta\zeta,\texttt{t})\hskip
0.8pt\widetilde{W}(\Delta\zeta)-P(\zeta,\texttt{t})\hskip
0.8pt\widetilde{W}(\Delta\zeta)\Big{]}\,.$ (3.4)

Here we used the shift symmetry, $\zeta\to\zeta+c$, to write the transition
amplitudes
$W(\zeta|\zeta^{\prime})=\widetilde{W}(\zeta-\zeta^{\prime}\equiv\Delta\zeta)$.
Taylor expanding $P(\zeta-\Delta\zeta,t)$ (_a.k.a_ performing a Kramers–Moyal
local expansion) reproduces Eq. 3.3, where

$\gamma_{n}\equiv\int{\rm
d}\Delta\zeta\,\big{(}\\!-\\!\Delta\zeta\big{)}^{n}\,\widetilde{W}(\Delta\zeta)\,.$
(3.5)

In this sense, we see that $\gamma_{n>2}$ corresponds to non-Gaussian
corrections to the transition amplitude, which is the same as the non-
Gaussianity of the probability for a step in a i.i.d. random walk.

The coefficients $\gamma_{n}$ are determined by computing the $n^{\text{th}}$
connected quantum field theory correlator through

$\langle\zeta^{n}({\vec{x}},\texttt{t})\rangle\supset\gamma_{n}\log aH\ .$
(3.6)

Explicit calculation shows that $\gamma_{1}=0$, which is a restatement of the
conservation of $\zeta$ outside the horizon. The quadratic term $\gamma_{2}$
is determined by the variance

$\langle\zeta^{2}({\vec{x}}=0)\rangle=\Delta_{\zeta}\int\frac{{\rm
d}^{3}k}{(2\pi)^{3}}\frac{1}{2k^{3}}=\frac{\Delta_{\zeta}}{4\pi^{2}}\log\frac{aH}{K_{\text{IR}}}\
,$ (3.7)

where $\Delta_{\zeta}=H^{4}/(2f_{\pi}^{4})$ sets the amplitude of the power
spectrum for $\zeta$, and we evaluated this integral by introducing a hard UV
cutoff $\Lambda=aH$ and an IR cutoff $K_{\text{IR}}$.101010While this is the
correct result, one might be concerned that this hard cutoff breaks spacetime
symmetry. For a discussion of a dimensional regularization-like regulator that
preserves the symmetries, see [38, 69]. Comparing Eq. 3.2 to Eq. 3.3, we see
that this term generates the noise term that appeared in the original
formulation of Stochastic Inflation, Eq. 3.1.

From the point of view of the quantum field theory correlators and the
resultant RG evolution, computing higher order corrections is completely
straightforward. Applying this approach to the EFT of inflation [70, 71], the
first non-trivial correction to the stochastic framework in single-field
inflation was found in [46]. Since $\zeta$ is derivatively coupled, we can
generically generate $\gamma_{n}$ by introducing an interaction of the form
$\dot{\zeta}^{n}/\Lambda^{4-n}$, where $\Lambda=f_{\pi}c_{s}$ is the
approximate UV cutoff111111When $c_{s}\to 1$, there are additional factors of
$(1-c_{s}^{2})$ so that $\Lambda\to\infty$ as $c_{s}\to 1$ in slow-roll
models. of the EFT of inflation when $c_{s}\ll 1$. Then by dimensional
analysis,

$\gamma_{n}=c_{n}\Delta_{\zeta}^{n/2}\left(\frac{H^{2}}{\Lambda^{2}}\right)^{2-n}\
,$ (3.8)

with $c_{n}={O}(1)$. This leads to the naive expectation that perturbation
theory should hold as long as one is working in the parameter space where
$H\ll\Lambda$. As was emphasized in [46], this is only true when the
observable of interest is insensitive to the tail of the probability
distribution. In the language of the LDP, these tails are dominated by a new
saddle point. The energy scale associated with the LDP saddle can be
significantly larger that $\Lambda$, signaling that one is sensitive to the
details of the UV completion, as in the case of the random walk examples
studied above.

###  3.2 Central Limit Theorem as a Resummation

We would now like to solve for the time evolution of $P(\zeta,t)$, assuming
$\zeta=\zeta_{0}$ at $t=0$. This will tell us the probability of different
possible values of $\zeta$, which should resemble a random walk. If the theory
is Gaussian, so that $\gamma_{n>2}=0$, then we are solving the heat equation

$\frac{\partial}{\partial\texttt{t}}P(\zeta,\texttt{t})=\frac{\sigma^{2}}{2}\frac{\partial^{2}}{\partial\zeta^{2}}P(\zeta,\texttt{t})\
,$ (3.9)

where $\sigma^{2}\equiv\gamma_{2}=\Delta_{\zeta}/(4\pi^{2})$ is the variance.
The solution to this equation is a Gaussian

$P_{\text{G}}(\zeta,\texttt{t};\zeta_{0})=\frac{1}{\sqrt{2\pi\sigma^{2}\texttt{t}}}e^{-(\zeta-\zeta_{0})^{2}/(2\sigma^{2}\texttt{t})}\
.$ (3.10)

We can use the Gaussian solution to construct the solution to Eq. 3.3 with
general $\gamma_{n}$, in terms of derivatives of $P_{\text{G}}$

$P(\zeta,\texttt{t};\zeta_{0})=\exp\left(\sum_{n>2}(-1)^{n}\,\frac{\gamma_{n}\texttt{t}}{n!}\frac{\partial^{n}}{\partial\zeta^{n}}\right)P_{\text{G}}(\zeta,\texttt{t};\zeta_{0})\
.$ (3.11)

Using this general form, we will show that the physics of the random walk is
reproduced by the solutions to this equation. First, let us consider the
behavior around the peak of the Gaussian solution where
$(\zeta-\zeta_{0})^{2}\simeq\sigma^{2}\texttt{t}$. If we expand the full
solution near the peak, we notice that as $\texttt{t}\to\infty$,

$\displaystyle\frac{\gamma_{n}\texttt{t}}{n!}\frac{\partial^{n}}{\partial\zeta^{n}}P_{\text{G}}(\zeta,\texttt{t};\zeta_{0})$
$\displaystyle\simeq\frac{\gamma_{n}\texttt{t}}{n!}\frac{(-1)^{n}(\zeta-\zeta_{0})^{n}}{(\sigma^{2}\texttt{t})^{n}}P_{\text{G}}(\zeta,\texttt{t};\zeta_{0})$
(3.12)
$\displaystyle={O}(\gamma_{n}\texttt{t}^{1-n/2}\sigma^{-n})P_{\text{G}}(\zeta,\texttt{t};\zeta_{0})\
.$ (3.13)

If we associated $\gamma_{n}$ with the $n^{\text{th}}$ cumulant of a random
walk, and $\texttt{t}\to N$ is the number of steps, then the suppression of
these terms precisely matches our expectations from the CLT as
$\texttt{t}\to\infty$, see Eq. 2.14.

Note that this implies that $\zeta\sim 1$ is under control for suitably large
t. In contrast, perturbative calculations of the probability distribution
using the Edgeworth series

$\displaystyle P(\zeta)\simeq\exp\left(-\int\frac{{\rm
d}^{3}k}{(2\pi)^{3}}\frac{\zeta({\vec{\scaleto{k}{7pt}}})\zeta(-{\vec{\scaleto{k}{7pt}}})}{2P(k)}+\int\frac{{\rm
d}^{3}k_{1}{\rm
d}^{3}k_{2}}{(2\pi)^{6}}\frac{B(k_{1},k_{2},k_{3})\zeta({\vec{\scaleto{k}{7pt}}}_{1})\zeta({\vec{\scaleto{k}{7pt}}}_{2})\zeta({\vec{\scaleto{k}{7pt}}}_{3})}{6P(k_{1})P(k_{2})P(k_{3})}+\ldots\right)\
$

where
${\vec{\scaleto{k}{7pt}}}_{3}=-{\vec{\scaleto{k}{7pt}}}_{1}-{\vec{\scaleto{k}{7pt}}}_{2}$,
breaks down for much smaller values of $\zeta$. This shows how Stochastic
Inflation improves the behavior of perturbation theory by resumming the
individual modes into a single random walk.

###  3.3 Large Deviations and the EFT of Inflation

Now let us consider the tail of the $P(\zeta,\texttt{t})$ distribution, where
$\zeta=\alpha\texttt{t}$ for some constant $\alpha$ in the limit
$\texttt{t}\to\infty$. The region $\alpha\geq 1$ corresponds to the regime of
eternal inflation, as the random fluctuations conspire to prevent the end of
inflation, even in the $\texttt{t}\to\infty$ limit. The transition at
$\alpha=1$ is where the quantum fluctuations exactly cancel the classical
evolution of the background field. Note that because the distance is linear in
t, rather than $\sqrt{\texttt{t}}$, we are considering a large deviation for
the probability distribution of $\zeta$.

It is straightforward to see that for these large deviations the CLT fails to
calculate dominant contribution to the tail, just as it did for the i.i.d.
random walk (see Fig. 2). Plugging $\zeta=\alpha\texttt{t}$ into Eq. 3.12, we
have

$\frac{\gamma_{n}\texttt{t}}{n!}\frac{\partial^{n}}{\partial\zeta^{n}}P_{\text{G}}(\zeta,\texttt{t};\zeta_{0})={O}(\gamma_{n}\texttt{t}\alpha^{n}\sigma^{-2n})P_{\text{G}}(\zeta,\texttt{t};\zeta_{0})\
.$ (3.14)

For $\alpha={O}(1)$, there is no suppression of the higher-order terms.
Concretely, the entire series in $\gamma_{n}$ will break down for sufficiently
large $\alpha$. For $\alpha=1$, this series will break down when
$\Lambda<f_{\pi}$, even though this parameter space is consistent with
condition that the EFT of inflation is weakly coupled at horizon crossing,
$\Lambda>H$ [46].

