NLO finite system size corrections to $2\to 2$ scattering in $\phi^{4}$ theory
using newly derived sum of sinc functions

J.F. Du Plessis1$\star$ and W. A. Horowitz2

1 Department of Physics, Stellenbosch University, Private Bag X1, Matieland
7602, South Africa

2 Department of Physics, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa

*<EMAIL_ADDRESS>
51st International Symposium on Multiparticle Dynamics (ISMD2022)
Pitlochry, Scottish Highlands, 1-5 August 2022 10.21468/SciPostPhysProc.?

## Abstract

Previously an equation of state for the relativistic hydrodynamics encountered
in heavy-ion collisions at the LHC and RHIC has been calculated using lattice
gauge theory methods. This leads to a prediction of very low viscosity, due to
the calculated trace anomaly. Finite system corrections to this trace anomaly
could challenge this calculation, since the lattice calculation was done in an
effectively infinite system. In order to verify this trace anomaly it is
sensible to add phenomenologically relevant finite system corrections. We
investigate massive $\phi^{4}$ theory with periodic boundary conditions on $n$
of the 3 spatial dimensions. $2\to 2$ NLO scattering is then computed. Using a
newly derived formula for an arbitrary dimensional sum of sinc functions, we
show that the NLO finite size corrections preserve unitarity.

##  1 Introduction

There is an apparent formation of Quark Gluon Plasma (QGP) in heavy ion
collisions [1, 2, 3], where the correlations between the outgoing low-momentum
particles appear to be well described by nearly inviscid relativistic
hydrodynamics. This calculation uses an Equation of State (EoS) provided by a
lattice QCD calculation that is extrapolated to infinite system size[4].

It is currently unclear what happens in QCD just above the transition
temperature $T=180$ MeV. There is strong evidence of a second order phase
transition, but the nature of the new phase is largely unknown. It is
therefore necessary to understand how reliably the experimental behaviour
found in the finite systems (such as heavy ion or parton collisions) can be
extrapolated to effectively infinite systems, such as the QGP found in the
$\sim 0.000001$ seconds after Big Bang.

A possibly significant assumption to be investigated is that heavy ion
collisions can be well approximated as infinite sized systems [5]. Indeed
quenched lattice QCD calculations have shown significant possible corrections
dependent on the size of the system [6]. An analytic derivation of the finite
size effects on the equation of state (or equivalently the trace anomaly) is
therefore sought. This work is a step in that direction, with the intention to
develop and understand the mathematical techniques necessary for a full
treatment necessary for finite temperature finite sized QCD.

##  2 Finite Sized $\phi^{4}$ Theory

Let us consider the $\phi^{4}$ Lagrangian

$\mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}$
(1)

in a system with periodic boundary conditions. If we consider $n$ compact
spatial dimensions, with the $i$th dimension being parameterized by $[-\pi
L_{i},\pi L_{i}]$ with periodic boundary conditions. This discretizes the
possible spatial momenta to
$\vec{p}=(\frac{k_{1}}{L_{1}},\frac{k_{2}}{L_{2}},\ldots,\frac{k_{n}}{L_{n}},)$
where $\vec{k}\in\mathbb{Z}^{n}$. In analogy with [7] we can define
$-i\lambda^{2}V(p^{2})\equiv\leavevmode\hbox to0.4pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex(a) at
(-0.55,0.5) {}; \vertex(b) at (-0.55,-0.5) {}; \vertex[dot] (c) at (-0.25,0)
{}; \vertex[dot] (d) at (0.25,0) {}; \vertex(e) at (0.55,0.5) {}; \vertex(f)
at (0.55,-0.5) {}; {}
{}{}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}
{}{}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{{}}{}{{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{0.0pt}{0.0pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{{}}{}{{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{0.0pt}{0.0pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}
{}{}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}
{}{}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ with $p$ being the
total incoming momentum. One then finds[8] in $n=3$ spatial dimensions that,
up to NLO, one gets the renormalized

$\overline{V}(p^{2},\\{L_{i}\\})=-\frac{1}{2(4\pi)^{2}}\int_{0}^{1}dx\Bigg{\\{}\log\left(\frac{\mu^{2}}{\Delta^{2}}\right)\\\
\cr+2\sideset{}{{}^{\prime}}{\sum}_{\vec{m}\in\mathbb{Z}^{3}}\cos(2\pi x\sum
m_{i}p^{i}L_{i})K_{0}\left(2\pi\sqrt{\Delta^{2}}\sqrt{\sum(m_{i}L_{i})^{2}}\right)\Bigg{\\}}.$
(2)

Here we recognize the $\log$ term as corresponding to the standard result in
infinite $\phi^{4}$ systems[7]. As one would then expect the second term in
the integral vanishes in the limit as all $L_{i}\to\infty$ since
$\lim_{x\to\infty}K_{0}(x)=0$. We can see that we could reduce the effective
number of finite dimensions by quite simply taking the corresponding
$L_{i}\to\infty$, since only terms in the sum with the corresponding $m_{i}=0$
will survive the limit. We then find that, as in the infinite system case,

$\mathcal{M}=\lambda\left[1+\lambda\left(\overline{V}(s)+\overline{V}(t)+\overline{V}(u)\right)\right]$
(3)

up to NLO, with $s,t$ and $u$ being the usual Mandelstam variables.

##  3 Unitarity

In order to verify that unitarity has stayed intact we will show that the
optical theorem holds, no matter how many dimensions $m$ are of finite size.
For the optical theorem to hold, we need that

$2\operatorname{Im}[\mathcal{M}]=\sigma_{\text{tot}}.$ (4)

It is a straight-forward calculation to find

$\sigma_{tot}=\frac{\lambda^{2}}{16\pi}\frac{\pi^{\frac{1-m}{2}}}{\Gamma\left(\frac{3-m}{2}\right)}\frac{1}{L\sqrt{s}}\,\sideset{}{{}^{*}}{\sum}_{0\leq
l<R^{2}}\frac{r_{m}(l)}{\sqrt{R^{2}-l}^{m-1}}.$ (5)

By following [8] one gets that

$2\operatorname{Im}[\mathcal{M}]=\frac{\lambda^{2}}{16\pi}\frac{2R}{L\sqrt{s}}\sum_{\vec{k}\in\mathbb{Z}^{m}}\operatorname{sinc}\left(2\pi
R\|\vec{k}\|\right)$ (6)

up to NLO. We can then use 14 to get

$2\operatorname{Im}[\mathcal{M}]=\frac{\lambda^{2}}{16\pi}\frac{2R}{L\sqrt{s}}\frac{1}{2R\pi^{\frac{m-1}{2}}\,\Gamma\left(\frac{3-m}{2}\right)}\,\sideset{}{{}^{*}}{\sum}_{0\leq
l<R^{2}}\frac{r_{m}(l)}{\sqrt{R^{2}-l}^{m-1}}.$ (7)

The equivalence of Equations 5 and 7 shows that the Optical Theorem, and
therefore Unitarity, holds independent of the amount of compact dimensions.

##  4 Conclusion

By passing all considered self-consistency checks, namely having the correct
infinite limit and preserving unitarity, we have shown the viability of the
mathematical tools developed and employed. Notably it greatly supports the
generalization of the number theoretic formula derived in A, which has
potential implications in number theory.

## Acknowledgements

W.A.H. wishes to thank the South African National Research Foundation and the
SA-CERN Collaboration for support and New Mexico State University for its
hospitality, where part of the work was completed. The authors wish to thank
Matt Sievert, Alexander Rothkopf, Bowen Xiao, Stan Brodsky, Andrea Shindler,
Kevin Bassler and Herbert Weigel for valuable discussions.

### Funding information

The authors wish to thank the South African National Research Foundation and
the SA-CERN Collaboration for support.

##  Appendix A Sum of Sinc functions

We will need a generalization of a formula for the multi-dimensional sum of
$\operatorname{sinc}$ functions, namely sums of the form

$\sum_{\vec{k}\in\mathbb{Z}^{m}}\operatorname{sinc}(2\pi R\|\vec{k}\|)$ (8)

where we define $\operatorname{sinc}(x)=\begin{cases}\frac{\sin(x)}{x}&x\neq
0\\\ 1&x=0\end{cases}$.
Since the sum only depends on the magnitude of $\vec{k}$, we can simplify the
sum using the Sum of Squares function $r_{d}(n)$ which gives the amount of
$\vec{k}\in\mathbb{Z}^{d}$ with $\|\vec{k}\|^{2}=n$. Allowing us to use

$\sum_{l=0}^{\infty}r_{d}(l)f(\sqrt{l})=\sum_{l=0}^{\infty}r_{d}(l)\hat{f}(\sqrt{l})$
(9)

from [9]. To employ their analytically continued version of the Poisson
summation formula we can first set $f(r)=\operatorname{sinc}(2\pi Rr)$. Then
we can calculate $\hat{f}$

$\displaystyle\hat{f}(p)$
$\displaystyle=\frac{2\pi^{\frac{m}{2}}}{\Gamma(\frac{m}{2})}\int_{0}^{\infty}\operatorname{sinc}(2\pi
Rr)\,{}_{0}F_{1}(\frac{m}{2};-\pi^{2}p^{2}r^{2})\,r^{m-1}dr$ (10)
$\displaystyle=\frac{1}{2R\pi^{\frac{m-1}{2}}\,\Gamma(\frac{3-m}{2})}(R^{2}-p^{2})^{\frac{1-m}{2}}\theta(R^{2}-p^{2})$
(11)

Now we can apply 9

$\displaystyle\sum_{\vec{k}\in\mathbb{Z}^{m}}\operatorname{sinc}(2\pi
R\|\vec{k}\|)$
$\displaystyle=\sum_{l=0}^{\infty}r_{m}(l)\operatorname{sinc}(2\pi R\sqrt{r})$
(12)
$\displaystyle=\sum_{l=0}^{\infty}r_{m}(l)\frac{(R^{2}-l)^{\frac{1-m}{2}}}{2R\pi^{\frac{m-1}{2}}\,\Gamma(\frac{3-m}{2})}\theta(R^{2}-l).$
(13)

Here we should note that if $r_{m}(R^{2})\neq 0$ we will get a term with
$\theta(0)=\frac{1}{2}$ which we note will give a $\frac{1}{0}$ term in the
sum for $m>1$. Therefore

$\sum_{\vec{k}\in\mathbb{Z}^{m}}\operatorname{sinc}(2\pi
R\|\vec{k}\|)=\frac{1}{2R\pi^{\frac{m-1}{2}}\,\Gamma(\frac{3-m}{2})}\,\sideset{}{{}^{*}}{\sum}_{0\leq
l<R^{2}}\frac{r_{m}(l)}{\sqrt{R^{2}-l}^{m-1}},$ (14)

where $\sideset{}{{}^{*}}{\sum}_{0\leq l<R^{2}}$ means that if
$R^{2}\in\mathbb{Z}$, then its term in the sum has weight $\frac{1}{2}$. The
$m=2$ case corresponds directly to a formula of Ramanujan [10], making this
result a generalization thereof.

