Primordial black holes from stochastic tunnelling Chiara AnimaliabVincent Venninb

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Primordial black holes from

stochastic tunnelling

Chiara Animali,a,b Vincent Venninb

aDipartimento di Fisica, Universit`a di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy,

and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy

bLaboratoire de Physique de l’´

Ecole Normale Sup´erieure, ENS, CNRS, Universit´e PSL,

Sorbonne Universit´e, Universit´e Paris Cit´e, F-75005 Paris, France

E-mail: chiara.animali@ens.fr,vincent.vennin@ens.fr

Abstract. If the inﬂaton gets trapped in a local minimum of its potential shortly before

the end of inﬂation, it escapes by building up quantum ﬂuctuations in a process known

as stochastic tunnelling. In this work we study cosmological ﬂuctuations produced in

such a scenario, and how likely they are to form Primordial Black Holes (PBHs). This

is done by using the stochastic-δN formalism, which allows us to reconstruct the highly

non-Gaussian tails of the distribution function of the number of e-folds spent in the false-

vacuum state. We explore two diﬀerent toy models, both analytically and numerically, in

order to identify which properties do or do not depend on the details of the false-vacuum

proﬁle. We ﬁnd that when the potential barrier is small enough compared to its width,

∆V/V < ∆φ2/M 2

Pl, the potential can be approximated as being ﬂat between its two

local extrema, so results previously obtained in a “ﬂat quantum well” apply. Otherwise,

when ∆V/V < V/M4

Pl, the PBH abundance depends exponentially on the height of

the potential barrier, and when ∆V/V > V/M 4

Pl it depends super-exponentially (i.e. as

the exponential of an exponential) on the barrier height. In that later case PBHs are

massively produced. This allows us to quantify how much ﬂat inﬂection points need to

be ﬁne-tuned. In a deep false vacuum, we also ﬁnd that slow-roll violations are typically

encountered unless the potential is close to linear. This motivates further investigations

to generalise our approach to non–slow-roll setups.

arXiv:2210.03812v2 [astro-ph.CO] 19 Feb 2023

Contents

1 Introduction and motivations 1

2 Statistics of ﬂuctuations in a false vacuum 5

2.1 Tail reconstruction 5

2.2 Linear toy model 6

2.3 Quadratic piecewise toy model 14

3 Implications for primordial black holes 19

4 Discussion and conclusion 23

A Additional formulas 26

1 Introduction and motivations

Cosmic inﬂation is, to date, the simplest and phenomenologically most successful

paradigm to describe the early universe [1–6]. Other than solving a number of puzzles in

the standard Big Bang cosmology, this early phase of almost exponential expansion also

provides the seeds for all cosmological structures in our universe, through the parametric

ampliﬁcation of quantum vacuum ﬂuctuations [7–11]. These cosmological perturbations

can be observed in the Cosmic Microwave Background (CMB) anisotropies [12–14], and

in the Large-Scale Structures (LSS) of the universe [15]. They are predicted to be almost

scale-invariant, quasi-Gaussian and quasi-adiabatic, which is in excellent agreement with

current observations [13].

However, we are still far from having a complete picture of the early universe. CMB

and LSS observations only give access to a restricted range of physical wavelengths, which

emerged from the Hubble radius during a short period of about 7 e-folds, over the ∼50

e-folds required to account for the observable universe. At these scales, perturbations

are constrained to be small, at the level of ζ'10−5, where ζis the so-called curvature

perturbation [16]. Even though this is enough to indicate that the inﬂationary potential

could be of the plateau type around 50 e-folds before the end of inﬂation (at least in

the minimal setting of single-ﬁeld slow-roll inﬂation) [17,18], the lack of observational

constraints at small scales makes the reconstruction of the inﬂationary potential close

to the end of inﬂation still elusive. On the one hand, this calls for new observational

windows at small scales; on the other hand, this prompts us to keep an open view about

possible deviations from “vanilla inﬂation” outside the constrained range [19], and to

investigate possible phenomenological consequences that might be looked for in those

new windows.

– 1 –

Figure 1. Sketch of the inﬂationary potential considered in this work. The large scales observed

in the CMB and the LSS exit the Hubble radius at large-ﬁeld value, where the potential is of the

plateau type (1). The inﬂaton φthen falls in a false vacuum state, i.e. a local minimum, from

which it escapes through quantum ﬂuctuations (2). It ﬁnally reaches the true vacuum state,

around which it oscillates during the preheating phase (3).