The breakdown of Stochastic Inflation is a precise reflection of what we found
in our analysis of large deviations for random walks. To see this more
clearly, we can write the solution to Stochastic Inflation in terms of the
Fourier transform of Eq. 3.11

$P(\zeta,\texttt{t})=\int_{-\infty}^{\infty}{\rm
d}k\,e^{-ik\zeta}\rho(k,\texttt{t})=C\int_{-\infty}^{\infty}{\rm
d}k\exp\left(-ik\zeta-k^{2}\frac{\sigma^{2}}{2}\texttt{t}-ik^{3}\frac{\gamma_{3}}{3!}\texttt{t}+\ldots\right)\
.$ (3.15)

We recognize this as precisely the result for a random walk we described
above, see Eq. 2.13. At the same time, we can identify

$\rho(k,\texttt{t})=C\exp\left(-k^{2}\frac{\sigma^{2}}{2}\texttt{t}+\sum_{n>2}(ik)^{n}\frac{\gamma_{n}}{n!}\texttt{t}\right)\equiv\langle\exp\left(ik\zeta\right)\rangle\
,$ (3.16)

so that when $\zeta=\texttt{t}\alpha$, we have $\rho(k,\texttt{t})\to
e^{\texttt{t}\lambda(\theta=ik)}$. Assuming $\texttt{t}\gg 1$, we can
calculate the integral over $k$ in Eq. 3.15 using the method of steepest
descents:

$P(\zeta=\alpha\texttt{t},\texttt{t})\simeq\exp\left(-\texttt{t}I(\alpha)\right)\
,$ (3.17)

where

$I(\alpha)=\left(-i\alpha
k_{\star}(\alpha)-k_{\star}^{2}(\alpha)\frac{\sigma^{2}}{2}+\sum_{n>2}\left(ik_{\star}(\alpha)\right)^{n}\frac{\gamma_{n}}{n!}\right)\
.$ (3.18)

The integrand has been expanded around $k=k_{\star}(\alpha)+\delta k$ defined
by a (complex) value of $k$ that is an extremum of the argument of the
exponential,

$\left(i\sum_{n\geq
2}\left(ik_{\star}(\alpha)\right)^{n}\frac{\gamma_{n+1}}{n!}\right)-k_{\star}\sigma^{2}=i\alpha\
.$ (3.19)

We see that using the method of steepest descents to calculate the inverse
Fourier transform is equivalent to using Cramér’s theorem, Eq. 2.20.
Furthermore, for large $\alpha$ the Gaussian solution,
$k_{\star}=(-i)\alpha/\sigma^{2}$, is a far from the true saddle as all the
terms in the $k_{\star}^{n}\gamma_{n}$ expansion will become equally
important. This is, of course, the Fourier transform of the result in Eq.
3.14.

When we applied the LDP to random walks in Sec. 2.2, it was clear that we
become sensitive to the microphysics. We would like to understand this
breakdown purely in terms of the EFT of Inflation. Concretely, the expansion
in $\gamma_{n}$ is under control at horizon crossing, which is the physical
energy scale where the fluctuations are produced. A natural guess is that
$\zeta\propto\texttt{t}$ behaves like a classical solution with
$\dot{\zeta}=H$ or $\dot{\phi}=f_{\pi}^{2}$, where $\phi$ is the inflaton. To
make sense of this, we can rewrite the evolution of $\zeta$ in terms of a
Langevin equation,

$\frac{{\rm d}}{{\rm d}\texttt{t}}\zeta(\texttt{t})=\xi(\texttt{t})\ ,$ (3.20)

where $\xi(\texttt{t})$ is a random variable that models a noise source.
Assuming that the noise is Gaussian, we have

$\langle\xi(\texttt{t})\xi(\texttt{t}^{\prime})\rangle=\sigma^{2}\delta(\texttt{t}-\texttt{t}^{\prime})\
.$ (3.21)

The probability of finding $\zeta=\zeta_{f}$ at $\texttt{t}=\texttt{t}_{f}$
given the initial condition $\zeta(\texttt{t}=0)=0$ is then

$P(\zeta_{f})=\int{\cal
D}\zeta(\texttt{t})\exp\left(-\int_{0}^{\texttt{t}_{f}}{\rm
d}\texttt{t}\,\frac{1}{2\sigma^{2}}\left(\frac{{\rm d}}{{\rm
d}\texttt{t}}\zeta(\texttt{t})\right)^{2}\right)\ .$ (3.22)

For large deviations, $\zeta=\alpha\texttt{t}$, and the probability is
determined by the saddle point $\frac{{\rm d}^{2}}{{\rm
d}\texttt{t}^{2}}\zeta=0$ or $\frac{{\rm d}}{{\rm d}\texttt{t}}\zeta=\alpha$,
so that

$P(\zeta_{f})\simeq\exp\left(-\texttt{t}\frac{\alpha^{2}}{2\sigma^{2}}\right)\
.$ (3.23)

When $\alpha={O}(1)$ (_i.e._ not ${O}(\texttt{t}^{-1})$) this satisfies the
LDP.

The key observation is that the probability of this large deviation is
determined by a classical solution where $\frac{{\rm d}}{{\rm
d}\texttt{t}}\zeta=\alpha$ (see Fig. 1). Translating this into the canonically
normalized field, $\zeta_{c}=\zeta f_{\pi}^{2}/H$, this is the condition that

$\frac{{\rm d}}{{\rm d}t}\zeta_{c}=\alpha f_{\pi}^{2}\ .$ (3.24)

The EFT of inflation is defined in terms of an expansion in
$\dot{\zeta}_{c}/\Lambda^{2}$ and therefore when $\dot{\zeta}_{c}>\Lambda$, we
cannot define these classical solutions within the EFT. Concretely, we can
modify the Langevin equation with nonlinear terms

$\left(1+\sum_{n>1}c_{n}\left(\frac{f_{\pi}^{2}}{\Lambda^{2}}\frac{{\rm
d}\zeta_{c}}{{\rm d}\texttt{t}}\right)^{n}\right)\frac{{\rm d}}{{\rm
d}\texttt{t}}\zeta_{c}(\texttt{t})=\xi(\texttt{t})\ ,$ (3.25)

where $c_{n}={O}(1)$ by the definition of $\gamma_{n}$ in terms of $\Lambda$
in Eq. 3.8. We can now calculate the probability distribution as before,

$P(\zeta_{f})=\int{\cal
D}\zeta_{c}(\texttt{t})\exp\left(-\int_{0}^{\texttt{t}_{f}}{\rm
d}\texttt{t}\,\frac{\left(\dot{\zeta}_{c}(\texttt{t})(1+\sum_{n>1}c_{n}\dot{\zeta}_{c}^{n}f_{\pi}^{2n}/\Lambda^{2n})\right)^{2}}{2\sigma^{2}}\right)\
.$ (3.26)

The saddle point is still $\ddot{\zeta}_{c}=0$, but we can see that the
probability distribution

$P(\zeta_{f})\simeq\exp\left(-\texttt{t}\frac{\alpha^{2}}{2\sigma^{2}}\left(1+\sum_{n>1}c_{n}\frac{\alpha^{n}f_{\pi}^{2n}}{\Lambda^{2n}}\right)^{2}\right)$
(3.27)

becomes ill-defined when $\alpha f_{\pi}^{2}>\Lambda^{2}$. It is also
noteworthy that the breakdown of EFT in this specific example is not
associated with the breakdown of Markovian dynamics, as higher time
derivatives vanish around the classical solution (see the discussion in Sec.
4.2). For single field inflation, it is a breakdown of the EFT of Inflation
itself, rather than SdSET, that is responsible for the ill-defined probability
distribution for sufficiently large deviations.

##  4 Random Walks with External Forces

In the previous sections, we focused on the application of the LDP to random
walks with no external deterministic forces. We argued that the late time
behavior for the typical fluctuations of these systems could be determined by
RG evolution. This analysis yielded the CLT, such that the resultant
probability distribution was a Gaussian with zero mean.

In this section, we will study the physics of a random walk that is driven by
an external deterministic force. This is easiest to understand in the case of
a constant force, which is equivalent to an i.i.d. random walk where the
average over steps is non-zero, $\langle x\rangle\neq 0$. Then from Eq. 2.15,
we see that the term proportional to the non-zero mean scales as $\sqrt{N}$,
and so this term grows as we take $N\to\infty$. In the RG language, this
implies that an external deterministic force has the effect of introducing a
relevant deformation into the theory.

###  4.1 Equilibrium Distributions

In the presence of confining forces, such as a potential with a local minimum,
we might expect to see ergodic behavior, such that the probability for being
at a given location at a fixed time approaches a time-independent
(equilibrium) distribution. Such behavior is also consistent with our
expectations from thermodynamics for large numbers of confined particles. In
fact, it turns out that the equilibrium distribution for this thermodynamic
system is itself a quantity that is calculable using the LDP. If we imagine
that a walk of length $N$ reaches equilibrium, then the probability of finding
the particle at location $y$ during the walk at a sufficiently large number of
steps, $1\ll n\leq N$, should simply follow the equilibrium distribution

$P(x_{n}=y)=P_{\rm eq}(y)\ .$ (4.1)

Now suppose that we calculate quantities averaged over the entire walk, such
as $e^{\theta X}$ where $X=\sum_{i}x_{i}$. Since we are averaging over the
locations of each step, assuming we have reached equilibrium, we have

$\bigg{\langle}\exp\bigg{(}\theta\sum_{i=1}^{N}x_{i}\bigg{)}\bigg{\rangle}\simeq\left(\int{\rm
d}y\,e^{\theta y}P_{\rm eq}(y)\right)^{N}=e^{N\lambda(\theta)}\ ,$ (4.2)

where $\lambda(\theta)$ is determined from the equilibrium distribution for a
single step in the walk. This is a qualitative argument that can be formalized
in terms of the eigenvalues of the transition amplitudes. The rate function
for the walk $X=\sum_{i=1}^{N}x_{i}$ again follows from Eq. 2.23 and Eq. 2.17,

$\lambda(\theta)=\sup_{\widetilde{X}}\left[\theta{\widetilde{X}}-I({\widetilde{X}})\right]\
.$ (4.3)

Importantly, in the limit $N\to\infty$, the probability for any average
quantity, $S=\sum_{i=1}^{n}f(x_{i})$, is just the $N^{\text{th}}$ power of
finding one particle with $f(x)=S/N$.