## References

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# Quantum diffusion and large primordial perturbations from inflation

Invited chapter to the book “Primordial Black Holes”, Springer 2024,
ed. Chris Byrnes, Gabriele Franciolini, Tomohiro Harada, Paolo Pani, Misao
Sasaki.

Vincent Vennin  David Wands

(today)

###### Abstract

Quantum diffusion describes the inflow of vacuum quantum fluctuations as they
get amplified by gravitational instability, and stretched to large distances
during inflation. In this picture, the dynamics of the universe’s expansion
becomes stochastic, and the statistics of the curvature perturbation is
encoded in the distribution of the duration of inflation. This provides a non-
perturbative framework to study cosmological fluctuations during inflation,
which is well-suited to the case of primordial black holes since they
originate from large fluctuations. We show that standard, perturbative
expectations for the primordial black hole abundance can be significantly
modified by quantum-diffusion effects, and we identify a few open challenges.

In memory of Alexei Starobinsky (1948-2023), the founding father of stochastic
inflation, and of so much more.

##  1 Introduction

PBHs may form in the early universe when large density fluctuations are
produced and gravitationally collapse into black holes. Although different
mechanisms have been proposed to amplify cosmological fluctuations above the
required threshold, one possibility is that the amplification takes place
during inflation. If large curvature perturbations are produced during
inflation, they may no longer be described by perturbative frameworks, and in
this chapter we show how the stochastic-inflation formalism, in combination
with the $\delta N$ formalism, can be used as a non-perturbative scheme to
keep track of large fluctuations. In this approach, quantum fluctuations
provide random kicks to the large-scale dynamics of the universe, which can
thus be described by means of stochastic systems. Different regions of space
inflate for different amounts of time, hence the number of inflationary
$e$-folds becomes itself a stochastic quantity, which can be identified with
the curvature perturbation. We show how its statistics can be studied using
first-passage-time techniques, and discuss the expected consequences for the
abundance of PBHs from inflation. This abundance is strongly enhanced compared
to perturbative predictions due to heavy tails in the probability
distribution, which are non-perturbative in nature and inevitable. We also
discuss the validity of the separate-universe approach, on which this
framework rests, in particular in the context of slow-roll violations, which
commonly arise in PBH-producing models. Finally, we identify a few open
challenges that remain to be addressed.

##  2 Stochastic inflation in a nutshell

The stochastic formalism of inflation [1, 2] is an effective theory for the
long-wavelength part of quantum fields living on an inflating background. It
can be obtained formally by integrating out the small-wavelength part of these
fields, using the language of effective-field theories and the Schwinger-
Keldysh formalism [3, 4, 5, 6]. Here, we present a heuristic derivation that
makes physical interpretation easier.

We consider a single scalar field $\phi$, minimally coupled to gravity in a
4-dimensional curved space-time with metric $g_{\mu\nu}$, described by the
action

$\displaystyle
S=\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{M_{\mathrm{\footnotesize{Pl}}}^{2}}{2}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right].$
(2.1)

In this expression, $R$ is the Ricci scalar curvature and $V(\phi)$ is the
potential energy of the scalar field.

###  2.1 Background

At the background level, space-time is homogeneous and isotropic, so the line-
element is of the Friedmann-Lemaître-Robertson-Walker (FLRW) form,
$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+a^{2}(t)\mathrm{d}x^{2}$. Here, $t$ is
cosmic time, $a$ is the scale factor and $x^{i}$ are spatial coordinates
comoving with the expansion. The Einstein equations yield a constraint
equation, called the Friedmann equation,

$\displaystyle
H^{2}=\frac{V(\phi)+\frac{\dot{\phi}^{2}}{2}}{3M_{\mathrm{\footnotesize{Pl}}}^{2}}$
(2.2)

where $H=\dot{a}/a$ is the Hubble expansion rate, a dot denotes a derivative
with respect to cosmic time and $M_{\mathrm{\footnotesize{Pl}}}=\sqrt{\hbar
c/(8\pi G)}$ is the reduced Planck mass. The dynamical equations for the
inflaton field, $\phi$, can be written as two coupled first-order equations

$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}N}=$
$\displaystyle\frac{\Pi}{a^{3}H}\,,$ (2.3)
$\displaystyle\frac{\mathrm{d}\Pi}{\mathrm{d}N}=$
$\displaystyle-\frac{a^{3}}{H}V^{\prime}(\phi)\,.$

where $\Pi$ is the momentum conjugate to $\phi$, $V^{\prime}$ is the first
derivative of the function $V(\phi)$, and time is labeled by the number of
$e$-folds $N\equiv\ln(a)$ for future convenience. When the potential function
is flat enough, $M_{\mathrm{\footnotesize{Pl}}}|V^{\prime}/V|\ll 1$ and
$M_{\mathrm{\footnotesize{Pl}}}^{2}|V^{\prime\prime}/V|\ll 1$, a phase of
slow-roll inflation takes place where $\ddot{a}>0$ and $H$ varies slowly.

###  2.2 Linear perturbations

Let us now introduce small fluctuations over the homogeneous and isotropic
background, by allowing $\phi$ to depend on space,
$\phi(\vec{x},t)=\bar{\phi}(t)+\delta\phi(\vec{x},t)$, where $\bar{\phi}$
follows the background dynamics given above and $|\delta\phi|\ll|\bar{\phi}|$.
Inserting this decomposition into eq. (2.1), at leading order in perturbation
theory one has

$\displaystyle\frac{\mathrm{d}\phi_{k}}{\mathrm{d}N}=$
$\displaystyle\frac{\pi_{k}}{a^{3}H}\,,$ (2.4)
$\displaystyle\frac{\mathrm{d}\pi_{k}}{\mathrm{d}N}=$
$\displaystyle-\frac{a^{3}}{H}V^{\prime\prime}(\bar{\phi})\phi_{k}-\frac{a}{H}k^{2}\phi_{k}\,.$

These equations are written in Fourier space, i.e.

$\displaystyle\delta\phi(\vec{x},t)=\left(2\pi\right)^{-3/2}\int\mathrm{d}\vec{k}\left[a_{\vec{k}}\phi_{k}(t)e^{-i\vec{k}\cdot\vec{x}}+a_{\vec{k}}^{\dagger}\phi^{*}_{k}(t)e^{i\vec{k}\cdot\vec{x}}\right]$
(2.5)

with a similar expression for $\delta\pi(\vec{x},t)$. Here, the field
$\delta\phi(\vec{x},t)$ is quantised, so it is expanded onto the creation and
annihilation operators $a^{\dagger}_{\vec{k}}$ and $a_{\vec{k}}$, which
satisfy the usual commutation relations
$[a_{\vec{k}},a^{\dagger}_{\vec{k}^{\prime}}]=\delta^{3}(\vec{k}-\vec{k}^{\prime})$
and
$[a_{\vec{k}},a_{\vec{k}^{\prime}}]=[a^{\dagger}_{\vec{k}},a^{\dagger}_{\vec{k}^{\prime}}]=0$.
They are independent of time, contrary to the mode functions $\phi_{k}(t)$
which satisfy eqs. (2.4). On isotropic backgrounds, they depend only on the
modulus $k=|\vec{k}|$ of the wave-vector, and they must satisfy
$\phi_{-\vec{k}}=\phi_{\vec{k}}^{\star}$ for the field $\delta\phi(\vec{x},t)$
to be Hermitian. In the asymptotic past, $k\gg aH$ and the above reduces to
the dynamics of a massless field on a Minkowski background. In this sub-Hubble
regime, each Fourier mode is described by a harmonic oscillator, and one can
set initial conditions in the so-called Bunch-Davies vacuum, i.e. the state
annihilated by $a_{\vec{k}}$. This corresponds to setting
$\phi_{k}=e^{-ik\eta}/(a\sqrt{2k})$ and $\pi_{k}=-ika^{2}\phi_{k}$ as
$k\eta\to-\infty$, where $\eta$ is the conformal time, related to the cosmic
time $t$ via $\mathrm{d}t=a\mathrm{d}\eta$. Together with eq. (2.4), this
specifies the state of the perturbations at any time. In particular, this is a
Gaussian state, given that the fundamental state of the harmonic oscillator is
Gaussian and that the evolution is linear.

###  2.3 Coarse-grained fields

In practice, cosmological perturbation theory provides an accurate description
of cosmological fluctuations so long as their amplitude remains small. In the
context of PBHs, one is precisely interested in situations where cosmological
perturbations grow on super-Hubble scales, hence non-perturbative frameworks
are required at large scales. One such framework is the stochastic-inflation
formalism, which describes the dynamics of fields coarse-grained at the
$\sigma$-Hubble radius, $(\sigma H)^{-1}$. Here, $\sigma$ is a fixed parameter
that sets the scale at which quantum fluctuations backreact onto the local
FLRW geometry. In practice, one must set $\sigma\ll 1$ in order for gradient
terms not to affect the dynamics of the coarse-grained field, see Sec. 3. One
thus decomposes

$\displaystyle\phi=\tilde{\phi}+\phi_{Q}$ (2.6)
$\displaystyle\Pi=\tilde{\pi}+\pi_{Q}$

where $\tilde{\phi}$ and $\tilde{\pi}$ are coarse-grained fields, and

$\displaystyle\phi_{Q}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\frac{\mathrm{d}^{3}\vec{k}}{(2\pi)^{3/2}}W\left(\frac{k}{k_{\sigma}}\right)\left[a_{\vec{k}}~{}\phi_{k}(\tau)e^{-i\vec{k}\cdot\vec{x}}+a^{\dagger}_{\vec{k}}~{}\phi^{\star}_{k}(\tau)e^{i\vec{k}\cdot\vec{x}}\right]$
(2.7) $\displaystyle\pi_{Q}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}\frac{\mathrm{d}^{3}\vec{k}}{(2\pi)^{3/2}}W\left(\frac{k}{k_{\sigma}}\right)\left[a_{\vec{k}}~{}\pi_{k}(\tau)e^{-i\vec{k}\cdot\vec{x}}+a^{\dagger}_{\vec{k}}~{}\pi^{\star}_{k}(\tau)e^{i\vec{k}\cdot\vec{x}}\right]$

are their small-wavelength complements. Here, $k_{\sigma}=\sigma aH$, and the
window function $W$ is such that $W\simeq 0$ for $k\ll k_{\sigma}$ and
$W\simeq 1$ for $k\gg k_{\sigma}$. The equations of motion for the coarse-
grained fields are given by

