# PBH in single field inflation: the effect of shape dispersion and non-Gaussianities

TL;DR: In this article, the authors considered the effect of non-Gaussianities on the threshold for overdensities to collapse into a primordial black hole (PBH) and showed that the effect is small in determining the threshold.

Abstract: Primordial black holes (PBHs) may result from high peaks in a random field of cosmological perturbations. In single field inflationary models, such perturbations can be seeded as the inflaton overshoots a small barrier on its way down the potential. PBHs are then produced through two distinct mechanisms, during the radiation era. The first one is the familiar collapse of large adiabatic overdensities. The second one is the collapse induced by relic bubbles where the inflaton field is trapped in a false vacuum, due to large backward fluctuations which prevented horizon sized regions from overshooting the barrier. We consider (numerically and analytically) the effect of non-Gaussianities on the threshold for overdensities to collapse into a PBH. Since typical high peaks have some dispersion in their shape or profile, we also consider the effect of such dispersion on the corresponding threshold for collapse. With these results we estimate the most likely channel for PBH production as a function of the non-Gaussianity parameter $f_{\rm NL}$. We also compare the threshold for collapse coming from the perturbative versus the non perturbative template for the non-Gaussianity arising in this model. We show that i) for $f_{\rm NL}\gtrsim 3.5$, the population of PBH coming from false vacuum regions dominates over that which comes from the collapse of large adiabatic overdensities, ii) the non-perturbative template of the non-Gaussianities is important to get accurate results. iii) the effect of the dispersion is small in determining the threshold for the compaction function, although it can be appreciable in determining the threshold amplitude for the curvature perturbation at low $f_{\rm NL}$. We also confirm that the volume averaged compaction function provides a very accurate universal estimator for the threshold.

## Summary (3 min read)

Jump to: [1 Introduction] – [2 Large and rare peaks from single field inflation] – [2.1 The typical high peak profiles] – [3.1 The Misner-Sharp equations] – [3.2 The long wavelength approximation and initial conditions] – [3.3 The criterion for BH production] – [3.4 A universal threshold for collapse] – [4 Results] – [4.1 Case A: Perturbative template] – [4.2 Case B: Non-perturbative template] and [5 Summary and conclusions]

### 1 Introduction

- Primordial Black Holes (PBH) may have formed during the radiation dominated era due to unusually high peaks in the distribution of cosmological density perturbations [1–4].
- There are strong observational constraints on the abundance of PBH over a wide range of mass scales [5].
- This point is particularly relevant when a mean profile for the perturbations cannot be defined, as it is the case for large overdensities coming from the model of single-field inflation with a barrier4 [25].
- Note, however, that the baby universe is not in the trapped region, or “interior” of the black hole.
- 5Refs. [44, 51–53] consider the non-Gaussianity in the density perturbation δ due to the non-linear relation between δ and ζ.

### 2 Large and rare peaks from single field inflation

- At cosmological scales, the power spectrum of primordial perturbations must be of the order of 10−9, in accordance with observations of the cosmic microwave background.
- This jump in the amplitude can be achieved if the inflaton field passes trough a transient period with φ̈/Hφ̇ ≈ const. < −3. Throughout this paper, the authors shall refer to this as “constant-roll” (CR).
- Parametrically, the fraction of dark matter in PBH is ΩPBH ∼ 109(M /MPBH)1/2β0, where the probability of PBH formation at the time when a large perturbation crosses the horizon can be roughly estimated as β0 ∼ exp[−ζ2th/2σ20], for some threshold value ζth ∼.
- The remaining factors in the estimate of ΩPBH account for the dilution of radiation relative to PBH, from the time of their formation until the time teq.
- Because these perturbations are very rare, the authors can use the theory of high peaks to describe them.

### 2.1 The typical high peak profiles

- Since the non-Gaussian curvature perturbation ζ is a local function of the Gaussian field ζg, let us start by reviewing the latter [57].
- This distribution is almost Gaussian, except for a Jacobian prefactor which relates the condition of being an extremum with the condition for the peak to be at a certain location.
- For a Gaussian distribution, the mean and the median coincide, and therefore for the rest of this paper the authors shall refer to (2.4) as the median Gaussian profile.
- Note that at low fNL < 5/3, the fall-off is not sharp enough to make σ2 [given in (2.7)] indepent on the ultraviolet details of the spectrum.
- In what follows, the authors will simply introduce a sharp cut-off at the peak value kp.

### 3.1 The Misner-Sharp equations

- The Misner-Sharp equations (MS) are the Einstein’s equations for a spherically symmetric spacetime, in a frame comoving with a perfect fluid [63].
- The metric components A(r, t), B(r, t) and R(r, t) are all positive, the latter one corresponding to the areal radius of the 2-spheres.
- Eq. (3.10) is the Hamiltonian constraint, a redundant equation which is useful in order to check the accuracy of the time evolution.
- Let us now discuss the initial conditions for evolution in terms of the random field ζ(r) of primordial curvature perturbations.