The appearance of an equilibrium quantity in the LDP calculation is not a
special feature of random walks, but is common to most statistical mechanics
problems [58]. In a precise sense, rate functions of the LDP are proportional
to the thermodynamic free energies for large numbers of particles. The overall
power of $N$ in the probability is just the familiar relationship between
extensive and intensive thermodynamic quantities. In fact, this connection was
already present when we calculate the rate function for the discrete walk in
Eq. 2.19b, which is (minus) the entropy associated with a binary random
variable.

This perspective helps explain why the LDP can be used calculate the
equilibrium probability distribution. This can be made concrete121212This
equation is formally meaningless, as the random variable is not
differentiable. One can formalize these results using the Itô or Stratonovich
prescriptions. This subtlety will not play an important role in our
discussion, see e.g. [37] for more details.  in terms of a Langevin equation:

$\dot{x}(t)=f\big{(}x(t),t\big{)}+\xi(t)\ ,$ (4.4)

where again $\xi(t)$ is a random variable that accounts for noise, and now
$f\big{(}x(t),t\big{)}$ is an external deterministic force. If we assume the
noise is Gaussian, then $\xi(t)$ obeys

$\big{\langle}\xi(t)\hskip
0.8pt\xi(t^{\prime})\big{\rangle}=\sigma^{2}\delta(t-t^{\prime})\ .$ (4.5)

In this case, the probability of a walk $x(t)$ is

$\displaystyle P\big{[}x(t)\big{]}$
$\displaystyle=C\exp\left(-\frac{1}{2\hskip 0.8pt\sigma^{2}}\int_{0}^{T}{\rm
d}t\hskip 0.8pt\big{(}\xi(t)\big{)}^{2}\right)$
$\displaystyle=C\exp\left(-\frac{1}{2\hskip 0.8pt\sigma^{2}}\int_{0}^{T}{\rm
d}t\hskip 0.8pt\big{(}\dot{x}(t)-f\big{(}x(t),t\big{)}\big{)}^{2}\right)\ ,$
(4.6)

where we used the equations of motion given in Eq. 4.4. One can confirm the
first line by taking functional derivatives with respect to $\xi(t)$ to
reproduce the two-point correlator in Eq. 4.5. For making future contact with
cosmology, we will assume the external force is due to a potential such that

$f\big{(}x(t),t\big{)}\to-V^{\prime}(x(t))\ ,$ (4.7)

where $V^{\prime}\equiv\partial_{x}V(x)$.

Now suppose we are given $x(0)=0$, and we want to integrate over all possible
paths to find the probability that $x(T)=L$. We can solve this problem using
the method of steepest descents. First, we must find the maximum likelihood
path, which is the same as finding the classical saddle for the ‘effective
action’

$I(L)=\frac{1}{2}\int_{0}^{T}{\rm d}t\hskip
0.8pt\big{(}\dot{x}+V^{\prime}\big{)}^{2}=\frac{1}{2}\int_{0}^{T}{\rm
d}t\hskip
0.8pt\big{(}\dot{x}^{2}+V^{\prime}{}^{2}\big{)}+\big{[}V(x)\big{]}\big{|}_{0}^{T}\
.$ (4.8)

From the equations of motion, we have

$\ddot{x}=V^{\prime\prime}V^{\prime}\quad\to\quad\frac{{\rm d}}{{\rm
d}t}\dot{x}^{2}=\dot{x}\frac{{\rm d}}{{\rm d}x}\big{(}V^{\prime
2}\big{)}\quad\to\quad\dot{x}^{2}=V^{\prime 2}\ .$ (4.9)

Using this result, we can write

$\frac{1}{2}\int_{0}^{T}{\rm d}t\hskip
0.8pt\big{(}\dot{x}^{2}+V^{\prime}{}^{2}\big{)}=\frac{1}{2}\int_{0}^{T}{\rm
d}t\,2\hskip 0.8pt\dot{x}\hskip 0.8ptV^{\prime}=\int_{0}^{T}{\rm
d}t\,\frac{{\rm d}}{{\rm d}t}V=\big{[}V(x)\big{]}\big{|}_{0}^{T}\ .$ (4.10)

Including the total-derivative term from Eq. 4.8, the probability of find
$x(T)=L$ is given in terms of

$I(L)=2\big{(}V(L)-V(0)\big{)}\qquad\text{with}\qquad
P(L)\simeq\exp(-I(L)/\sigma^{2})\ .$ (4.11)

Here our determination of the probability, $P(L)$, is a LDP result in the
sense that $I(L)$ is a rate function

$\lim_{\sigma\to 0}\sigma^{2}\log P(L)=-I(L)\ .$ (4.12)

In the same sense as for the path integral, it is the paths near the classical
solution that yield the dominant probability, while the contribution from the
fluctuations about the classical path determine the sub-leading terms in the
expansion with respect to $\sigma$.

This result matches the equilibrium probability distribution we derive from
the Fokker-Planck equation by setting ${\rm d}P/{\rm d}t=0$:

$\frac{{\rm d}}{{\rm
d}x}\big{(}V^{\prime}P(x)\big{)}+\frac{\sigma^{2}}{2}\frac{{\rm d}^{2}}{{\rm
d}x^{2}}P(x)=0\ .$ (4.13)

Integrating twice with respect to $x$ gives

$\log\frac{P(L)}{P(0)}=-\frac{2}{\sigma^{2}}\big{(}V(L)-V(0)\big{)}\ .$ (4.14)

We see that the LDP reproduces the equilibrium distribution that is predicted
by the Fokker-Planck equation.

###  4.2 Markovian Evolution

Within this framework, we can also easily understand the role of Markovian
evolution when assessing the validity of the calculation of the equilibrium
solution. Markovian refers to a class of theories where the next time step is
fully determined by the state of the system at the previous time step. In
terms of differential equations, Markovian evolution therefore is equivalent
to the statement that we have a first-order (in time) equation of motion. If
there were higher derivatives, then one would need to know about the state of
the velocity field along with the state of the system itself to determine the
next step in the evolution [72].

We can therefore model non-Markovian evolution by adding a small acceleration
term to our equations of motion given in Eq. 4.4:

$\vartheta\ddot{x}(t)+\dot{x}(t)=-V^{\prime}\big{(}x(t)\big{)}+\xi(t)\ ,$
(4.15)

so that the dynamics are Markovian in the $\vartheta\to 0$ limit. Repeating
our previous calculation, we find an effective action

$\displaystyle I(L)$ $\displaystyle=\frac{1}{2}\int_{0}^{T}{\rm
d}t\big{(}\vartheta\hskip 0.8pt\ddot{x}(t)+\dot{x}+V^{\prime}\big{)}^{2}$
$\displaystyle=\int_{0}^{T}{\rm d}t\big{(}\vartheta^{2}\hskip
0.8pt\ddot{x}^{2}+\dot{x}^{2}\big{(}1-\vartheta
V^{\prime\prime}\big{)}+V^{\prime}{}^{2}\big{)}+\big{[}V(x)+2\hskip
0.8pt\vartheta\hskip 0.8pt\dot{x}\hskip 0.8ptV^{\prime}(x)+\vartheta\hskip
0.8pt\dot{x}\big{]}\big{|}_{0}^{T}\ ,$ (4.16)

so that $P(L)=\exp\big{(}-I(L)/\sigma^{2}\big{)}$. There are two new non-
Markovian terms that contribute to the action, which are small corrections
when

$\displaystyle\vartheta^{2}\ddot{x}^{2}=\vartheta^{2}(V^{\prime\prime}V^{\prime})^{2}$
$\displaystyle\ll\dot{x}^{2}=V^{\prime 2}$ (4.17a) $\displaystyle\vartheta
V^{\prime\prime}$ $\displaystyle\ll 1\ ,$ (4.17b)

where the first and second lines correspond the first and second
$\vartheta$-dependent terms in Eq. 4.16, and the equalities in Eq. 4.17a are
due to the Eq. 4.9. Notice that both terms remain small when we impose the
condition $\vartheta V^{\prime\prime}\ll 1$ everywhere along the path. If we
enforce Eqs. (4.17), then the effective action is first order in time, and
hence the evolution is Markovian.

##  5 Light Scalar Fields in de Sitter

Light scalar fields in de Sitter with non-trivial potentials present an
additional complication beyond single-field inflation. The stochastic
framework applied to these models is known to give rise to a non-Gaussian
equilibrium probability distribution, acting as a kind of non-trivial fixed
point of the dynamical RG. This presents a vastly different situation,
compared to single-field inflation, where interactions are negligible for
typical fluctuations due to the CLT. This section will show how the dynamics
of these models can be mapped onto the language of random walks with external
forces that we developed in the previous section.

###  5.1 Effective Potentials and Markovian Dynamics

Stochastic Inflation provides a compelling framework with which to understand
the dynamics of light scalar fields in dS. In its original form, it describes
the probability distribution for an interacting scalar in dS, via the Fokker-
Planck equation

$\frac{\partial}{\partial
t}P(\phi,t)=\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}P(\phi,t)+\frac{1}{3H}\frac{\partial}{\partial\phi}\big{[}V^{\prime}(\phi)P(\phi,t)\big{]}\
.$ (5.1)

The equilibrium probability distribution is given by

$P_{\rm eq}(\phi)\simeq\exp\left(-\frac{8\pi^{2}V(\phi)}{3H^{4}}\right)\ .$
(5.2)

If we take $V(\phi)=\lambda\phi^{4}/4!$, the most likely field values are
$|\phi|\lesssim H\lambda^{-1/4}$. In light of the discussion of non-Gaussian
noise in Sec. 3, one would naturally wonder about the regime of validity and
corrections to this formula.