False vacuum

Among the panorama of possible features that could alter the inﬂationary potential at

small scales, in this work we consider the possibility of a false vacuum state, i.e. a local

minimum that is up-shifted compared to the global vacuum in which (p)reheating takes

place. The situation is depicted in Fig. 1. This is somehow reminiscent of the “old

inﬂation” proposal [3], where inﬂation occurs while the universe is trapped in a false

vacuum state, from which a graceful exit to the true vacuum is enabled by quantum

tunnelling [20,21]. It was then realised that reheating through percolation of true-

vacuum bubbles was challenging in this setup [22]. This led to the “new inﬂation”

proposal [4,5], in which inﬂation is driven by a scalar ﬁeld φalong a smooth potential

V(φ), and terminates by violation of the slow-roll conditions. The situation depicted in

Fig. 1may thus be seen as a hybrid setup mixing old and new inﬂation (see Refs. [23,24]

for other setups combining the two mechanisms).

False vacua may appear naturally in various high-energy constructions of inﬂa-

tionary potentials: when embedded in supersymmetry or supergravity constructions,

plateau-like potentials at CMB scales often include features like local minima at smaller

scales [25]. Moreover, in models yielding inﬂection points in the potentials, as usually

encountered in string- or supersymmetry-inspired constructions [26–29] (and as often

studied in the context of primordial black hole production [30–32]), the breaking of the

ﬂat-inﬂection point condition through radiative corrections [33–35] may create an ad-

ditional local minimum, depending on its sign. Finally, false vacua are found in more

speciﬁc scenarios such as the critical Higgs inﬂationary model [36,37].

When the inﬂaton encounters a false vacuum, there are two ways it can climb up

the potential barrier and reach the global minimum of the potential, close to which

– 2 –

inﬂation ends. This ﬁrst possibility, as recently investigated in Refs. [25,38–40], is that

the ﬁeld’s classical velocity is large enough to overshoot the local minimum. In that

case, the slow-roll conditions are necessarily violated, which leads to a sharp increase in

the amplitude of curvature perturbations that may later collapse into primordial black

holes (PBHs) [41–43]. The second possibility is that the inﬂaton gets trapped in the

false vacuum. In that case, quantum ﬂuctuations jiggle the inﬂaton and after some time

they shall push it outwards. In that case, large ﬂuctuations have to build up, and as a

consequence one may also expect PBHs to form.

The goal of this paper is to put this statement under closer scrutiny, and investigate

how quantum diﬀusion proceeds in a false-vacuum state during inﬂation. The reason is

that, while non-linear structure formation makes primordial ﬂuctuations of small ampli-

tude diﬃcult to reconstruct at small scales, the presence of PBHs is a distinct feature

that can be speciﬁcally looked for (and, if not detected, at least constrained). There-

fore, they constitute an important window into small-scales ﬂuctuations of large-enough

amplitude, hence into the potential presence of a false-vacuum state during inﬂation.

Stochastic-δN formalism

During inﬂation, quantum diﬀusion can be described by means of the stochastic-inﬂation

formalism [9,44], where small-scale ﬂuctuations behave as a random noise acting on the

large-scale evolution as they cross out the Hubble radius. These “quantum kicks” make

the inﬂaton wriggle away from the false vacuum, a process known as stochastic tunnelling

(note that it is diﬀerent from standard quantum tunnelling that proceeds below the

potential barrier and for which there is no classical description). Stochastic tunnelling

has been studied in various contexts in Refs. [24,45–57], and here we investigate how

this mechanism aﬀects cosmological perturbations. This can be done by following the

stochastic-δN approach, which we now brieﬂy summarise in the case of a single ﬁeld φ

slowly rolling on a potential V(φ) (see e.g. Ref. [58] for a more extensive review).