$\displaystyle\frac{\mathrm{d}{\tilde{\phi}}}{\mathrm{d}N}=$
$\displaystyle\frac{\tilde{\pi}}{a^{3}H}-\frac{\mathrm{d}\phi_{Q}}{\mathrm{d}N}+\frac{\pi_{Q}}{Ha^{3}}\,,$
(2.8) $\displaystyle\frac{\mathrm{d}{\tilde{\pi}}}{\mathrm{d}N}=$
$\displaystyle-\frac{a^{3}}{H}V^{\prime\prime}(\tilde{\phi})-\frac{\mathrm{d}\pi_{Q}}{\mathrm{d}N}-\frac{a^{3}}{H}V^{\prime\prime}(\tilde{\phi})\phi_{Q}+\frac{a}{H}\Delta\phi_{Q}.$

In these expressions, the Laplacian of $\tilde{\phi}$ has been dropped since
it is suppressed by $\sigma$, and a linear expansion in the small-wavelength
perturbations $\phi_{Q}$ and $\pi_{Q}$ has been performed. Replacing
$\phi_{Q}$ and $\pi_{Q}$ by eqs. (2.7), and making use of the fact that the
mode functions $\phi_{k}$ and $\pi_{k}$ satisfy eqs. (2.4), the above can be
rewritten as [7, 8, 9, 10, 11, 12]

$\displaystyle\frac{\mathrm{d}{\tilde{\phi}}}{\mathrm{d}N}=$
$\displaystyle\frac{\tilde{\pi}}{a^{3}H}+\xi_{\phi}(N)\,,$ (2.9)
$\displaystyle\frac{\mathrm{d}{\tilde{\pi}}}{\mathrm{d}N}=$
$\displaystyle-\frac{a^{3}}{H}V^{\prime\prime}(\tilde{\phi})+\xi_{\pi}(N)\,,$

where the quantum sources $\xi_{\phi}$ and $\xi_{\pi}$ are given by

$\displaystyle\xi_{\phi}$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{3}}\frac{\mathrm{d}^{3}\vec{k}}{(2\pi)^{3/2}}\frac{\mathrm{d}{W}}{\mathrm{d}N}\left(\frac{k}{k_{\sigma}}\right)\left[a_{\vec{k}}\phi_{k}(\tau)e^{-i\vec{k}\cdot\vec{x}}+a^{\dagger}_{\vec{k}}\phi^{\star}_{k}(\tau)e^{i\vec{k}\cdot\vec{x}}\right],$
(2.10) $\displaystyle\xi_{\pi}$ $\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{3}}\frac{\mathrm{d}^{3}\vec{k}}{(2\pi)^{3/2}}\frac{\mathrm{d}{W}}{\mathrm{d}N}\left(\frac{k}{k_{\sigma}}\right)\left[a_{\vec{k}}\pi_{k}(\tau)e^{-i\vec{k}\cdot\vec{x}}+a^{\dagger}_{\vec{k}}\pi^{\star}_{k}(\tau)e^{i\vec{k}\cdot\vec{x}}\right].$

They act as source terms in the equations of the motion for the coarse-grained
fields. Since they only involve sub-Hubble fluctuations, they can be evaluated
in cosmological perturbation theory where they obey Gaussian statistics. Their
properties are thus entirely captured by their two-point functions,

$\displaystyle\left\langle\xi_{f}(\vec{x}_{1},N)\xi_{g}(\vec{x}_{2},N^{\prime})\right\rangle=\frac{\mathrm{d}\ln
k_{\sigma}}{\mathrm{d}N}\mathcal{P}_{fg}\left[k_{\sigma}(N),N\right]\mathrm{sinc}\left[k_{\sigma}(N)|\vec{x}_{2}-\vec{x}_{1}|\right]\delta\left(N-N^{\prime}\right)$
(2.11)

where $f$ and $g$ denote either $\phi$ or $\pi$. Here, we have assumed that
the window function is sharp in Fourier space,
$W(k/k_{\sigma})=\Theta(k/k_{\sigma}-1)$. This explains why the quantum noises
are white, i.e. uncorrelated over time. In the above expression, we have also
introduced the reduced power spectrum,

$\displaystyle\mathcal{P}_{fg}(k,N)=\frac{k^{3}}{2\pi^{2}}f_{k}(N)g_{k}^{\star}(N)\,.$
(2.12)

###  2.4 Langevin equations

The next step involves describing spatial hypersurfaces of the universe as
ensembles of $\sigma$-Hubble patches. Between two such patches, the
correlations between the quantum sources are suppressed by the cardinal sine
function appearing in eq. (2.11). Since gradient interactions have been
already dropped when discarding the Laplacian of $\tilde{\phi}$ in eq. (2.8),
such correlations must be neglected at the order at which the calculation is
performed. This implies that $\sigma$-Hubble patches evolve independently,
which is usually referred to as the “separate-universe” picture.

Before elaborating more on the validity of this assumption in Sec. 3, let us
note that, in this approach, each patch is endowed with a local FLRW geometry,
in which the (local) expansion rate is still given by the Friedmann equation
(2.2), where the right-hand side must now be evaluated with the (local) values
of $\tilde{\phi}$ and $\tilde{\pi}$. Such an equation may however seem
problematic since, although $H$ is a geometrical classical quantity, at this
stage $\tilde{\phi}$ and $\tilde{\pi}$ are still quantum fields. The present
approach thus requires $\tilde{\phi}$ and $\tilde{\pi}$ to be seen as
classical, random fields, rather than quantum fields. The quantum sources
$\xi_{\phi}$ and $\xi_{\pi}$ become stochastic noises, with statistical
correlation functions that are identified with the quantum expectation values
(2.11).

Since $\xi_{\phi}$ and $\xi_{\pi}$ are Hermitian quantum fields placed in a
Gaussian state, and given that they enter linearly in the equations of motion
(2.9), they can be equivalently described by classical random variables
sharing the same two-point functions, and the two descriptions are perfectly
equivalent [13, 14, 12]. In contrast, the coarse-grained fields $\tilde{\phi}$
and $\tilde{\pi}$ may be placed in non-Gaussian states as an effect of non-
linear evolution, and in this case the stochastic and quantum descriptions are
not equivalent. It is indeed well-known that quantum systems can display
correlations that cannot be accounted for using classical setups, a famous
example being the celebrated Bell inequalities [15]. In fact, even if
$\tilde{\phi}$ and $\tilde{\pi}$ remain in Gaussian states, non-classical
correlations appear when considering “improper” combinations,111In this
context, “improper” denotes operators for which the Wigner-Weyl transform
takes values outside their spectrum [16]. for which Bell inequalities can be
explicitly violated [17, 18]. In practice however, these quantum signatures
are often hidden in the amplitude of the so-called decaying mode [19, 20, 21,
22], which is again gradient suppressed, and may be further concealed by
quantum decoherence [23, 24, 25, 26].

When such a “quantum-to-classical” transition takes place, the stochastic
approach is valid, and eqs. (2.9) may be seen as Langevin equations. The white
Gaussian noises $\xi_{\phi}$ and $\xi_{\pi}$ model the inflow of sub-Hubble
quantum fluctuations into the super-Hubble, coarse-grained sector of the
theory, which behaves classically.

##  3 Primordial perturbations from inflation

Traditionally when studying the evolution of inhomogeneous fluctuations in
cosmology, consistently perturbing the spacetime as well as the field content,
one allows for arbitrary, but small perturbations about the homogeneous and
isotropic background and then systematically expands order-by-order in a small
parameter controlling the deviation from FLRW symmetries [27]. However this
perturbative approach is liable to break down where fluctuations become large,
either due to the accumulation of many small perturbations over a long time
period, e.g., due to quantum diffusion, or a sudden large enhancement in the
variance of those perturbations at a particular time or scale, for example,
due to a localised feature in the scalar field potential.

###  3.1 Separate-universe approach

An important simplification occurs when studying perturbations on sufficiently
large scales where anisotropy and spatial gradients can be neglected. In that
case the evolution of the fields locally can be described using ordinary
differential equations for the time-dependence of effectively homogeneous and
isotropic fields in an FLRW cosmology, considerably simplifying the
calculation, and enabling us to track the nonlinear as well as linear
evolution. This is the separate-universe approach [28, 29]. Formally we can
interpret this as the leading-order behaviour in a gradient expansion [30, 31,
32, 33].

For example, if we consider linear perturbations about an FLRW cosmology of
the scalar field and the lapse function (allowing for local dilation of the
proper time, $\tau$ relative to the background cosmic time $t$):

$\phi\to\phi+\delta\phi\,,\quad\partial/\partial
t\to\partial/\partial\tau=(1-A)\partial/\partial t\,,$ (3.1)

then, at first order in the perturbations, the Klein-Gordon equation for the
scalar field at each point can be written as [34]

$\frac{\partial^{2}}{\partial\tau^{2}}(\phi+\delta\phi)+3(H+\delta
H)\frac{\partial}{\partial\tau}(\phi+\delta\phi)+V_{,\phi}(\phi+\delta\phi)=\nabla^{2}(\delta\phi)\,,$
(3.2)

where $3(H+\delta H)$ is the local expansion rate determined by the local
energy density. This has exactly the same form as the Klein-Gordon equation
for a homogeneous scalar field in an FLRW spacetime up to the spatial gradient
term, $\nabla^{2}\delta\phi$. Typically we require the spatial gradients of
the scalar field to be small relative to the Hubble rate appearing in the
Klein-Gordon equation (3.2), which suggests that the separate-universe
approach should be valid if we coarse-grain the fields on scales somewhat
larger than the Hubble scale. The separate universe approach turns out to be
particularly powerful during inflation, since the accelerated expansion of the
universe ensures that the spatial gradients of comoving wavemodes decrease
more quickly than the Hubble rate ($H>|\dot{H}/H|$). Thus spatial gradients
rapidly decrease as modes are stretched beyond the Hubble scale.

Intuitively we consider the coarse-grained field in each super-Hubble sized
patch of the universe to be evolving like a separate FLRW universe, where the
fields may take different values in different patches, but are treated as
homogeneous within each patch. After sewing back together the different
patches, we can reconstruct the inhomogeneous evolution of our entire
observable universe above the coarse-graining scale. To relate local
quantities to a global coordinate system amounts to a choice of gauge. This
cannot be determined by the local FLRW quantities but requires the use of
additional constraint equations from cosmological perturbation theory. For
example in the spatially-flat gauge the momentum constraint imposes a simple
relation between the lapse function and the scalar-field perturbation [35, 27]

$A=\frac{\dot{\phi}}{2M_{\mathrm{\footnotesize{Pl}}}^{2}H}\delta\phi\,.$

In the slow-roll limit $\dot{\phi}\to 0$ and the local proper time coincides
with the background cosmic time, but beyond slow roll one must consistently
account for local variations in the proper time interval in different patches
to be able to relate the separate-universe evolution locally to the evolution
of inhomogeneous perturbations about the global background [34].