It is useful to discuss Stochastic Inflation and its corrections in the
context of SdSET. The relationship between the variables of SdSET and scalar
field theory in dS can be understood from the SdSET ansatz for a free massless
scalar:

$\phi({\vec{\scaleto{k}{7pt}}},\texttt{t})=H\left(\varphi_{+}({\vec{\scaleto{k}{7pt}}},\texttt{t})+\varphi_{-}({\vec{\scaleto{k}{7pt}}},\texttt{t})[a(\texttt{t})H]^{-3}\right)\
.$ (5.3)

Here we have rewritten the scalar in terms of the two power law solutions as
$k\to 0$, where $\varphi_{+}$ is the constant (or growing) mode and
$\varphi_{-}$ is the decaying mode.

In SdSET, Stochastic Inflation is a consequence of Callan-Symanzik-like
equations for the dynamical RG of the $\varphi_{+}^{n}$ operators. This
information can be rewritten as a master equation for the probability
distribution $P(\varphi_{+},\texttt{t})$, which at lowest order reproduces Eq.
5.1, while also containing an infinite series of corrections:

$\displaystyle\frac{\partial}{\partial\texttt{t}}P(\varphi_{+},\texttt{t})$
$\displaystyle=\frac{1}{3}\frac{\partial}{\partial\varphi_{+}}\left[V^{\prime}_{\rm
eff}(\varphi_{+})P(\varphi_{+},\texttt{t})\right]$ $\displaystyle\hskip
14.0pt+\frac{\partial^{2}}{\partial\varphi_{+}^{2}}\left[\sum_{m=0}^{\infty}\frac{b_{m}}{2!}\varphi_{+}^{2m}P(\varphi_{+},\texttt{t})\right]+\frac{\partial^{3}}{\partial\varphi_{+}^{3}}\left(\varphi_{+}\sum_{m=0}^{\infty}\frac{d_{m}}{3!}\varphi_{+}^{2m}P(\varphi_{+},\texttt{t})\right)$
$\displaystyle\hskip
14.0pt+\frac{\partial^{4}}{\partial\varphi_{+}^{4}}\left(\sum_{m=0}^{\infty}\frac{e_{m}}{4!}\varphi_{+}^{2m}P(\varphi_{+},\texttt{t})\right)+\ldots\
.$ (5.4)

For the UV example of a massless scalar with $V(\phi)=\lambda\phi^{4}/4!$, the
leading corrections (as defined below) were calculated in [39], resulting in

$\displaystyle\frac{\partial}{\partial\texttt{t}}P({\bar{\varphi}}_{+},\texttt{t})$
$\displaystyle=\frac{1}{3}\frac{\partial}{\partial{\bar{\varphi}}_{+}}\big{[}V^{\prime}_{\rm
eff}({\bar{\varphi}}_{+})P({\bar{\varphi}}_{+},\texttt{t})\big{]}+\frac{1}{8\pi^{2}}\frac{\partial^{2}}{\partial{\bar{\varphi}}_{+}^{2}}P({\bar{\varphi}}_{+},\texttt{t})$
$\displaystyle\hskip 14.0pt+\frac{\lambda_{\rm
eff}}{1152\pi^{2}}\frac{\partial^{3}}{\partial{\bar{\varphi}}_{+}^{3}}\big{(}{\bar{\varphi}}_{+}P(b{\bar{\varphi}}_{+},\texttt{t})\big{)}\
,$ (5.5)

with

$\displaystyle V^{\prime}_{\rm eff}$ $\displaystyle=\frac{\lambda_{\rm
eff}}{3!}\bigg{(}{\bar{\varphi}}_{+}^{3}+\frac{\lambda_{\rm
eff}}{18}{\bar{\varphi}}_{+}^{5}+\frac{\lambda_{\rm
eff}^{2}}{162}{\bar{\varphi}}_{+}^{7}+\,...\,\bigg{)}\ ,$ (5.6)

where we redefined

$\lambda_{\rm
eff}=\lambda+18b_{2}\qquad\text{and}\qquad\varphi_{+}={\bar{\varphi}}_{+}+\frac{b_{1}}{6b_{0}}{\bar{\varphi}}_{+}^{3}$
(5.7)

to remove the scheme-dependent corrections $b_{1}={O}(\lambda)$ and
$b_{2}={O}(\lambda^{2})$.

First, let us establish in what sense these are small corrections to the
original Fokker-Planck equation. If we ignore the corrections to the evolution
Eq. 5.5 and the potential Eq. 5.6, so that $V_{\rm
eff}(\varphi_{+})=\lambda\varphi_{+}^{4}/4!$, then the equilibrium solution is
the same as in Eq. 4.14:

$P^{\rm LO}_{\rm eq}(\varphi_{+})=\exp\left(-\frac{2V_{\rm
eff}(\varphi_{+})}{3\sigma^{2}}\right)=\exp\left(-\frac{\pi^{2}\lambda\varphi_{+}^{4}}{9}\right)\
,$ (5.8)

where we substituted $\sigma^{2}=\gamma_{2}=(4\pi)^{-1}$ and $V^{\prime}\to
V^{\prime}_{\rm eff}/3$ to match Eq. 5.5. Notice that the typical fluctuations
reside in the region $|\varphi_{+}|\lesssim\lambda^{-1/4}$. We can determine
the scaling behavior of the solutions using $\varphi_{+}\sim\lambda^{-1/4}$
such that the corrections to $V_{\rm eff}(\varphi_{+})$ are ${O}(\lambda_{\rm
eff}^{1/2})$ and ${O}(\lambda_{\rm eff})$, which we will call next-to leading
order (NLO) and next-to-next-to leading order (NNLO) respectively. The cubic-
derivative term, on the second line of Eq. 5.5, is similarly NNLO. By the same
$\lambda$-scaling argument, the equilibrium solution can be written as $P_{\rm
eq}=CP_{\rm LO}(\varphi_{+})P_{\rm NLO}(\varphi_{+})P_{\rm NNLO}(\varphi_{+})$
with

$\displaystyle P_{\rm LO}$
$\displaystyle=\exp\left(-\frac{\pi^{2}}{9}\lambda_{\rm
eff}\varphi_{+}^{4}\right)$ (5.9a) $\displaystyle P_{\rm NLO}$
$\displaystyle=\exp\left(-\frac{\pi^{2}}{243}\lambda_{\rm
eff}^{2}\varphi_{+}^{6}\right)$ (5.9b) $\displaystyle P_{\rm NNLO}$
$\displaystyle=\exp\left(\frac{5}{10368}\lambda_{\rm
eff}^{2}\varphi_{+}^{4}-\frac{17\pi^{2}}{46656}\lambda_{\rm
eff}^{3}\varphi_{+}^{8}\right)\,.$ (5.9c)

The terms in $P_{\text{NNLO}}$ come with different powers of $\varphi_{+}$ but
have the same $\lambda_{\rm eff}$ counting for typical fluctuations.
Importantly, the second term, which is ${O}(\varphi_{+}^{8})$, receives
contributions from both the change to $V_{\rm eff}(\varphi_{+})$ and the
higher derivative term.

Given that our UV theory only has a marginal coupling, $\lambda\phi^{4}$, it
is not obvious that there should be a breakdown of the stochastic framework
akin to what happened for the EFT of Inflation in Sec. 3. Furthermore, we saw
in Sec. 4 that the equilibrium distribution is itself a result of the LDP, and
therefore it is not a given that the framework could break down.

However, as we saw in Sec. 4.2, a critical assumption for the validity of the
stochastic framework is that the evolution is Markovian. Therefore, the
stochastic description can fail when the acceleration terms become important.
The condition for non-Markovian terms to be negligible was given in Eqs.
(4.17) above. In the language of SdSET, non-Markovian evolution would arise
from nontrivial mixing between $\varphi_{+}$ and $\varphi_{-}$, defined in Eq.
5.3.

Evaluating these conditions requires that we identify the parameter
$\vartheta$. For a light scalar field $\phi$ in dS, the equations of motion in
the limit that ${\vec{\scaleto{k}{7pt}}}\to 0$ are

$\ddot{\phi}(t)+3H\dot{\phi}(t)=-V^{\prime}(\phi)\ .$ (5.10)

In terms of dimensionless time $\texttt{t}=Ht$, we would therefore131313The
non-Markovian form of Stochastic Inflation has been use in the literature,
where $\vartheta=1/3$ as stated, e.g. [73, 74, 75]. It would be interesting to
understand these non-Markovian terms in SdSET where the higher derivative
corrections have been computed.  expect $\vartheta\simeq 1/3$. Assuming that
$\vartheta^{-1}=O(1)$, Eq. 4.17b implies that we should worry that the
evolution becomes non-Markovian when

$V^{\prime\prime}_{\rm eff}={O}(1)\quad\to\quad\lambda\varphi_{+}^{2}={O}(1)\
.$ (5.11)

Using the explicit form of the corrections provided in Eq. 5.9, we see that
this is precisely where our expansion in powers of $\lambda$ breaks down:

$P_{\rm LO}\left(\varphi_{+}\sim\lambda^{-1/2}\right)\simeq P_{\rm
NLO}\left(\varphi_{+}\sim\lambda^{-1/2}\,\right)\simeq P_{\rm
NNLO}\left(\varphi_{+}\sim\lambda^{-1/2}\,\right)\sim\exp\big{(}-\lambda^{-1}\big{)}\
.$ (5.12)

Similarly, this is the scale where the infinite series of corrections to the
effective potential in Eq. 5.6 become equally important. It is therefore
natural to conclude that the breakdown in our perturbative expansion in
$\lambda$ when $\varphi_{+}>\lambda^{-1/2}$ is due to the failure of the
Markovian assumption.

###  5.2 Light Scalars in de Sitter with Derivative Interactions

We argued in Sec. 3.3 that the breakdown of Stochastic Inflation for large
fluctuations in single-field inflation was associated with the breakdown of
the EFT of inflation. In this section, we will expore if a similar breakdown
occurs for the equilibrium distribution of a scalar field $\phi$ described by
an EFT with higher-derivative interactions in addition to a potential.