On the slow-roll attractor, the long-wavelength part of the inﬂaton is driven by the

Langevin equation

dφ

dN=−V0(φ)

3H2(φ)+H(φ)

2πξ , (1.1)

where N= ln(a) is the number of e-folds [59,60] with athe scale factor, a prime

denotes a derivative with respect to φ,H= ˙a/a is the Hubble parameter where a dot

denotes derivation with respect to cosmic time, and ξis a white Gaussian noise with

vanishing mean and unit variance, i.e. hξ(N)ξ(N0)i=δ(N−N0). This noise describes

the inﬂow of small wavelength scales as the universe expansion stretches them into the

long-wavelength sector, and hereafter h·i denotes stochastic average. At leading order in

slow roll, H2'V /(3M2

Pl) where MPl is the reduced Planck mass, because of Friedmann

equation. This Langevin equation can then be turned into a Fokker-Planck equation for

the probability density function (PDF), P(φ, N), associated with the ﬁeld value at time

∂

∂N P(φ, N) = M2

∂

∂φ v0(φ)

v(φ)P(φ, N)+M2

∂2

∂φ2[v(φ)P(φ, N)] ,(1.2)

– 3 –

where for convenience we have introduced the rescaled potential v=V/(24π2M4

Pl).

Starting from a certain initial ﬁeld value φ, let Ndenote the number of e-folds that

is realised until inﬂation ends. This is a random quantity, since it is diﬀerent for each

realisation of the stochastic process (1.1). It is therefore endowed with a distribution

function P(N, φ), which can be shown to follow the adjoint Fokker-Planck equation [61,

62], namely

∂

∂NP(N, φ) = −M2

v0(φ)

v(φ)

∂

∂φP(N, φ) + M2

Plv(φ)∂2

∂φ2P(N, φ).(1.3)

This equation needs to be solved with the boundary condition P(N, φend) = δ(N),

assuming that inﬂation ends at φend (an additional boundary condition is sometimes

needed at large-ﬁeld values [63,64]). The statistics of cosmological perturbations can

then be extracted using the δN formalism [65–68], which states that on super-Hubble

scales, the curvature perturbation ζis related to the integrated local amount of expansion

of a homogeneous patch, treated as a separate universe [60,65,68–73], i.e.

ζ(x) = N(x)− hNi .(1.4)

In this expression, which is valid even at the non-perturbative level, ζis coarsed-grained

at the Hubble radius at the end of inﬂation, but the statistics of the curvature per-

turbation (and of related quantities such as the density contrast or the compaction

function) when coarse-grained at arbitrary scales can be inferred using the techniques

introduced in Ref. [74]. In this way, solving the ﬁrst-passage time problem described by

Eq. (1.3) allows one to reconstruct the statistics of cosmological ﬂuctuations on large

scales, by taking into account the backreaction of small-scales quantum diﬀusion in a

non-perturbative way. This is the so-called stochastic-δN program [61,75,76], which

we intend to apply to the false-vacuum setup in this work.

Let us stress that the above equations are written in the slow-roll regime, which

at the classical level assumes that the acceleration term ¨

φis subdominant in the Klein-

Gordon equation ¨

φ+ 3H˙

φ+V0= 0. One may be concerned that this condition cannot

be satisﬁed around a local minimum of the potential, since there V0= 0. However,

slow roll being a dynamical attractor, if the potential function satisﬁes the slow-roll

conditions then the system does not leave the attractor even when approaching a local

minimum.1In other words, although it is true that V0decreases to 0 when reaching

the local minimum, so do 3H˙

φand ¨

φ, at a rate such that the acceleration term remains

negligible. Our use of the slow-roll approximation is therefore fully justiﬁed, as long as

one makes sure that the potential function satisﬁes the slow-roll conditions all along,

which we will carefully check in what follows.

1For explicitness, let us expand V'V0+m2φ2/2 around a local minimum located at φ= 0. Upon

linearising the Klein-Gordon and Friedmann equations around the phase-space point (φ= 0,˙

φ= 0), in

the regime m2H2

0=V0/(3M2

Pl ), one ﬁnds that φ(t) is attracted towards the solution φ∝e−m2t/(3H0),

which is such that 3H˙

φ' −V0(φ) and ¨

φ'm2/(9H2

0)V0(φ)V0(φ), and which therefore corresponds

to the slow-roll attractor.

– 4 –

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PrimordialblackholesfromstochastictunnellingChiaraAnimali,a;bVincentVenninbaDipartimentodiFisica,UniversitadiPisa,LargoB.Pontecorvo3,56127Pisa,Italy,andIstitutoNazionalediFisicaNucleare,SezionediPisa,Pisa,ItalybLaboratoiredePhysiquedel'EcoleNormaleSuperieure,ENS,CNRS,UniversitePSL,SorbonneUniver...

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Primordial black holes from stochastic tunnelling Chiara AnimaliabVincent Venninb

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