Since the Langevin equations (2.9) use the logarithmic expansion, $N$, to
describe the evolution of the scalar field and its canonical momentum in local
patches, when we describe the ensemble of many different patches at a given
$N$, we are implictly using a gauge in which the integrated local expansion

$N\equiv\int(H+\delta H)\,(1+A)\,\mathrm{d}t\,,$ (3.3)

remains unperturbed. Quantum field fluctuations crossing outside the Hubble
scale during inflation are commonly calculated using the Sasaki-Mukhanov [36,
37] mode equation for scalar field fluctuations in the spatially-flat gauge
[35, 27]. In slow-roll inflation the integrated local expansion is unperturbed
in the spatially-flat gauge, but in general there is a gauge transformation
required to consistently describe scalar field fluctuations in a uniform-$N$
gauge [34].

While the separate universe approach works well on super-Hubble scales in many
situations, it has recently been shown [38] that it breaks down on a finite
range of super-Hubble scales in inflationary models with features, such as an
inflection point, leading to a sudden, non-adiabatic transition from slow roll
to a transient, ultra-slow-roll phase. This boosts the amplitude of density
perturbations on small scales, which exit the Hubble-scale after the
transition, which could lead to the formation of primordial black holes after
inflation, or a stochastic background of gravitational waves induced at
second-order in perturbation theory. But sudden transitions can also lead to
spatial gradients generating non-adiabatic perturbations on super-Hubble
scales at the transition [39], breaking the usual assumption that the separate
universe approach can safely be used soon after Hubble exit. In such a
situation one should be cautious about applying the usual stochastic-inflation
formalism, for example, by waiting until after a sudden transition [38], or
until the transient, non-adiabatic perturbations have decayed, before using
the Langevin equations [40].

###  3.2 $\delta N$ formalism

Primordial density perturbations at the end of inflation are commonly
characterised in terms of the gauge-invariant variable, $\zeta$, which
corresponds to the scalar metric perturbation, ${\cal C}$, on uniform-density
hypersurfaces ($\delta\rho=0$). At first-order this is written in terms of the
curvature and density perturbations in an arbitrary gauge as [27]

$\zeta\equiv{\cal C}-\frac{H\delta\rho}{\dot{\rho}}\,.$ (3.4)

In particular it can be identified with the dimensionless density perturbation
on spatially-flat hypersurfaces

$\zeta=-\left.\frac{H\delta\rho}{\dot{\rho}}\right|_{{\cal C}=0}\,.$ (3.5)

It can be shown that $\zeta$ is conserved for adiabatic perturbations,
irrespective of the field or matter content of the universe, in the large-
scale limit where spatial gradients can be neglected [29, 41]. Hence it is a
particularly useful tool to link scalar-field fluctuations during inflation to
perturbations in the primordial plasma some time after inflation [42].

More generally $\zeta$ can be identified with the nonlinearly perturbed
expansion between a spatially-flat hypersurface (${\cal C}=0$) and the
uniform-density hypersurface [28, 43, 44]

$\zeta=\delta N\equiv N\left[\vec{\Phi}(\vec{x})\right]-\langle
N(\vec{\Phi})\rangle\,,$ (3.6)

where the local expansion $N$ was defined in equation (3.3). In our definition
of $\zeta$ in (3.7) we require that the lower limit of the integral (3.3) is a
spatially-flat hypersurface, where the initial field-phase space vector at
some point is given by $\vec{\Phi}(\vec{x})$, and the upper limit of the
integral is a uniform-density hypersurface at the end of inflation
corresponding to the field-phase space vector $\vec{\Phi}_{\mathrm{end}}$ at
the end of inflation.

Thus we are able to calculate the distribution of primordial density
perturbations after inflation in terms of the distribution of values of the
perturbed local expansion, $N$, from an initial spatially-flat hypersurface
during inflation to a uniform-density hypersurface at the end of inflation. If
we employ the separate-universe approach to determine the nonlinear evolution
on large scales, then we can use the background FLRW equations of motion to
determine local expansion, $N(\vec{\Phi})$, as a function of the local coarse-
grained field-phase space values $\vec{\Phi}(\vec{x})$ [1, 45, 46, 47].

In what we will call the classical $\delta N$ formalism the fields are coarse-
grained on one initial hypersurface such that all comoving scales of interest
are above the Hubble scale at that time. If we further restrict ourselves to
slow-roll evolution where the local field momenta are a function the local
field values, we can then formally expand eq. (3.7) to give [47]

$\zeta=\sum_{I}\frac{\partial
N}{\partial\phi^{I}}\delta\phi_{I}+\frac{1}{2}\sum_{I,J}\frac{\partial^{2}N}{\partial\phi^{I}\partial\phi^{I}}\delta\phi_{I}\delta\phi_{J}+\ldots\,.$
(3.7)

Furthermore for single-field slow-roll inflation we have $\partial
N/\partial\phi=-H/\dot{\phi}$ and hence eq. (3.7) reduces to eq. (3.5) at
first order, where $\rho=\rho(\phi)$.

The classical $\delta N$ formalism assumes that the subsequent evolution above
one fixed comoving coarse-graining scale is independent of further
fluctuations on smaller scales and can be described by the unperturbed
background equations. In the stochastic $\delta N$ formalism [48] we choose a
time-dependent coarse-graining scale, determined by the physical Hubble scale
at any given time, thus incorporating the effect of quantum diffusion due to
fluctuations on all scales that cross the coarse-graining scale before the end
of inflation. The integrated expansion up to the end of inflation becomes a
stochastic variable, ${\cal N}[\vec{\Phi}(\vec{x})]$, whose statistical
properties are a function of the initial field-phase space vector.

##  4 First-passage-time analysis

The Langevin equations (2.9) can be written more generally as

$\displaystyle\frac{\mathrm{d}{\Phi}^{i}}{\mathrm{d}N}=F^{i}(\vec{\Phi})+G^{i}_{j}(\vec{\Phi})\xi^{j}(N)\,,$
(4.1)

where $\vec{\Phi}$ is the field-phase space vector that comprises all fields
and conjugate momenta. The classical, homogeneous equations of motion are
contained in $\vec{F}$, while $\vec{G}$ is obtained by evolving the field
fluctuations from the Bunch-Davies vacuum to the $\sigma$-Hubble crossing
time. The white Gaussian noises $\xi_{i}$ are normalised such that
$\langle\xi^{i}(N)\xi^{j}(N^{\prime})\rangle=\delta^{i,j}\delta(N-N^{\prime})$.
In the case of single-field inflation with a canonical kinetic term, $\vec{F}$
and $\vec{G}$ can be read off eqs. (2.9). Along the slow-roll attractor,
quantum diffusion only occurs along the classical flow lines [12], hence field
phase space reduces to $\vec{\Phi}=(\phi)$ and one has
$F=-V^{\prime}/(3H^{2})$ and $G=H/(2\pi)$, with
$H^{2}=V/(3M_{\mathrm{\footnotesize{Pl}}}^{2})$. For now, let us keep the
generic form (4.1), since it allows us to describe any setup, including multi-
fields models in non-trivial field-space geometries [6].

###  4.1 Fokker-Planck equation

From the Langevin equation (4.1), one can derive a Fokker-Planck equation that
drives the probability
$P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})$ to find the
configuration $\vec{\Phi}$ at time $N$ knowing that the system was initiated
as $\vec{\Phi}_{\mathrm{in}}$ at time $N_{\mathrm{in}}$. It is given by [49]

$\displaystyle\frac{\partial}{\partial
N}P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})=$
$\displaystyle\left(-\frac{\partial}{\partial\Phi^{i}}F^{i}+\frac{1}{2}\frac{\partial^{2}}{\partial\Phi^{i}\partial\Phi^{j}}G^{j}_{k}G^{ki}\right)P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})$
(4.2) $\displaystyle\equiv$ $\displaystyle\mathcal{L}(\vec{\Phi})\cdot
P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})\,,$

where we use Itô’s discretisation convention. This defines the Fokker-Planck
operator $\mathcal{L}(\vec{\Phi})$, which is a differential operator acting in
the $\vec{\Phi}$ direction. The Fokker-Planck equation needs to be solved from
the initial condition
$P(\vec{\Phi},N_{\mathrm{in}}|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})=\delta(\vec{\Phi}-\vec{\Phi}_{\mathrm{in}})$,
and with an absorbing boundary condition on the end-of-inflation hypersurface
$\mathcal{C}_{\mathrm{end}}$, such that
$P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})=0$ when
$\vec{\Phi}\in\mathcal{C}_{\mathrm{end}}$, where $\mathcal{C}_{\mathrm{end}}$
is of uniform energy density.

The Langevin equation (4.1) describes a Markovian process, hence for any fixed
intermediate time $\bar{N}$, using the rules of conditional probabilities one
has

$\displaystyle
P\left(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}}\right)=\int_{\Omega}\mathrm{d}\bar{\vec{\Phi}}P\left(\vec{\Phi},N|\bar{\vec{\Phi}},\bar{N}\right)P\left(\bar{\vec{\Phi}},\bar{N}|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}}\right),$
(4.3)

where $\Omega$ is the inflating domain of field-phase space, bounded by
$\mathcal{C}_{\mathrm{end}}$. By differentiating both hands of this relation
with respect to $\bar{N}$, one obtains

$\displaystyle\frac{\partial}{\partial\bar{N}}P\left(\vec{\Phi},N|\bar{\vec{\Phi}},\bar{N}\right)=\mathcal{L}^{\dagger}(\bar{\vec{\Phi}})\cdot
P\left(\vec{\Phi},N|\bar{\vec{\Phi}},\bar{N}\right),$ (4.4)

where $\mathcal{L}^{\dagger}$ is adjoint to the Fokker-Planck operator
$\mathcal{L}$ in the sense that, for any two field-phase space functions
$f(\vec{\Phi})$ and $g(\vec{\Phi})$,
$\int\mathrm{d}\vec{\Phi}f(\vec{\Phi})[\mathcal{L}(\vec{\Phi})\cdot
g(\vec{\Phi})]=\int\mathrm{d}\vec{\Phi}[\mathcal{L}^{\dagger}(\vec{\Phi})\cdot
f(\vec{\Phi})]g(\vec{\Phi})$. This is why eq. (4.4) is called the adjoint
Fokker-Planck equation, or Kolmogorov backward equation, where, using
integration by parts

$\displaystyle\mathcal{L}^{\dagger}(\vec{\Phi})=F^{i}\frac{\partial}{\partial\Phi^{i}}+\frac{1}{2}G^{i}_{k}G^{ki}\frac{\partial^{2}}{\partial\Phi^{i}\partial\Phi^{j}}\,.$
(4.5)

Our goal is to reconstruct the distribution of first-passage times
$\mathcal{N}$ through the end-of-inflation hypersurface
$\mathcal{C}_{\mathrm{end}}$, starting from a given field configuration
$\vec{\Phi}_{\mathrm{in}}$, and identify it with the coarse-grained curvature
perturbation using eq. (3.7). This is the so-called stochastic-$\delta N$
program [50, 51, 48, 52]. The first-passage time distribution is denoted
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})$, and
is a central topic in stochastic analysis, see for instance Ref. [53]. Here,
we only give a few key results, following Refs. [48, 52, 54] in the context of
inflationary cosmology.