We will take our action for our scalar to be an EFT that includes arbitrarily
high powers of derivatives141414This is not to be confused with SdSET, which
only describes the long wavelength modes in dS. to take the form

$S=\int{\rm
d}^{4}x\,\sqrt{-g}\left(\frac{1}{2}\partial_{\mu}\phi\partial_{\mu}\phi-V(\phi)+\sum_{n>1}\frac{y_{n}}{\Lambda^{4(n-1)}}(\partial_{\mu}\phi\partial_{\mu}\phi)^{n}+\ldots\right)\
,$ (5.13)

where $\Lambda$ is the UV cutoff of the EFT and the $\ldots$ include operators
with more than one derivative per field. This description is under control in
de Sitter when $\Lambda\gg H$. We will again take $V(\phi)=\lambda\phi^{4}/4!$
such that $\phi$ is massless and its growing mode $\varphi_{+}$ will evolve at
zeroth order in $y_{n}$ according to $\lambda$-corrected equations of
Stochastic Inflation given in Eqs. 5.5 and 5.6.

The impact of the higher-derivative couplings $y_{n}$ on Stochastic Inflation
is nearly identical to the corrections in single-field inflation. The leading
corrections in $y_{n}$ survive the $\lambda\to 0$ limit and therefore can be
determined independent of the potential. In this limit, $\phi$ has a shift
symmetry $\phi\to\phi+c$. Repeating the argument used for single field
inflation in Sec. 3, one finds corrections

$\frac{\partial}{\partial\texttt{t}}P(\varphi_{+},\texttt{t})\supset\sum_{n>1}\frac{\gamma_{n}}{(2n)!}\frac{\partial^{2n}}{\partial\varphi_{+}^{2n}}P(\varphi_{+},\texttt{t})\
,$ (5.14)

where

$\gamma_{n}\propto y_{n}\left(\frac{H}{\Lambda}\right)^{4(n-1)}\ ,$ (5.15)

at leading order in $y_{n}$.

The presence of higher-derivative terms in the scalar EFT introduces an
infinite series of derivatives in the effective Fokker-Planck equation. For
this to be under control, we expect that the equilibrium solution with
$\gamma_{n>2}=0$ should be corrected by an expansion in powers of a small
parameter. If we write $P(\varphi_{+},\texttt{t})=P^{\text{eq}}_{\rm
LO}(\varphi_{+})Q(\varphi_{+},\texttt{t})$, then in the limit
$\varphi_{+}\gg\lambda^{-1/4}$, Eq. 5.14 becomes

$\frac{\partial}{\partial\texttt{t}}\log
Q\simeq\sum_{n>2}\frac{\gamma_{n}}{(2n)!}\left(-\frac{2V_{\rm
eff}^{\prime}(\varphi_{+})}{3\sigma^{2}}\right)^{2n}\ ,$ (5.16)

where we used Eq. 5.2 for $P_{\text{LO}}^{\text{eq}}(\varphi_{+})$. For
$y_{n}={O}(1)$, this series is under control when

$\frac{8\pi^{2}H^{2}}{3\Lambda^{2}}V^{\prime}_{\rm eff}(\varphi_{+})\ll 1\ .$
(5.17)

Taking $V_{\rm eff}\simeq\lambda\varphi_{+}^{4}$, this tells us that the
equilibrium solution is under control for
$H\varphi_{+}\lesssim\Lambda^{2}/(\lambda H)$, which is parametrically larger
than $\Lambda$. We can make sense of the regime of validity of this result
using Eq. 4.9 to relate

$V^{\prime}_{\rm eff}(\varphi_{+})=|{\dot{\varphi}}_{+}|\ ,$ (5.18)

along the classical trajectory. Now using $\phi\simeq H\varphi_{+}$ and
$\texttt{t}=Ht$, we can rewrite this condition for the expansion to be under
control:

$\frac{|\dot{\phi}|}{\Lambda^{2}}\ll 1\ .$ (5.19)

We see that the breakdown is precisely where we would expect from the
derivative expansion of the microscopic theory. Critically, the derivation of
this breakdown required knowledge of the LDP to see that the equilibrium
distribution could be derived from a classical saddle, in the regime where
${\dot{\varphi}}_{+}$ was larger than allowed by the UV cutoff of the EFT of
inflation.

##  6 Models of Primordial Black Hole Generation

The impact of rare fluctuations is particularly important for models of
primordial black hole (PBH) generation. The PBHs are formed from order-one
fluctuations directly in the primordial distribution and, as such, the
fluctuations are exponentially unlikely for scale-invariant Gaussian random
fields. Models for the generation of PBHs therefore exploit breakdowns of both
scale-invariance and Gaussianity, see _e.g._ [76, 77] for recent reviews.

The typical approach to estimating the abundance of PBHs follows from the
critical collapse model [76]. In the conventional description, one takes a
smoothed density field,

$\delta_{R}({\vec{x}})=\int{\rm d}^{3}k\hskip
0.8pte^{-i{\vec{\scaleto{k}{7pt}}}\cdot{\vec{x}}\,}W(kR)\hskip
0.8pt\delta({\vec{\scaleto{k}{7pt}}})\ ,$ (6.1)

where $W(\tilde{k})$ is a filter that removes power on scales $\tilde{k}\gg
1$, and $R$ is some distance scale. The critical collapse model assumes that
any region where $\delta_{R}>\delta_{\rm cr}$, for some constant threshold
$\delta_{\rm cr}={O}(1)$, will form a collapsed object with a total mass
determined by the size of the region $R$.

Within this framework, the abundance of primordial black holes is determined
by the probability of finding $\delta_{R}>\delta_{\rm cr}={O}(1)$, where the
precise value of $\delta_{\rm cr}$ is model-dependent. This threshold can be
also be written as a critical value of $\zeta_{R}({\vec{x}})$ [78, 79],
$\zeta_{\rm cr}$, defined as in Eq. 6.1 such that the probability of finding
$\zeta_{R}({\vec{x}})>\zeta_{\rm cr}$ determines that production of PBHs. For
concreteness, a value of $\zeta_{\rm cr}=0.1-0.2$ arises in some analytic
collapse models in radiation domination [79]. In comparison with the LDP, note
that the relevant time-scale for the random walk is the number of e-folds of
inflation after horizon crossing for a model with $kR\sim 1$, or
$N_{e}(R)=\log Ra(t_{\rm end})H$, where $t_{\rm end}$ is the time when
inflation ends. For scales $N_{e}(R)={O}(10)$, fluctuations above the
$\zeta_{\rm cr}$ threshold would correspond to $\alpha\gtrsim 10^{-2}$ using
the parameterization of large deviations described in Sec. 3.3. In models of
inflation consistent with observations, these values of $\alpha$ may still lie
outside the domain of the EFT of inflation.

Non-Gaussian tails arise in a variety of contexts, included single- and multi-
field inflation. In light of the connection between tails of distributions and
the LDP explored in this paper, we would like to understand when such large
non-Gaussian contributions can be calculated reliably given only an effective
description at horizon crossing. We will argue that framing these questions in
the language of RG for a random walk provides useful intuition.

###  6.1 Non-Gaussian Tails

We have explained how the stochastic approach to inflation translates the
problem of finding the distribution of scalar fluctuations onto characterizing
the behavior of a random walk. The CLT tells us that the Gaussian probability
distribution is a fixed point of the conventional random walk. Just as with RG
flows in quantum field theories, we can classify the deformations that could
produce a non-Gaussian tail, in analogy with Sec. 2.1, into three types:
relevant, marginal, or irrelevant. Relevant: A non-zero mean, _e.g._ due to a
deterministic force, takes us away from the Gaussian fixed point of the CLT.
For inflation, this corresponds to a potential $V(\phi)$ such that the
equilibrium probability distribution takes the form of Eq. 4.14, namely

$P(\phi)=e^{-2V(\phi)/2\sigma^{2}}\ .$ (6.2)

If the potential includes any operators other than a mass term, this
distribution is non-Gaussian. Yet, since it is due to the presence of the
unique relevant deformation, a large deviation from Gaussianity does not
indicate a breakdown of the effective description.

In practice, non-trivial production rates for PBHs require some more
complicated and possibly non-analytic potential $V(\phi)$. Some UV models may
motivate particular non-perturbative shapes for $V(\phi)$, but in practice the
formation of PBHs has mostly been explored using phenomenological models for
the inflationary potential, see  _e.g._ [80, 81, 82]. Marginal: A marginal
deformation of a random walk corresponds to changing the covariance matrix
that governs the steps in the walk. In the context of inflation, this means
changing the amplitude of scalar fluctuations, $P_{\zeta}(k)$, or mixing the
inflaton with additional fields. The former is a common strategy but only
leads to enhanced Gaussian tails. Mixing with additional fields can give rise
to non-Gaussian tails in a variety of ways.

A canonical example of models that use mixing are the curvaton or modulated
reheating scenarios, where the late time adiabatic mode is determined by a
spectator field $\chi$, so that

$\zeta\simeq F(\chi)\to\langle e^{J\zeta}\rangle=\langle e^{JF(\chi)}\rangle\
,$ (6.3)

for some model-dependent function $F$. As a concrete example, suppose our
spectator field has a potential $V(\chi)=\lambda\phi^{4}$ so that at leading
order we have

$P(\chi)=\exp\left(-\frac{\pi^{2}\lambda\chi^{4}}{9}\right)\ .$ (6.4)

Now suppose that by some process after inflation, the adiabatic mode is
determined by $\zeta=\kappa\chi^{3}$ for some constant $\kappa$. By the change
of variables (integrating over $\chi$ subject to the mixing with $\zeta$), we
have

$P(\zeta)=\exp\left(-\frac{\pi^{2}\lambda|\zeta|^{4/3}}{9\kappa^{4/3}}\right)\
.$ (6.5)

In this way, we can produce non-analytic behavior in the tail from otherwise
local interactions. Of course, this assumes that the functions $V(\chi)$ and
$F(\chi)$ are known exactly, when in fact they are themselves expansions in
$\chi$. For a given model, one must check the self-consistency of truncating
these expansions, including the corrections to $V_{\rm eff}(\chi)$ discussed
in Sec. 5.