A direct way to reconstruct the distribution of first passage times is through
numerical sampling of the Langevin equation (2.9): for each realisation,
record the duration of inflation $\mathcal{N}$, and draw a histogram (or use
more advanced kernel reconstruction methods). This approach is numerically
expensive, since a large number of realisations needs to be simulated to reach
satisfactory levels of statistical noise, especially in the tail of the first-
passage-time distribution where the statistics is scarce. Importance sampling
can be used to circumvent this issue, see Ref. [55], or methods based on the
formalism of stochastic excursions, see Ref. [56]. However, for analytical
insight it is useful to derive a driving equation for
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})$,
analogous to the Fokker-Planck equation for
$P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})$. This is the goal
of this section.

###  4.2 A warm up: free diffusion

Let us start with the simple toy model consisting of a scalar field $\phi$
diffusing along a constant potential, along the slow-roll attractor. Upon
introducing the rescaled variable $x=2\pi(\phi-\phi_{\mathrm{end}})/H$, the
Langevin equation reads $\mathrm{d}x/\mathrm{d}N=\xi(N)$, and the Fokker-
Planck equation is given by

$\displaystyle\frac{\partial}{\partial
N}P(x,N|x_{\mathrm{in}},N_{\mathrm{in}})=\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}P(x,N|x_{\mathrm{in}},N_{\mathrm{in}})\,,$ (4.6)

which admits Gaussian solutions of the form

$\displaystyle
f_{\mathrm{x_{0},\sigma_{0}}}(x,N)=\frac{e^{-\frac{(x-x_{0})^{2}}{2\sigma^{2}(N)}}}{\sqrt{2\pi\sigma^{2}(N)}}\quad\quad\text{with}\quad\quad\sigma^{2}(N)=\sigma_{0}^{2}+N-N_{\mathrm{in}}\,.$
(4.7)

Since it is linear, any linear combination of such functions is still a
solution. The linear combination that allows one to satisfy both the initial
condition
$P(x,N_{\mathrm{in}}|x_{\mathrm{in}},N_{\mathrm{in}})=\delta(x-x_{\mathrm{in}})$
and the absorbing condition $P(0,N|x_{\mathrm{in}},N_{\mathrm{in}})=0$ is
given by

$\displaystyle
P(x,N|x_{\mathrm{in}},N_{\mathrm{in}})=\left[f_{\mathrm{x_{\mathrm{in}},0}}(x,N)-f_{\mathrm{-x_{\mathrm{in}},0}}(x,N)\right]\Theta(x)\,,$
(4.8)

where the Heaviside function restricts the solution to the inflating domain
$x>0$. This solution is obtained by adding a (negative) image of the
unconstrained solution mirrored through the absorbing boundary, which is often
referred to as the “method of images”.

From the solution to the Fokker-Planck equation, one can compute the survival
probability $S_{x_{\mathrm{in}}}(N)$, i.e. the probability for the field to be
still inflating at time $N$

$\displaystyle
S_{x_{\mathrm{in}}}(N)=\int_{0}^{\infty}\mathrm{d}xP(x,N|x_{\mathrm{in}},N_{\mathrm{in}})=\mathrm{erf}\left[\frac{x_{\mathrm{in}}}{\sqrt{2(N-N_{\mathrm{in}})}}\right].$
(4.9)

Alternatively, this can be written as the probability that, starting from
$x_{\mathrm{in}}$, the first-passage time through the end of inflation
$x_{\mathrm{end}}$ exceeds $N-N_{\mathrm{in}}$, hence

$\displaystyle
S_{x_{\mathrm{in}}}(N)=\int_{N-N_{\mathrm{in}}}^{\infty}P_{\mathrm{\footnotesize{FPT}},x_{\mathrm{in}}}(\mathcal{N})\mathrm{d}\mathcal{N}\,.$
(4.10)

By equating the two expressions above, and differentiating with respect to
$N$, one obtains
$P_{\mathrm{\footnotesize{FPT}},x_{\mathrm{in}}}(\mathcal{N})=-\left.\partial
S_{x_{\mathrm{in}}}/\partial N\right|_{N=N_{\mathrm{in}}+\mathcal{N}}$, hence

$\displaystyle
P_{\mathrm{\footnotesize{FPT}},x_{\mathrm{in}}}(\mathcal{N})=\frac{x_{\mathrm{in}}}{\sqrt{2\pi}\mathcal{N}^{3/2}}\exp\left(-\frac{x_{\mathrm{in}}^{2}}{2\mathcal{N}}\right).$
(4.11)

This is called a Lévy distribution. One can see that it features a very heavy
tail, since
$P_{\mathrm{\footnotesize{FPT}},x_{\mathrm{in}}}(\mathcal{N})\propto\mathcal{N}^{-3/2}$
at large $\mathcal{N}$. This implies that the probability to realise large
excursion times $\mathcal{N}$, hence large curvature perturbations according
to Sec. 3, is much larger than what would be obtained with Gaussian
statistics. Although this conclusion is obtained with a toy, arguably
unrealistic model, it is in fact entirely generic, as we shall now see.

###  4.3 Adjoint Fokker-Planck equation

In general, the survival probability starting from a given configuration
$\vec{\Phi}_{\mathrm{in}}$ can be defined in a way analogous to eq. (4.9),
i.e.

$\displaystyle
S_{\vec{\Phi}_{\mathrm{in}}}(N)=\int_{\Omega}\mathrm{d}\vec{\Phi}P\left(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}}\right).$
(4.12)

Since $P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})$ satisfies the
adjoint Fokker-Planck equation (4.4), so does the survival probability,
$(\partial/\partial
N_{\mathrm{in}})S_{\vec{\Phi}_{\mathrm{in}}}(N)=\mathcal{L}^{\dagger}(\vec{\Phi}_{\mathrm{in}})\cdot
S_{\vec{\Phi}_{\mathrm{in}}}(N)$. Since $\vec{F}(\vec{\Phi})$ and
$\vec{G}(\vec{\Phi})$ do not depend on time explicitly, the survival
probability only depends on the time difference $N-N_{\mathrm{in}}$, hence
$(\partial/\partial
N_{\mathrm{in}})S_{\vec{\Phi}_{\mathrm{in}}}(N)=-(\partial/\partial
N)S_{\vec{\Phi}_{\mathrm{in}}}(N)$. Moreover, similarly to eq. (4.10) one can
write
$S_{\vec{\Phi}_{\mathrm{in}}}(N)=\int_{N-N_{\mathrm{in}}}^{\infty}P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})\mathrm{d}\mathcal{N}$.
Combining these results, one obtains

$\displaystyle\frac{\partial}{\partial\mathcal{N}}P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})=\mathcal{L}^{\dagger}(\vec{\Phi}_{\mathrm{in}})\cdot
P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})\,,$
(4.13)

i.e. the first-passage time distribution obeys the adjoint Fokker-Planck
equation. The only difference with the Kolmogorov backward equation (4.4) is
the boundary conditions: here, eq. (4.13) needs to be solved with
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})=\delta(\mathcal{N})$
when $\vec{\Phi}_{\mathrm{in}}\in\mathcal{C}_{\mathrm{end}}$. In the free-
diffusion problem, one can check that, indeed, the Lévy distribution (4.11)
satisfies the adjoint Fokker-Planck equation, which in that case coincides
with the Fokker-Planck equation (4.6) since $F=0$ and $G$ is uniform.

##  5 Heavy tails and non-perturbative non-Gaussianities

The first-passage-time distribution depends on the details of the inflationary
model under consideration, and a variety of profiles can be obtained, see for
instance Refs. [52, 57, 58, 59, 54, 60, 61, 62, 63, 64, 55, 65, 66, 67, 68,
69]. In all cases however, the tail of the distribution, which is relevant for
the formation of PBHs, is heavier than the Gaussian behaviour predicted by
linear perturbation theory. In this section, we briefly review why.

###  5.1 Characteristic function

The adjoint Fokker-Planck equation (4.13) is a linear partial differential
equation, hence it can be turned into a set of ordinary differential equations
by introducing the characteristic function

$\displaystyle\chi\left(t,\vec{\phi}\right)=\left\langle
e^{it\mathcal{N}}\right\rangle=\int_{-\infty}^{\infty}\mathrm{d}\mathcal{N}e^{it\mathcal{N}}P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}}(\mathcal{N})\,.$
(5.1)

This is nothing but the Fourier transform of the first-passage-time
distribution, which is a function of the dummy parameter $t$. Using eq.
(4.13), it satisfies the differential equation

$\displaystyle\mathcal{L}^{\dagger}\left(\vec{\Phi}\right)\cdot\chi(t,\vec{\Phi})=-it\chi\left(t,\vec{\Phi}\right),$
(5.2)

together with the boundary condition $\chi(t,\vec{\Phi})=1$ when
$\vec{\Phi}\in\mathcal{C}_{\mathrm{end}}$.

###  5.2 Pole expansion

In general, the presence of non-trivial boundary conditions results in the
existence of poles, and the characteristic function can be expanded as

$\displaystyle\chi\left(t,\vec{\Phi}\right)=f\left(t,\vec{\Phi}\right)+\sum_{n}\frac{a_{n}(\vec{\Phi})}{\Lambda_{n}-it}$
(5.3)

where $f$ is a regular function, and we have assumed that the poles form a
discrete set. The form (5.3) follows from the structure of the Fokker-Planck
operator and can be justified formally, see Ref. [54], where the parameters
$\Lambda_{n}$ are shown to correspond to the eigenvalues of the adjoint
Fokker-Planck operator, hence they do not depend on $\vec{\Phi}$.

Let us illustrate this decomposition with the example of free diffusion
studied in Sec. 4.2, where an additional reflective boundary is added at
$\phi_{\mathrm{r}}$. This boundary can be removed by evaluating the formulas
below in the limit $\phi_{\mathrm{r}}\to\infty$ but at this stage
$\phi_{\mathrm{r}}$ needs to be kept finite, since only when adding the
boundary condition
$(\partial/\partial\phi)\chi(t,\phi)|_{\phi=\phi_{\mathrm{r}}}=0$ is the
solution to eq. (5.2) fully specified, and reads

$\displaystyle\chi(t,\phi)=\frac{\cos\left[\sqrt{it}\mu\left(\frac{\phi}{\phi_{\mathrm{r}}}-1\right)\right]}{\cos\left(\sqrt{it}\mu\right)}\quad\quad\text{where}\quad\quad\mu^{2}=\frac{24\pi^{2}M_{\mathrm{\footnotesize{Pl}}}^{2}\phi^{2}_{\mathrm{r}}}{V}$
(5.4)

and where we have set $\phi_{\mathrm{end}}=0$ without loss of generality. The
partial fraction decomposition of $\chi(t,\phi)$ is of the form (5.3), with
$\Lambda_{n}=(\pi/\mu)^{2}(n+1/2)^{2}$ and
$a_{n}(\phi)=(-1)^{n}\pi(2n+1)\cos[\frac{\pi}{2}(2n+1)(\frac{\phi}{\phi_{\mathrm{r}}}-1)]/\mu^{2}$.

The inverse Fourier transform that is required to derive the first-passage-
time distribution can be performed by means of the residue theorem, and one
obtains

$\displaystyle
P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}}\left(\mathcal{N}\right)=\sum_{n}a_{n}(\vec{\Phi})e^{-\Lambda_{n}\mathcal{N}}\,.$
(5.5)

On the tail, i.e. at large values of $\mathcal{N}$, the lowest eigenvalues
$\Lambda_{n}$ dominate, hence if the poles are ordered such that
$\Lambda_{0}<\Lambda_{1}<\cdots$, the pole expansion is essentially a tail
expansion.

###  5.3 Tail behaviour

Figure 1: Probability density functions of the number of $e$-folds generated
by an inflection-point inflationary potential (see inset, with
$V_{0}/(24\pi^{2}M_{\mathrm{\footnotesize{Pl}}}^{4})=10^{-3}$,
$\alpha=0.24/M_{\mathrm{\footnotesize{Pl}}}$,
$\beta=9M_{\mathrm{\footnotesize{Pl}}}^{3}$ and
$\Delta\phi_{\mathrm{well}}=5.83M_{\mathrm{\footnotesize{Pl}}}$ – the value
for $V_{0}$ is unrealistically large but it is used for convenience of the
illustration). Solid lines correspond to the full distribution functions
computed by mean of the stochastic-$\delta N$ formalism, where different
colours correspond to different initial conditions. The dotted lines
correspond to the Gaussian approximation provided by linear perturbation
theory, which provides a good fit for the maximum of the distribution but
however fails to describe the heavy tails, where PBHs are nonetheless
produced. Adapted from Ref. [54].

In Fig. 1, we display the first-passage-time distribution obtained in a
tilted-inflection point model, for a few initial conditions. The solid lines
correspond to the numerical solutions of eq. (4.13) (in practice, we solved
eq. (5.2) on a grid of $t$ values and inverse-Fourier transformed the result),
while the dotted lines correspond to the prediction of linear cosmological
perturbation theory. In perturbation theory, one obtains Gaussian
distributions, which provide good fits to the full distributions close to the
maximum. This is because the statistics of the most likely fluctuations are
almost Gaussian. However, in the tail of distribution, the Gaussian
approximation breaks down. Indeed, when $\mathcal{N}$ becomes large, eq. (5.5)
reduces to
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}}\left(\mathcal{N}\right)\simeq
a_{0}(\phi)e^{-\Lambda_{0}\mathcal{N}}$, i.e. the tail has an exponential
profile. Compared to Gaussian suppression,
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}}\left(\mathcal{N}\right)\propto
e^{-\propto\mathcal{N}^{2}}$, the tail is much heavier, and since large values
of $\mathcal{N}$ correspond to large values of $\zeta$, this can have a strong
impact on the abundance of PBHs [70].

If $\zeta$ has Gaussian statistics, the probability to realise
$\zeta>\zeta_{\mathrm{c}}\sim 1$ is of order
$p_{\mathrm{c}}=\mathrm{erfc}[\zeta_{\mathrm{c}}/(\sqrt{2\mathcal{P}_{\zeta}})]/2\propto
e^{-\zeta_{\mathrm{c}}^{2}/(2\mathcal{P}_{\zeta})}$, where $\mathrm{erfc}$ is
the complementary error function, which we have expanded in the regime
$\zeta_{\mathrm{c}}\gg\sqrt{\mathcal{P}_{\zeta}}$. PBH dark-matter typically
requires the PBH mass fraction to be of order $p_{\mathrm{c}}\sim 10^{-20}$ at
formation, see Part LABEL:part:constraints, which thus corresponds to
$\mathcal{P}_{\zeta}\simeq 10^{-2}$. For this reason it is generally
considered that, for dark matter to be comprised of PBHs, such large values of
the power spectrum are required, which can be challenging to achieve from a
model-building perspective. However, in the presence of exponential tails, the
situation is quite different. For instance, consider the case where the PDF of
$\zeta$ is Gaussian across $n_{\mathrm{SD}}$ standard deviations around its
maximum, and then exponential. In that case $p_{\mathrm{c}}\simeq
e^{-n_{\mathrm{SD}}^{2}/2-\Lambda_{0}}/(\Lambda_{0}\sqrt{2\pi}\mathcal{P}_{\zeta})$,
which clearly signals that cosmologically-relevant values of $p_{\mathrm{c}}$
do not require large values of the power-spectrum. This is why it is crucial
to take quantum diffusion into account when it comes to computing the
abundance and statistics of PBHs.

Finally, let us note that a compact inflating domain is required for the
eigenvalues $\Lambda_{n}$ to form a discrete set. Indeed, separated poles in
the characteristic function follow from imposing boundary conditions at finite
field distances, in the same way that standing waves can assume only certain
frequencies in a finite-sized cavity. If the cavity becomes infinite, any
frequency is allowed, and likewise, if the inflating domain is not bounded,
$n$ becomes a continuous rather than discrete index in eq. (5.5). An explicit
example of this phenomenon is given by the pure-diffusion example: in that
case, $\Lambda_{n+1}-\Lambda_{n}=2(n+1)\pi^{2}/\mu^{2}$, which goes to zero,
since $\mu^{2}\to\infty$, when $\phi_{r}\to\infty$, hence the set of poles
becomes continuous. When this occurs the tail of
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}}(\mathcal{N})$ can no longer be
approximated by $a_{0}(\vec{\Phi})e^{-\Lambda_{0}\mathcal{N}}$, hence it
becomes even heavier than exponential. In the free-diffusion example, in the
limit $\phi_{\mathrm{r}}\to\infty$ we found a Lévy distribution, see eq.
(4.11), for which the tail is indeed monomial rather than exponential. This
makes the conclusions derived above even more relevant in that case.

###  5.4 Non-perturbative non-Gaussianities

Figure 2: Probability density function of the number of $e$-folds generated by
a flat well starting from $\phi=\phi_{\mathrm{r}}$, see eqs. (5.4)-(5.5). The
orange line corresponds to the Gaussian approximation, while the dashed green
line stand for the local $f_{\mathrm{\footnotesize{NL}}}$ expansion, see eq.
(5.6). The inset zooms in on the maximum of the distribution, where the
Gaussian and quasi-Gaussian approximations provide good fits. However, they
both fail to properly describe the heavy tails.

The importance of non-Gaussianities in shaping the detailed PBH statistics is
well-known, but so far it has been mostly investigated using perturbative
techniques, such as $f_{\mathrm{\footnotesize{NL}}}$-type parametrisations.
These expansions are perturbative in nature, i.e. they expand around Gaussian
statistics, introducing corrections arising from the bispectrum (the three-
point correlation function), the trispectrum (the four-point correlation
function), etc. For instance, if $\mathcal{N}_{\mathrm{G}}$ is a Gaussian
random variable, the local-$f_{\mathrm{\footnotesize{NL}}}$ parametrisation
corresponds to the approximation

$\displaystyle\mathcal{N}\simeq\mathcal{N}_{\mathrm{G}}+f_{\mathrm{\footnotesize{NL}}}\left(\mathcal{N}_{\mathrm{G}}^{2}-\left\langle\mathcal{N}_{\mathrm{G}}^{2}\right\rangle\right).$
(5.6)

The parameter $f_{\mathrm{\footnotesize{NL}}}$ can be extracted from the third
moment of $\mathcal{N}$, which consists of Taylor expanding the PDF of
$\mathcal{N}$ around its maximum. This approximation is compared to the full
distribution in Fig. 2 in the free-diffusion toy model. This illustrates that,
although the $f_{\mathrm{\footnotesize{NL}}}$ approximation improves the
quality of the fit close to the maximum, it fails to capture the tail
behaviour, which is crucial for the PBH abundance.

The reason for this failure is that the non-Gaussianities produced by quantum
diffusion are non-perturbative in nature. Thus, they require non-perturbative
parametrisations. A straightforward non-perturbative generalisation of eq.
(5.6) consists of treating the local-$f_{\mathrm{\footnotesize{NL}}}$
parametrisation as a Taylor expansion of an underlying local ansatz
$\mathcal{N}(\vec{x})=f[\mathcal{N}_{\mathrm{G}}(\vec{x}),\vec{x}]$, where $f$
is any (in general non-linear) function. This approach was recently followed
in Refs. [62, 70]. It has the advantage of allowing one to derive the
statistics of the compaction function, hence the mass fraction of PBHs, in a
consistent non-perturbative way, which is otherwise difficult (see Sec. 6.2
below). However, it is not yet clear whether or not non-Gaussianities
inherited from quantum diffusion are well-approximated by the above local
ansatz, and further investigations are required.

##  6 Prospects

Let us end this chapter by reviewing a few of the open challenges that remain
to be addressed to better understand the role of quantum diffusion in shaping
the statistics of PBHs. Our goal is not to list all of them, but simply to
highlight a few conceptual and technical issues that we think deserve further
attention in the years to come.

###  6.1 The stochastic-$\delta N$ formalism beyond slow roll

Most applications of the stochastic-$\delta N$ formalism to-date have been
restricted to slow-roll inflation in which the coarse-grained fields follow an
attractor trajectory where the fields’ momenta are uniquely determined by the
field values, reducing the phase-space to 1D for a single scalar field, for
example. However models of inflation where there is a significant enhancement
of primordial density perturbations on small scales necessarily break the
single-field and/or slow-roll assumptions and include transient, non-attractor
behaviour, such as ultra-slow roll.

Extending the phase-space beyond the slow-roll trajectory has several
important consequences. Firstly, and most obviously, it raises the
dimensionality of the probability distribution function
$P(\vec{\Phi},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}})$ and the Fokker-
Planck equation (4.2), and hence makes more challenging the first-passage time
problem to find
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})$. In
the case of a single field, for example, extending the phase space beyond slow
roll [12] leads to the the characteristic function (5.1) obeying a partial
differential equation [59] rather than an ordinary differential equation as in
the case of slow roll for a single field [52, 54]. The first-passage-time
problem in 2D has been studied in the ultra-slow roll limit, assuming the
usual Bunch-Davies vacuum state for a massless field in de Sitter [71, 59]. In
the small-velocity (diffusion-dominated) expansion one recovers at leading
order the previous slow-roll results [52], but at next-to-leading order we
find an additional series of poles in the characteristic function. This does
not affect the lowest pole, and hence the leading exponential term in the tail
of the probability distribution for the first passage time, but it can
suppress the amplitude of the tail.