The derivation of Eq. 6.3, while a trivial restatement of the mixing, has an
important interpretation in the context of the LDP. In the LDP literature,
this change of variables is known as the correspondence principle, which says
that when $\zeta=F(\chi)$ and the rate function for $\chi$ is known, then the
rate function for $\zeta$ is given by

$I(\zeta)=\inf_{\chi:F(\chi)=\zeta}\,\tilde{I}_{\chi}(\chi)\ ,$ (6.6)

where $\tilde{I}(\chi)$ is the rate function that determines the large
deviations of the $\chi$ field.

These types of probability distributions can arise from interactions that mix
the adiabatic and isocurvature modes during or after inflation [83, 84, 80,
85, 86, 87, 74, 88, 89, 90]. The probability distributions found in these
examples match the discussion given here as they are well-described by the
LDP. In SdSET, one can remove these types of mixing interactions via a field
redefinition, which effectively introduces a transformation of the form Eq.
6.3 on the observable fluctuations.

Irrelevant: Non-Gaussian noise is an irrelevant perturbation of a random walk.
The CLT ensures that even a highly non-Gaussian probability distribution will
produce a Gaussian distribution for the total distance of the walk. We saw
that this is not true for large deviations, which lie outside the regime of
the CLT, but also require exact knowledge of the non-Gaussian probability
distribution.

It is tempting to use non-Gaussian statistics for quantum fluctuations as a
mechanism to produce PBHs. However, as discussed in Sec. 3 (and [46]), when
the non-Gaussian terms in Stochastic Inflation become important, both the
stochastic framework and the EFT of inflation are breaking down. In principle
one can use the LDP techniques to calculate the rate of functions within the
microscopic model of inflation; however, given that the stochastic framework
does not apply to the microscopic theory, one must go beyond the classical
probabilistic description we have used here.

###  6.2 Relation to Factorial Enhancement

The increased probability distribution for large fluctuations has been tied to
the factorial enhancement of higher-order correlators in a number of examples
[91, 92, 85, 93]. Concretely, if one is calculating the correlators of some
field $\chi$ as a perturbative expansion in a parameter $\kappa\ll 1$, then
the $M$-point correlators are factorial enhanced when [93]

$\langle\chi^{M}...\rangle\propto\kappa^{m}M!\propto
M!\,\langle\chi^{m}\rangle^{M/m}\ .$ (6.7)

This scaling with $M$ is significant as it implies that there is more
information encoded in the largest $M$-point functions.

The factorial enhancement can be understood more directly from the generating
function $W[J]$, which is defined by the partition function with a source $J$
as151515The notation in this section follows [93]. It can be related to the
symbols introduced in Sec. 2.2 via the following map:
$J\to\theta,Z[J]\to\langle e^{\theta
X}\rangle,W[J]\to-N\lambda[\theta],V(\chi)\to NI({\widetilde{X}})$.

$Z[J]=e^{-W[J]}=\left\langle\exp\left(-\int{\rm
d}^{d}x\,J({\vec{x}})\chi({\vec{x}})\right)\right\rangle\ .$ (6.8)

The $M^{\text{th}}$ connected correlator is then determined from the
generating functional as

$\langle\chi({\vec{x}}_{1})..\chi({\vec{x}}_{M})\rangle_{C}=\frac{\delta^{M}}{\delta
J({\vec{x}}_{1})..\delta J({\vec{x}}_{M})}W[J]|_{J=0}\ .$ (6.9)

If we can locally expand $W[J]$ as

$W[J]=\sum_{m}a_{m}J^{m}\ ,$ (6.10)

then the correlators will be factorially enhanced unless $a_{M}\propto 1/M!$.
This condition implies that the expansion in Eq. 6.10 converges everywhere in
the complex plane and therefore $W[J]$ is an entire function. It was confirmed
by explicit calculation in [93] that $W[J]$ has a logarithmic branch
cut161616As discussed in [93], the existence of a point in the complex plane
where $Z[J]=0$ is sufficient to demonstrate the $W[J]$ is not entire. for
$V(\chi)\propto|\chi|^{p}$ when $p>2$. In this precise sense, the non-Gaussian
tails that are calculable via Stochastic Inflation also imply a factorial
enhancement of the large-$M$-point correlators.

For the models described by Eq. 6.3, the generating functional was calculated
in [93] using the method of steepest descents. It is noteworthy that $W[J]$ is
the same quantity as $\lambda[\theta]\to W[J]$ that appears in Cramér’s
theorem, see Eq. 2.23. Concretely, $W[J=ik]$ is simply the Fourier transform
of $P(\chi)$ so that

$P(\chi)=\int_{-\infty}^{\infty}{\rm d}k\,e^{-ik\chi}\exp(-W[ik])\ .$ (6.11)

As we saw in Eq. 3.18, when the LDP holds, the inverse Fourier transform can
be calculated from the method of steepest descents and reproduces Cramér’s
theorem. Given that $W[J]$ is itself calculable by the non-trivial saddle, Eq.
6.11 can be interpreted as a Legendre transform from the generating functional
to the rate function $W[J]\to I(\chi)$. The appearance of the Legendre
transform in calculating this rate function using the LDP is equivalent to the
role of the Legendre transform in relating free energies in statistical
mechanics.

The role of the Fourier transform in relating the language of Stochastic
Inflation and rare fluctuations also appears in the tail expansion of the
probability distribution reviewed in _e.g._ [81]. In that case, one is Fourier
transforming the time variable, rather than the field $\chi$, but the $\delta
N$-formalism ultimately relates the two at the end of inflation. In principle,
the techniques of the LDP should also apply directly to the tail expansion and
might offer insights into the regime of control of those calculations.

Finally, the above discussion is also related to the factorial enhancement of
scattering amplitudes at high multiplicity [94, 95, 96, 97, 98, 99]. In that
context, semi-classical solutions have also proven to be important and are
closely related to the semi-classical calculation of $W[J]$ described above.
It is likely there is a deeper connection to the LDP, as we have seen in the
case of cosmological correlators.

##  7 Conclusions

In this paper, we demonstrated that the Large Deviation Principle can be used
to diagnose the validity of the underlying Effective Field Theory expansion
being used to derive the evolution equations of Stochastic Inflation. We
showed how to interpret the dynamical Renormalization Group equations that
derive Stochastic Inflation as coarse-graining a random walk. When the
potential is essentially zero, for example in the case of the inflaton, we
argued that this procedure leads to a Gaussian distribution as a consequence
of the Central Limit Theorem. In this case, EFT expectations hold and
everything is under perturbative control. However, if one asks questions that
are sensitive to the tails of the probability distribution, then the LDP tells
us that a new saddle point of the action dominates, and making a reliable
prediction requires knowledge of the EFT to all orders (or equivalently one
must appeal to the UV completion). We then showed how the LDP applies for
models with a non-trivial potential, and again explored the regime of EFT
validity. Finally, we showed how the LDP could be used to diagnose the
validity of models that were introduced with the goal of yielding a non-
trivial production of primordial black holes.

There are many important future directions to explore. It would be of great
interest to apply the LDP to compute the stochastic evolution equations in UV
complete examples in such a way that the impact on the tails of the
distributions was completely under control. This would provide a test case
analog of the random walk examples that were presented in Sec. 2 above. It
would also be interesting to explore other applications of the LDP in
cosmology and quantum field theory. For example, the appearance of additional
saddles describing the tail of the distribution is reminiscent of scattering
amplitudes with high-multiplicity [94, 95, 96, 97, 98, 99] and the large
charge expansion of conformal field theories [100, 101, 102, 103, 104]. These
are natural settings where one might expect the LDP to play a role, and it
would be exciting to make this precise. We anticipate that having connected
the validity of Stochastic Inflation to the LDP will yield many new insights
into the nature of quantum field theory, both in dS spacetime and beyond.

#### Acknowledgements

We are grateful to Cliff Burgess, Paolo Creminelli, Paolo Glorioso, John
McGreevy, Mehrdad Mirbabayi, Eva Silverstein, and Raman Sundrum. TC is
supported by the US Department of Energy under Grant DE-SC0011640. DG is
supported by the US Department of Energy under Grant DE-SC0009919. AP was
supported in part by the Kavli Institute for Cosmological Physics at the
University of Chicago through an endowment from the Kavli Foundation. This
work was completed at the Aspen Center for Physics, which is supported by the
NSF grant PHY-1607611.