There are several other challenges. As we move away from the massless limit
the power spectrum for the stochastic noise due to quantum fluctuations
crossing outside the coarse-graining scale deviates from the simple result for
a massless field in de Sitter, $\langle\xi_{\varphi}^{2}\rangle=(H/2\pi)^{2}$,
and one must also include the noise affecting the field momentum, $\xi_{\pi}$
[12, 58, 72, 73, 74]. In slow roll, the fluctuations are overdamped on super-
horizon scales leading to a squeezed state that can be treated as classical
noise, but in the absence of an overdamped attractor then the question arises
as to whether a classical treatment remains sufficient.

Finally, we note that single-field models leading to a rapid rise in the
primordial power spectrum on small scales require a sudden, non-adiabatic
transition to force the background field out of slow-roll into an ultra-slow-
roll phase. The sudden transition leads to deviations from the Bunch-Davies
vacuum state (i.e. particle production) on sub-Hubble scales, and non-
adiabatic perturbations, sourced by residual spatial gradients, on a finite
range of super-Hubble scales at the transition [39]. These gradient terms are
not included in the usual separate-universe approximation and require a re-
evaluation of the classical and stochastic $\delta N$ formalisms at a sudden
transition [38].

###  6.2 From the curvature perturbation to the compaction function

So far in this chapter, the production of PBHs has been attributed to large
local values of $\zeta_{R}$, the curvature perturbation coarse-grained at the
scale $R$, where the mass of the resulting black hole scales with the Hubble
mass at the time the scale $R$ re-enters the Hubble radius. This is however
only an approximation, and modern PBH formation criteria rather depend on the
smoothed density contrast or the compaction function [75, 76, 77] to determine
whether a PBH forms, and use critical scaling to infer its mass [78, 79, 80].

The curvature perturbation is a direct outcome of the stochastic $\delta N$
formalism but the statistics of other relevant cosmological fields, at the
non-perturbative level, remain unexplored. One reason why the curvature
perturbation is not ideal to predict the formation of PBHs is that it is
sensitive to power at large scales. Indeed, from the coincident two-point
function, $\langle\zeta_{R}^{2}(\vec{x})\rangle=\int\mathrm{d}\ln
k\mathcal{P}_{\zeta}(k)\mathrm{sinc}(kR/a)$, one can see that power at scales
$k\ll a/R$ may contribute significantly to $\zeta_{R}$, depending on the power
spectrum. However, at large scales, $\zeta$ is conserved and may be
interpreted as a renormalisation of the scale factor seen by a local observer,
hence it should not affect the collapse dynamics of a local over-density. In
contrast, at the linear level the comoving density contrast is related to the
curvature perturbation via

$\displaystyle\delta_{\mathrm{lin}}=\frac{2(1+w)}{5+3w}\frac{k^{2}}{a^{2}H^{2}}\zeta\,,$
(6.1)

where $w$ is the equation-of-state parameter. The density contrast thus
features an additional $k^{2}$ suppression at large scales, which ensures that
only the scales $k\sim a/R$ contribute to the formation of black holes with
size $R$. If one accounts for the non-linear relationship between the
curvature perturbation and the density contrast, one is specifically led to
consider the compaction function, but the argument remains the same.

One possible approach to compute the statistics of the density contrast or the
compaction function is to consider the coarse-shelled curvature perturbation,
where instead of averaging $\zeta$ over a sphere of radius $R$, one considers
its average between two spheres of radii $R_{1}$ and $R_{2}$,
$\Delta\zeta_{R_{1},R_{2}}(\vec{x})=\zeta_{R_{2}}(\vec{x})-\zeta_{R_{1}}(\vec{x})$.
This selects scales in the interval $a/R_{2}<k<a/R_{1}$, so by adjusting the
parameters $R_{1}$ and $R_{2}$ around $R$, one can try to mimic
$\delta_{R}(\vec{x})$. Since the coarse-shelled curvature perturbation can be
written as the difference between two coarse-grained curvature perturbations,
its statistics can be computed in the stochastic-$\delta N$ formalism, and in
this way one may approximate the statistics of the comoving density contrast.
This approach is developed in Ref. [81], where it is also applied to the
compaction function. This requires us to compute the two-point distribution of
the curvature perturbation,
$P[\zeta_{R_{1}}(\vec{x}),\zeta_{R_{2}}(\vec{x})]$, but no higher statistics.

In those rare cases where all $n$-point functions are known, exact
calculations can also be performed. For instance, under the generalised local
ansatz mentioned in Sec. 5.4,
$\zeta(\vec{x})=f[\zeta_{\mathrm{G}}(\vec{x}),\vec{x}]$ where
$\zeta_{\mathrm{G}}$ is a Gaussian field, the statistics of the compaction
function $\mathcal{C}_{R}$ can be derived using

$\displaystyle\mathcal{C}_{R}=\frac{2}{3}\left[1-\left(1+R\zeta_{R}^{\prime}\right)^{2}\right]\,,$
(6.2)

where a prime denotes differentiation with respect to $R$. This approach was
followed in Refs. [62, 70], where the importance of heavy tails was confirmed.
Similarly, in the setup recently considered in Ref. [82], an explicit local
expression is given for $\zeta_{R}(\vec{x})$ in terms of a set of Gaussian
fields, which makes the computation of the compaction function tractable.

Beyond those simple cases however, the derivation of the compaction function
in the stochastic-$\delta N$ formalism remains an open issue. The use of more
advanced structure-formation tools, such as the excursion set [83, 84, 85] or
peak theory [86], together with critical scaling, also remains to be explored.

###  6.3 Volume weighting and backward observables

In the standard picture of cosmological perturbation theory, along a given
field-phase space classical trajectory, there is a one-to-one relationship
between comoving scales $k$ and the system’s configuration $\vec{\Phi}_{*}(k)$
when that scale crosses out the Hubble radius during inflation. Therefore, by
performing measurements of the statistics at the scale ${k}$, one is able to
probe the inflationary setup (the potential function, the field geometry, etc)
only around that location $\vec{\Phi}_{*}$. In the stochastic picture however,
such a one-to-one relationship does not exist. Assuming that $H$ is quasi-
constant during inflation, a given scale $k$ exits the Hubble radius at the
time $N_{*}(k)=N_{\mathrm{end}}-\ln(k_{\mathrm{end}}/k)$, where
$k_{\mathrm{end}}$ denotes the comoving Hubble scale at the end of inflation.
Therefore, if $\vec{\Phi}_{*}$ denotes the system’s configuration at the time
$N_{*}$, then it is a random quantity, and its distribution function is given
by [87]

$\displaystyle
P_{\mathrm{bw}}\left(\vec{\Phi}_{*}\right)=P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{*}}\left(N_{\mathrm{bw}}\right)\frac{\int_{0}^{\infty}P\left(\vec{\Phi}_{*},N|\vec{\Phi}_{\mathrm{in}},N_{\mathrm{in}}\right)\mathrm{d}N}{\int_{N_{\mathrm{bw}}}^{\infty}P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}\left(N_{\mathrm{tot}}\right)\mathrm{d}N_{\mathrm{tot}}}\,.$
(6.3)

Here, $N_{\mathrm{bw}}=\ln(k_{\mathrm{end}}/k)$ is the number of $e$-folds
corresponding to the Hubble exit of $k$, counted backwards from the end of
inflation. When the model under consideration features regions with large
quantum diffusion, this distribution function may have a wide support, which
implies that a broad region of field-phase space can be probed even if a
limited range of scales is observationally accessible. Note that this is the
case even if the scales being measured emerge prior to the diffusion-dominated
region, which allows one to place constraints on PBH-compliant models from CMB
measurements only [87].

Figure 3: Space-time structure of a stochastically-inflating universe (adapted
from Ref. [88]).

In practice, the relationship between field values and physical distances is
even more involved, and is encoded in the recursive geometry of a
stochastically inflating universe. Consider indeed the situation sketched in
Fig. 3. Here, each disk stands for a Hubble patch, and as inflation proceeds
each patch gives rise to new Hubble patches, until the end-of-inflation
hypersurface $\mathcal{C}_{\mathrm{end}}$ is being crossed. In this picture,
the inflating space-time has the structure of a tree in the language of graph
theory. From a given initial patch, denoted $\mathcal{P}_{*}$ in the figure
and with field value $\vec{\Phi}_{*}$, a certain volume $V$ is produced on the
final hypersurface, which corresponds to the number of leaves originating from
the root $\mathcal{P}_{*}$. The distribution function $P(V|\vec{\Phi}_{*})$ is
not straightforward to compute, since all branches originating from the root
$\mathcal{P}_{*}$ coincide until they split, hence the leaves cannot be seen
as the end-point of independent stochastic realisations of the Langevin
equation. Methods can nonetheless be developed to reconstruct the distribution
of the final volume [88], which contains the relevant information to match
distances as measured by local observers on the final hypersurface to field
configurations when the corresponding spatial region emerges during inflation.

Subtleties may however arise when considering volume weighting. Consider
indeed a local observer on the final hypersurface of the tree in Fig. 3, and
task them with measuring the mean density on that hypersurface. What they
would do is to measure the density in each leaf, and compute the ensemble
average. As mentioned above, ensemble averages over subsets of leaves
correspond to volume averages from the viewpoint of the root patch
$\mathcal{P}_{*}$. The reason is that regions of the universe that inflate
more give rise to a larger number of leaves on the final hypersurface, hence
they contribute more to ensemble averages computed by local observers. As a
consequence, when it comes to observable quantities, distributions should be
volume-weighted by $e^{3\mathcal{N}}$. For instance, for the curvature
perturbation coarse-grained at $R_{\mathrm{end}}$, the Hubble radius at the
end of inflation, one has

$\displaystyle P\left(\zeta_{R_{\mathrm{end}}}\right)\propto
P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}\left(\left\langle\mathcal{N}_{\vec{\Phi}_{\mathrm{in}}}\right\rangle_{V}+\zeta_{R_{\mathrm{end}}}\right)e^{3\zeta_{R_{\mathrm{end}}}}\,,$
(6.4)

where the overall factor is set such that the distribution is normalised, and
$\langle\cdot\rangle_{V}$ is the volume-weighted stochastic average, i.e.