## References

  * [1] X. Chen, M.-x. Huang, S. Kachru, and G. Shiu, “Observational signatures and non-Gaussianities of general single field inflation,” JCAP 01 (2007) 002, arXiv:hep-th/0605045.
  * [2] D. Seery, “One-loop corrections to a scalar field during inflation,” JCAP 11 (2007) 025, arXiv:0707.3377 [astro-ph].
  * [3] L. Leblond and S. Shandera, “Simple Bounds from the Perturbative Regime of Inflation,” JCAP 08 (2008) 007, arXiv:0802.2290 [hep-th].
  * [4] S. Shandera, “The structure of correlation functions in single field inflation,” Phys. Rev. D 79 (2009) 123518, arXiv:0812.0818 [astro-ph].
  * [5] C. P. Burgess, L. Leblond, R. Holman, and S. Shandera, “Super-Hubble de Sitter Fluctuations and the Dynamical RG,” JCAP 1003 (2010) 033, arXiv:0912.1608 [hep-th].
  * [6] D. Baumann and D. Green, “Equilateral Non-Gaussianity and New Physics on the Horizon,” JCAP 09 (2011) 014, arXiv:1102.5343 [hep-th].
  * [7] D. Baumann and D. Green, “A Field Range Bound for General Single-Field Inflation,” JCAP 05 (2012) 017, arXiv:1111.3040 [hep-th].
  * [8] R. Flauger, D. Green, and R. A. Porto, “On squeezed limits in single-field inflation. Part I,” JCAP 08 (2013) 032, arXiv:1303.1430 [hep-th].
  * [9] D. Baumann, D. Green, and R. A. Porto, “B-modes and the Nature of Inflation,” JCAP 01 (2015) 016, arXiv:1407.2621 [hep-th].
  * [10] D. Baumann, D. Green, H. Lee, and R. A. Porto, “Signs of Analyticity in Single-Field Inflation,” Phys. Rev. D 93 no. 2, (2016) 023523, arXiv:1502.07304 [hep-th].
  * [11] P. Adshead, C. P. Burgess, R. Holman, and S. Shandera, “Power-counting during single-field slow-roll inflation,” JCAP 02 (2018) 016, arXiv:1708.07443 [hep-th].
  * [12] I. Babic, C. P. Burgess, and G. Geshnizjani, “Keeping an eye on DBI: power-counting for small-cs cosmology,” JCAP 05 (2020) 023, arXiv:1910.05277 [gr-qc].
  * [13] T. Grall and S. Melville, “Inflation in motion: unitarity constraints in effective field theories with (spontaneously) broken Lorentz symmetry,” JCAP 09 (2020) 017, arXiv:2005.02366 [gr-qc].
  * [14] R. Bousso, B. Freivogel, and I.-S. Yang, “Eternal Inflation: The Inside Story,” Phys. Rev. D 74 (2006) 103516, arXiv:hep-th/0606114.
  * [15] N. Arkani-Hamed, S. Dubovsky, A. Nicolis, E. Trincherini, and G. Villadoro, “A Measure of de Sitter entropy and eternal inflation,” JHEP 05 (2007) 055, arXiv:0704.1814 [hep-th].
  * [16] D. Seery, “A parton picture of de Sitter space during slow-roll inflation,” JCAP 0905 (2009) 021, arXiv:0903.2788 [astro-ph.CO].
  * [17] H. Matsui and F. Takahashi, “Eternal Inflation and Swampland Conjectures,” Phys. Rev. D 99 no. 2, (2019) 023533, arXiv:1807.11938 [hep-th].
  * [18] W. H. Kinney, “Eternal Inflation and the Refined Swampland Conjecture,” Phys. Rev. Lett. 122 no. 8, (2019) 081302, arXiv:1811.11698 [astro-ph.CO].
  * [19] S. Brahma and S. Shandera, “Stochastic eternal inflation is in the swampland,” JHEP 11 (2019) 016, arXiv:1904.10979 [hep-th].
  * [20] A. Vilenkin, “The Birth of Inflationary Universes,” Phys. Rev. D 27 (1983) 2848.
  * [21] A. A. Starobinsky, “Stochastic de Sitter (Inflationary) Stage in the Early Universe,” Lect. Notes Phys. 246 (1986) 107–126.
  * [22] M. Aryal and A. Vilenkin, “The Fractal Dimension of Inflationary Universe,” Phys. Lett. B 199 (1987) 351–357.
  * [23] Y. Nambu and M. Sasaki, “Stochastic Stage of an Inflationary Universe Model,” Phys. Lett. B 205 (1988) 441–446.
  * [24] H. E. Kandrup, “Stochastic Inflation as a Time Dependent Random Walk,” Phys. Rev. D 39 (1989) 2245.
  * [25] D. S. Salopek and J. R. Bond, “Nonlinear evolution of long wavelength metric fluctuations in inflationary models,” Phys. Rev. D42 (1990) 3936–3962.
  * [26] A. A. Starobinsky and J. Yokoyama, “Equilibrium state of a selfinteracting scalar field in the De Sitter background,” Phys. Rev. D 50 (1994) 6357–6368, arXiv:astro-ph/9407016.
  * [27] D. Wands, K. A. Malik, D. H. Lyth, and A. R. Liddle, “A New approach to the evolution of cosmological perturbations on large scales,” Phys. Rev. D 62 (2000) 043527, arXiv:astro-ph/0003278.
  * [28] N. C. Tsamis and R. P. Woodard, “Stochastic quantum gravitational inflation,” Nucl. Phys. B724 (2005) 295–328, arXiv:gr-qc/0505115 [gr-qc].
  * [29] D. Seery and J. E. Lidsey, “Primordial non-Gaussianities from multiple-field inflation,” JCAP 09 (2005) 011, arXiv:astro-ph/0506056.
  * [30] D. Wands, “Local non-Gaussianity from inflation,” Class. Quant. Grav. 27 (2010) 124002, arXiv:1004.0818 [astro-ph.CO].
  * [31] F. Finelli, G. Marozzi, A. A. Starobinsky, G. P. Vacca, and G. Venturi, “Stochastic growth of quantum fluctuations during slow-roll inflation,” Phys. Rev. D 82 (2010) 064020, arXiv:1003.1327 [hep-th].
  * [32] Y. Tada and V. Vennin, “Squeezed bispectrum in the $\delta N$ formalism: local observer effect in field space,” JCAP 02 (2017) 021, arXiv:1609.08876 [astro-ph.CO].
  * [33] A. Achúcarro, V. Atal, C. Germani, and G. A. Palma, “Cumulative effects in inflation with ultra-light entropy modes,” JCAP 02 (2017) 013, arXiv:1607.08609 [astro-ph.CO].
  * [34] T. Markkanen, A. Rajantie, S. Stopyra, and T. Tenkanen, “Scalar correlation functions in de Sitter space from the stochastic spectral expansion,” JCAP 08 (2019) 001, arXiv:1904.11917 [gr-qc].
  * [35] M. Baumgart and R. Sundrum, “De Sitter Diagrammar and the Resummation of Time,” JHEP 07 (2020) 119, arXiv:1912.09502 [hep-th].
  * [36] M. Baumgart and R. Sundrum, “Manifestly Causal In-In Perturbation Theory about the Interacting Vacuum,” JHEP 03 (2021) 080, arXiv:2010.10785 [hep-th].
  * [37] L. Pinol, S. Renaux-Petel, and Y. Tada, “A manifestly covariant theory of multifield stochastic inflation in phase space,” JCAP 04 (2021) 048, arXiv:2008.07497 [astro-ph.CO].
  * [38] T. Cohen and D. Green, “Soft de Sitter Effective Theory,” JHEP 12 (2020) 041, arXiv:2007.03693 [hep-th].
  * [39] T. Cohen, D. Green, A. Premkumar, and A. Ridgway, “Stochastic Inflation at NNLO,” JHEP 09 (2021) 159, arXiv:2106.09728 [hep-th].
  * [40] D. Green, “EFT for de Sitter Space,” arXiv:2210.05820 [hep-th].
  * [41] C. P. Burgess, R. Holman, and G. Tasinato, “Open EFTs, IR effects & late-time resummations: systematic corrections in stochastic inflation,” JHEP 01 (2016) 153, arXiv:1512.00169 [gr-qc].
  * [42] V. Gorbenko and L. Senatore, “$\lambda\phi^{4}$ in dS,” arXiv:1911.00022 [hep-th].
  * [43] M. Mirbabayi, “Infrared dynamics of a light scalar field in de Sitter,” JCAP 12 (2020) 006, arXiv:1911.00564 [hep-th].
  * [44] M. Mirbabayi, “Markovian dynamics in de Sitter,” JCAP 09 (2021) 038, arXiv:2010.06604 [hep-th].
  * [45] M. Celoria, P. Creminelli, G. Tambalo, and V. Yingcharoenrat, “Beyond perturbation theory in inflation,” JCAP 06 (2021) 051, arXiv:2103.09244 [hep-th].
  * [46] T. Cohen, D. Green, and A. Premkumar, “A Tail of Eternal Inflation,” arXiv:2111.09332 [hep-th].
  * [47] S. R. S. Varadhan, “Special invited paper. Large deviations,” arXiv e-prints (Apr., 2008) arXiv:0804.2330, arXiv:0804.2330 [math.PR].
  * [48] H. Touchette, “A basic introduction to large deviations: Theory, applications, simulations,” arXiv e-prints (June, 2011) arXiv:1106.4146, arXiv:1106.4146 [cond-mat.stat-mech].
  * [49] F. Bernardeau and P. Reimberg, “Large deviation principle at play in large scale structure cosmology,” Phys. Rev. D 94 no. 6, (2016) 063520, arXiv:1511.08641 [astro-ph.CO].
  * [50] C. Uhlemann, S. Codis, C. Pichon, F. Bernardeau, and P. Reimberg, “Back in the saddle: Large-deviation statistics of the cosmic log-density field,” Mon. Not. Roy. Astron. Soc. 460 no. 2, (2016) 1529–1541, arXiv:1512.05793 [astro-ph.CO].
  * [51] P. Reimberg and F. Bernardeau, “Large deviation principle at work: Computation of the statistical properties of the exact one-point aperture mass,” Phys. Rev. D 97 no. 2, (2018) 023524, arXiv:1708.00252 [astro-ph.CO].
  * [52] S. Codis, “Beyond the power spectrum with large deviation theory,” in SF2A-2018: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, P. Di Matteo, F. Billebaud, F. Herpin, N. Lagarde, J. B. Marquette, A. Robin, and O. Venot, eds., p. Di.  Dec., 2018.
  * [53] M. M. Ivanov, A. A. Kaurov, and S. Sibiryakov, “Non-perturbative probability distribution function for cosmological counts in cells,” JCAP 03 (2019) 009, arXiv:1811.07913 [astro-ph.CO].
  * [54] A. Barthelemy, S. Codis, C. Uhlemann, F. Bernardeau, and R. Gavazzi, “A nulling strategy for modelling lensing convergence in cones with large deviation theory,” Mon. Not. Roy. Astron. Soc. 492 no. 3, (2020) 3420–3439, arXiv:1909.02615 [astro-ph.CO].
  * [55] A. Barthelemy, S. Codis, and F. Bernardeau, “Probability distribution function of the aperture mass field with large deviation theory,” Mon. Not. Roy. Astron. Soc. 503 no. 4, (2021) 5204–5222, arXiv:2012.03831 [astro-ph.CO].
  * [56] A. Barthelemy, S. Codis, and F. Bernardeau, “Post-Born corrections to the one-point statistics of (CMB) lensing convergence obtained via large deviation theory,” Mon. Not. Roy. Astron. Soc. 494 no. 3, (2020) 3368–3382, arXiv:2002.03625 [astro-ph.CO].
  * [57] M. Cataneo, C. Uhlemann, C. Arnold, A. Gough, B. Li, and C. Heymans, “The matter density PDF for modified gravity and dark energy with Large Deviations Theory,” Mon. Not. Roy. Astron. Soc. 513 no. 2, (2022) 1623–1641, arXiv:2109.02636 [astro-ph.CO].
  * [58] H. Touchette, “The large deviation approach to statistical mechanics,” Physics Reports 478 no. 1, (2009) 1–69.
  * [59] J. McGreevy, “Physics 217: The renormalization group.” https://mcgreevy.physics.ucsd.edu/f18/2018F-217-lectures.pdf, 2018.
  * [60] M. Kardar, Statistical Physics of Particles.  Cambridge University Press, 2007.
  * [61] S. R. S. Varadhan, “Large deviations,” The Annals of Probability 36 no. 2, (2008) 397 – 419. https://doi.org/10.1214/07-AOP348.
  * [62] A. D. Linde, “Eternally Existing Selfreproducing Chaotic Inflationary Universe,” Phys. Lett. B 175 (1986) 395–400.
  * [63] A. S. Goncharov, A. D. Linde, and V. F. Mukhanov, “The Global Structure of the Inflationary Universe,” Int. J. Mod. Phys. A 2 (1987) 561–591.
  * [64] P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore, and M. Zaldarriaga, “The Phase Transition to Slow-roll Eternal Inflation,” JHEP 09 (2008) 036, arXiv:0802.1067 [hep-th].
  * [65] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 05 (2003) 013, arXiv:astro-ph/0210603 [astro-ph].
  * [66] P. Creminelli and M. Zaldarriaga, “Single field consistency relation for the 3-point function,” JCAP 0410 (2004) 006, arXiv:astro-ph/0407059 [astro-ph].
  * [67] P. Creminelli, J. Noreña, and M. Simonović, “Conformal consistency relations for single-field inflation,” JCAP 07 (2012) 052, arXiv:1203.4595 [hep-th].
  * [68] K. Hinterbichler, L. Hui, and J. Khoury, “Conformal Symmetries of Adiabatic Modes in Cosmology,” JCAP 1208 (2012) 017, arXiv:1203.6351 [hep-th].
  * [69] A. Premkumar, “Regulating Loops in dS,” arXiv:2110.12504 [hep-th].
  * [70] P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore, “Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies,” JHEP 12 (2006) 080, arXiv:hep-th/0606090.
  * [71] C. Cheung, P. Creminelli, A. Fitzpatrick, J. Kaplan, and L. Senatore, “The Effective Field Theory of Inflation,” JHEP 03 (2008) 014, arXiv:0709.0293 [hep-th].
  * [72] N. Van Kampen, “Elimination of fast variables,” Physics Reports 124 no. 2, (1985) 69–160. https://www.sciencedirect.com/science/article/pii/037015738590002X.
  * [73] R. P. Woodard, “A Leading logarithm approximation for inflationary quantum field theory,” Nucl. Phys. B Proc. Suppl. 148 (2005) 108–119, arXiv:astro-ph/0502556.
  * [74] A. Achucarro, S. Cespedes, A.-C. Davis, and G. A. Palma, “The hand-made tail: non-perturbative tails from multifield inflation,” JHEP 05 (2022) 052, arXiv:2112.14712 [hep-th].
  * [75] C. P. Burgess, R. Holman, G. Kaplanek, J. Martin, and V. Vennin, “Minimal decoherence from inflation,” arXiv:2211.11046 [hep-th].
  * [76] B. Carr and F. Kuhnel, “Primordial Black Holes as Dark Matter: Recent Developments,” Ann. Rev. Nucl. Part. Sci. 70 (2020) 355–394, arXiv:2006.02838 [astro-ph.CO].
  * [77] S. Bird et al., “Snowmass2021 Cosmic Frontier White Paper:Primordial Black Hole Dark Matter,” arXiv:2203.08967 [hep-ph].
  * [78] M. Kopp, S. Hofmann, and J. Weller, “Separate Universes Do Not Constrain Primordial Black Hole Formation,” Phys. Rev. D 83 (2011) 124025, arXiv:1012.4369 [astro-ph.CO].
  * [79] T. Harada, C.-M. Yoo, and K. Kohri, “Threshold of primordial black hole formation,” Phys. Rev. D 88 no. 8, (2013) 084051, arXiv:1309.4201 [astro-ph.CO]. [Erratum: Phys.Rev.D 89, 029903 (2014)].
  * [80] G. Franciolini, A. Kehagias, S. Matarrese, and A. Riotto, “Primordial Black Holes from Inflation and non-Gaussianity,” JCAP 03 (2018) 016, arXiv:1801.09415 [astro-ph.CO].
  * [81] V. Vennin, “Stochastic inflation and primordial black holes,” habilitation thesis, Université Paris-Saclay, 2020.
  * [82] Y.-F. Cai, X.-H. Ma, M. Sasaki, D.-G. Wang, and Z. Zhou, “Highly non-Gaussian tails and primordial black holes from single-field inflation,” arXiv:2207.11910 [astro-ph.CO].
  * [83] J. R. Bond, A. V. Frolov, Z. Huang, and L. Kofman, “Non-Gaussian Spikes from Chaotic Billiards in Inflation Preheating,” Phys. Rev. Lett. 103 (2009) 071301, arXiv:0903.3407 [astro-ph.CO].
  * [84] X. Chen, G. A. Palma, W. Riquelme, B. Scheihing Hitschfeld, and S. Sypsas, “Landscape tomography through primordial non-Gaussianity,” Phys. Rev. D 98 no. 8, (2018) 083528, arXiv:1804.07315 [hep-th].
  * [85] G. Panagopoulos and E. Silverstein, “Primordial Black Holes from non-Gaussian tails,” arXiv:1906.02827 [hep-th].
  * [86] G. A. Palma, B. Scheihing Hitschfeld, and S. Sypsas, “Non-Gaussian CMB and LSS statistics beyond polyspectra,” JCAP 02 (2020) 027, arXiv:1907.05332 [astro-ph.CO].
  * [87] G. A. Palma, S. Sypsas, and C. Zenteno, “Seeding primordial black holes in multifield inflation,” Phys. Rev. Lett. 125 no. 12, (2020) 121301, arXiv:2004.06106 [astro-ph.CO].
  * [88] S. Hooshangi, M. H. Namjoo, and M. Noorbala, “Rare events are nonperturbative: Primordial black holes from heavy-tailed distributions,” Phys. Lett. B 834 (2022) 137400, arXiv:2112.04520 [astro-ph.CO].
  * [89] S. Pi and M. Sasaki, “Logarithmic Duality of the Curvature Perturbation,” arXiv:2211.13932 [astro-ph.CO].
  * [90] A. D. Gow, H. Assadullahi, J. H. P. Jackson, K. Koyama, V. Vennin, and D. Wands, “Non-perturbative non-Gaussianity and primordial black holes,” arXiv:2211.08348 [astro-ph.CO].
  * [91] R. Flauger, M. Mirbabayi, L. Senatore, and E. Silverstein, “Productive Interactions: heavy particles and non-Gaussianity,” JCAP 10 (2017) 058, arXiv:1606.00513 [hep-th].
  * [92] M. Münchmeyer and K. M. Smith, “Higher N-point function data analysis techniques for heavy particle production and WMAP results,” Phys. Rev. D 100 no. 12, (2019) 123511, arXiv:1910.00596 [astro-ph.CO].
  * [93] G. Panagopoulos and E. Silverstein, “Multipoint correlators in multifield cosmology,” arXiv:2003.05883 [hep-th].
  * [94] J. M. Cornwall, “On the High-energy Behavior of Weakly Coupled Gauge Theories,” Phys. Lett. B 243 (1990) 271–278.
  * [95] H. Goldberg, “Breakdown of perturbation theory at tree level in theories with scalars,” Phys. Lett. B 246 (1990) 445–450.
  * [96] L. S. Brown, “Summing tree graphs at threshold,” Phys. Rev. D 46 (1992) R4125–R4127, arXiv:hep-ph/9209203.
  * [97] A. S. Gorsky and M. B. Voloshin, “Nonperturbative production of multiboson states and quantum bubbles,” Phys. Rev. D 48 (1993) 3843–3851, arXiv:hep-ph/9305219.
  * [98] M. V. Libanov, V. A. Rubakov, D. T. Son, and S. V. Troitsky, “Exponentiation of multiparticle amplitudes in scalar theories,” Phys. Rev. D 50 (1994) 7553–7569, arXiv:hep-ph/9407381.
  * [99] D. T. Son, “Semiclassical approach for multiparticle production in scalar theories,” Nucl. Phys. B 477 (1996) 378–406, arXiv:hep-ph/9505338.
  * [100] S. Hellerman, D. Orlando, S. Reffert, and M. Watanabe, “On the CFT Operator Spectrum at Large Global Charge,” JHEP 12 (2015) 071, arXiv:1505.01537 [hep-th].
  * [101] G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, “The Epsilon Expansion Meets Semiclassics,” JHEP 11 (2019) 110, arXiv:1909.01269 [hep-th].
  * [102] A. Grassi, Z. Komargodski, and L. Tizzano, “Extremal correlators and random matrix theory,” JHEP 04 (2021) 214, arXiv:1908.10306 [hep-th].
  * [103] G. Badel, G. Cuomo, A. Monin, and R. Rattazzi, “Feynman diagrams and the large charge expansion in $3-\varepsilon$ dimensions,” Phys. Lett. B 802 (2020) 135202, arXiv:1911.08505 [hep-th].
  * [104] L. A. Gaumé, D. Orlando, and S. Reffert, “Selected topics in the large quantum number expansion,” Phys. Rept. 933 (2021) 1–66, arXiv:2008.03308 [hep-th].