$\displaystyle\left\langle\mathcal{N}_{\vec{\Phi}_{\mathrm{in}}}\right\rangle_{V}=\frac{\int
P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})e^{3\mathcal{N}}\mathcal{N}\mathrm{d}\mathcal{N}}{\int
P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})e^{3\mathcal{N}}\mathrm{d}\mathcal{N}}\,.$
(6.5)

This is what a local observer would measure by recording
$\mathcal{N}_{\vec{\Phi}_{\mathrm{in}}}$ in each leaf and computing the
ensemble average. Recalling that the first-passage-time distribution has an
exponential tail,
$P_{\mathrm{\footnotesize{FPT}},\vec{\Phi}_{\mathrm{in}}}(\mathcal{N})\propto
e^{-\Lambda_{0}\mathcal{N}}$, the above formula fails to converge when
$\Lambda_{0}\leq 3$. This is the case where “eternal inflation” takes place,
and the mean volume originating from a given patch $\mathcal{P}_{*}$ diverges.
Note that this issue cannot be avoided by starting inflation at sufficiently
low energy: as mentioned in Sec. 5.3, $\Lambda_{0}$ is independent of
$\vec{\Phi}$, hence if volume divergences arise somewhere then they arise
everywhere. Note also that, as mentioned in Sec. 5.3, when the field-phase
space is not compact, the tail of the first-passage time distribution is even
heavier than exponential, hence volume divergences necessarily occur. This is
the case for most inflationary models considered in the literature, hence
deriving properly volume-weighted observables in this context remains an
important challenge to be addressed.

## Acknowledgement

We would like to thank all the colleagues with whom we have investigated
quantum diffusion and primordial black holes over the past few years, in
particular Kenta Ando, Chiara Animali, Hooshyar Assadullahi, Vadim Briaud,
Jose-María Ezquiaga, Juan García-Bellido, Andrew Gow, Joseph Jackson, Kazuya
Koyama, Chris Pattison, Yuichiro Tada, Koki Tokeshi and Eemeli Tomberg.

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$V$-band extinction | $A_{\rm V}$ | mag | $0.062^{+0.082}_{-0.044}$ | EXOFASTv2 | –
SED photometry error scaling | $\sigma_{SED}$ |  | $14.8^{+3.9}_{-2.6}$ | EXOFASTv2 | –
Parallax | $\varpi$ | mas | $6.999\pm 0.020$ | EXOFASTv2 | –
Distance | $d$ | pc | $142.88^{+0.42}_{-0.40}$ | EXOFASTv2 | $134^{+18}_{-8}$
Age |  | Gyr | $5.9_{-3.8}^{+4.8}$ | EXOFASTv2 | $10.2\pm 2.5$
Transit parameters |  |  |  |  |
Sum of the fractional radii | $r_{\rm p}+r_{\star}$ |  | $0.0962312^{+0.0070}_{-0.0068}$ | JKTEBOP | –
Ratio of the fractional radii | $r_{\rm p}/r_{\star}$ |  | $0.0732\pm 0.0011$ | JKTEBOP | $0.0737\pm 0.0012$
Impact parameter | $b$ |  | $0.55^{+0.17}_{-0.13}$ | JKTEBOP | $0.30^{+0.11}_{-0.12}$
Planetary parameters |  |  |  |  |
Mass | $M_{\mathrm{p}}$ | $M_{\mathrm{Jup}}$ | $0.0577\pm 0.0069\pm 0.0013$ | ABSDIM | $0.059\pm 0.007$
Radius | $R_{\mathrm{p}}$ | $R_{\mathrm{Jup}}$ | $0.652\pm 0.055\pm 0.007$ | ABSDIM | $0.565^{+0.072}_{-0.032}$
Mean density | $\rho_{\mathrm{p}}$ | $\rho_{\mathrm{Jup}}$ | $0.195\pm 0.055\pm 0.002$ | ABSDIM | $0.32\pm 0.08$
Surface gravity | $g_{\mathrm{p}}$ |  m s-2 | $3.37\pm 0.69$ | ABSDIM | $4.47^{+0.90}_{-0.92}$
Equilibrium temperature | $T_{\mathrm{eq}}$ | K | $1080\pm 39$ | ABSDIM | $1001^{+66}_{-37}$
Safronov number | $\Theta$ |  | $0.0106\pm 0.0015\pm 0.0001$ | ABSDIM | $0.012\pm 0.002$
Orbital parameters |  |  |  |  |
Reference epoch of mid-transit | $T_{0}$ | BJD(TDB) | $2\,455\,304.65234\,(35)$ | Timing fit | 2 455 304.65122 (35)
Period | $P_{\mathrm{orb}}$ | days | 4.23450213 (76) | Timing fit | 4.234516 (15)
Inclination | $i$ | degree | $87.20\pm 0.86$ | JKTEBOP | $88.6^{+0.5}_{-0.9}$
Semi-major axis | $a$ | au | $0.04748\pm 0.00030\pm 0.00052$ | ABSDIM | $0.0479\pm 0.0006$
RV-curve semi-amplitude | $K_{\rm A}$ |  m s-1 | – | – | $8.5\pm 1.0$
Barycentric RV | $\gamma$ |  km s-1 | $Unconstrained$ | RM fit | $14.72\pm 0.10$
Projected spin-orbit angle | $\lambda$ | degree | $Unconstrained$ | RM fit | –
Table 20: Physical parameters of the planetary system HAT-P-29 derived in this work. Parameter | Nomen. | Unit | This Work | Source | Buchhave et al. (2011)
---|---|---|---|---|---
Stellar parameters |  |  |  |  |
Spectra class |  |  |  |  | G
Effective temperature | $T_{\mathrm{eff}}$ | K |  | $6140^{+29}_{-30}$
---
$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ 6140\pm 30$
| EXOFASTv2
---
HARPS-N
$6087\pm 88$
Metallicity | [Fe/H] |  |  | $+0.240^{+0.078}_{-0.080}$
---
$\leavevmode\nobreak\ +0.25\pm 0.08$
| EXOFASTv2
---
HARPS-N
$0.21\pm 0.08$
Projected rotational velocity | $v\,\sin{i_{\star}}$ |  km s-1 |  | $5.2\pm 0.7$
---
$4.5\pm 0.8$
| RM fit
---
HARPS-N
$3.9\pm 0.5$
Convective blue-shift velocity | CBV |  km s-1 | $>-0.58$ | RM fit | –
Luminosity | $L_{\star}$ | $L_{\sun}$ | $2.05^{+0.13}_{-0.12}$ | EXOFASTv2 | $1.84^{+0.47}_{-0.26}$
Mass | $M_{\star}$ | $M_{\sun}$ | $1.206_{-0.052\,-0.016}^{+0.048\,+0.011}$ | ABSDIM | $1.207\pm 0.046$
Radius | $R_{\star}$ | $R_{\sun}$ | $1.272_{-0.042\,-0.006}^{+0.032\,+0.004}$ | ABSDIM | $1.224^{+0.133}_{-0.075}$
Mean density | $\rho_{\star}$ | $\rho_{\sun}$ | $0.586_{-0.039}^{+0.049}$ | ABSDIM | –
Logarithmic surface gravity | $\log{g_{\star}}$ | cgs | $4.311_{-0.020\,-0.002}^{+0.026\,+0.001}$ | ABSDIM | $4.34\pm 0.06$
Equal Evolutionary Phase | $EEP$ |  | $347^{+41}_{-28}$ | EXOFASTv2 | –
$V$-band extinction | $A_{\rm V}$ | mag | $0.58\pm 0.11$ | EXOFASTv2 | –
SED photometry error scaling | $\sigma_{SED}$ |  | $4.87^{+1.40}_{-0.88}$ | EXOFASTv2 | –
Parallax | $\varpi$ | mas | $3.135^{+0.020}_{-0.019}$ | EXOFASTv2 | –
Distance | $d$ | pc | $318.9^{+1.9}_{-2.0}$ | EXOFASTv2 | $322^{+35}_{-21}$
Age |  | Gyr | $2.5_{-1.7\,-0.4}^{+1.2\,+0.4}$ | ABSDIM | $2.2\pm 1.0$
Transit parameters |  |  |  |  |
Sum of the fractional radii | $r_{\rm p}+r_{\star}$ |  | $0.0964^{+0.0030}_{-0.0020}$ | JKTEBOP | –
Ratio of the fractional radii | $r_{\rm p}/r_{\star}$ |  | $0.08777\pm 0.00063$ | JKTEBOP | $0.0927\pm 0.0028$
Impact parameter | $b$ |  | $0.37\pm 0.12$ | JKTEBOP | $0.591^{+0.062}_{-0.094}$
Planetary parameters |  |  |  |  |
Mass | $M_{\mathrm{p}}$ | $M_{\mathrm{Jup}}$ | $0.773_{-0.050\,-0.007}^{+0.050\,+0.005}$ | ABSDIM | $0.778^{+0.076}_{-0.040}$
Radius | $R_{\mathrm{p}}$ | $R_{\mathrm{Jup}}$ | $1.105_{-0.037\,-0.005}^{+0.031\,+0.003}$ | ABSDIM | $1.107^{+0.136}_{-0.082}$
Mean density | $\rho_{\mathrm{p}}$ | $\rho_{\mathrm{Jup}}$ | $0.536_{-0.053\,-0.002}^{+0.056\,+0.002}$ | ABSDIM | $0.54\pm 0.14$
Surface gravity | $g_{\mathrm{p}}$ |  m s-2 | $15.7_{-1.2}^{+1.3}$ | ABSDIM | $15.84\pm 2.55$
Equilibrium temperature | $T_{\mathrm{eq}}$ | K | $1281_{-26}^{+23}$ | ABSDIM | $1260^{+64}_{-45}$
Safronov number | $\Theta$ |  | $0.0772_{-0.0050\,-0.0002}^{+0.0052\,+0.0003}$ | ABSDIM | $0.077\pm 0.007$
Orbital parameters |  |  |  |  |
Reference epoch of mid-transit | $T_{0}$ | BJD(TDB) | $2\,455\,838.59462\,(65)$ | Timing fit | $2\,455\,197.5754\,(18)$
Period | $P_{\mathrm{orb}}$ | days | 5.7233746 (34) | Timing fit | 5.723186 (49)
Inclination | $i$ | degree | $88.82\pm 0.50$ | JKTEBOP | $87.1^{+0.5}_{-0.7}$
Semi-major axis | $a$ | au | $0.06667_{-0.00096\,-0.00029}^{+0.00089\,+0.00021}$ | ABSDIM | $0.0667\pm 0.0008$
RV-curve semi-amplitude | $K_{\rm A}$ |  m s-1 | – | – | $78.3\pm 5.9$
Barycentric RV | $\gamma$ |  km s-1 | $-21.651\pm 0.002$ | RM fit | $-21.670\pm 0.08$
Projected spin-orbit angle | $\lambda$ | degree | $-26\pm 16$ | RM fit | –