### 3.2 The long wavelength approximation and initial conditions

- Initially, at early times, perturbations have a physical wavelength L much larger than the Hubble radius H−1 [46].
- Hence, the authors are going to consider the long wavelength approximation to determine the form of their initial metric and hydrodynamical variables.
- The authors have also restricted to spherical symmetry, which excludes the presence of tensor modes (gravitational waves).
- As mentioned below Eq. (3.11), all remaining variables can be obtained from the set (U, ρ,M,R), and so it is not necessary to specify any additional initial conditions.

### 3.3 The criterion for BH production

- The formation of a black hole for a given initial condition can be inferred from the behaviour of perturbations which do not dissipate after entering the horizon but continue growing until a trapped surface [64] is formed.
- This signals the onset of gravitational collapse.
- If both expansions are negative, the surface is called “trapped”, while if both are positive, the surface is “anti-trapped”.
- Here the authors use the definition of the compaction function given originally in [46], which has also been used in most of the subsequent literature.
- From a practical point of view, both criteria perform with similar efficiency in the simulations the authors have run.

### 3.4 A universal threshold for collapse

- In the long wavelength limit, the compaction function is time independent, and can be expressed in terms of the curvature perturbation as [47] C(r) = 1 3 (1− (1 + rζ ′(r))2), (3.22) where the prime denotes the partial derivative with respect to the radial coordinate.
- Since this window is relatively narrow, spanning less than a factor of 2, the use of a threshold Cth has been popular in phenomenological studies of PBH 11Note that this differs from the proper volume element on t = const.
- Nonetheless, the precise value of Cth still depends on the profile of the perturbation, and so in this approach the authors cannot completely dispense with numerical simulations of collapse in order to obtain accurate results.
- Here the authors shall further confirm its validity by checking that it holds to very good accuracy in the class of profiles that they will study.

### 4 Results

- For s = 1 this includes 68% of all realizations, including generic profiles which are not spherically symmetric.
- Since their numerical code assumes spherical symmetry, here the authors shall restrict attention to profiles with such symmetry.
- These are, roughly speaking, the envolvent of all realizations within one standard deviation from the median.
- This is beyond the scope of the present work, and is left for further research.
- The dispersions in the thresholds given in Eq. (4.3) are represented in Fig.

### 4.1 Case A: Perturbative template

- This case corresponds to the Dirac delta function power spectrum (2.8), together with the perturbative local template (2.15) for the relation between ζg and the curvature perturbation ζ.
- As soon as the dominant peak enters the horizon, at the time tH , the width of the secondary peaks will also be within the horizon, and the authors see that these secondary structures disipate due to pressure gradients.
- This is another reason why the authors do not expect these additional structures to form bigger PBHs.
- Their treatment of the dispersion by considering the profiles ζ± is only indicative, since it ignores the effect of non-sphericity, which is expected to shift the threshold to slightly higher values.
- 2 tends to further enhance the abundance of PBH with growing fNL, relative to the Gaussian case.

### 4.2 Case B: Non-perturbative template

- Let us now consider the single field model where the background inflaton overshoots a barrier in the slope of the potential, as in Fig.
- Assuming, for the sake of argument, that this is the case, one should then question what is the point of focussing on the average profile, as opposed to a more representative sample of all realizations.
- As a result, the dispersion in the threshold becomes small when µth becomes close to µ∗.
- Whether this can be done realistically is an interesting open question.

### 5 Summary and conclusions

- And of the statistical dispersion in the shape of high peaks, on the threshold for PBH formation.the authors.
- In the context of PBH formation, the curvature perturbation ζ is sizable, and it is important to consider the full non-perturbative relation between ζ and ζg.
- In particular, this reveals a new regime for PBH formation through the collapse of false vacuum bubbles.
- Numerically, it is hard to probe larger values of fNL because µth approaches µ∗, and the profiles become extremely peaked near the origin.
- Here the authors have estimated such dispersion, illustrated by the shaded regions in Fig. 4, by using the spherically symmetric profiles ζ±, which are the envelope of all realizations at one standard deviation from the median.

Did you find this useful? Give us your feedback

Prepared for submission to JCAP

PBH in single ﬁeld inﬂation: the

eﬀect of shape dispersion and

non-Gaussianities.

Vicente Atal, Judith Cid, Albert Escriv`a and Jaume Garriga

Departament de F´ısica Qu`antica i Astroﬁ´sica, i Institut de Ci`encies del Cosmos, Universitat

de Barcelona, Mart´ı i Franqu´es 1, 08028 Barcelona, Spain.

E-mail: vicente.atal@icc.ub.edu, jcidgime7@alumnes.ub.edu,

albert.escriva@fqa.ub.edu, jaume.garriga@ub.edu

Abstract. Primordial black holes (PBHs) may result from high peaks in a random ﬁeld

of cosmological perturbations. In single ﬁeld inﬂationary models, such perturbations can

be seeded as the inﬂaton overshoots a small barrier on its way down the potential. PBHs

are then produced through two distinct mechanisms, during the radiation era. The ﬁrst

one is the familiar collapse of large adiabatic overdensities. The second one is the collapse

induced by relic bubbles where the inﬂaton ﬁeld is trapped in a false vacuum. The latter

are due to rare backward ﬂuctuations of the inﬂaton which prevented it from overshooting

the barrier in horizon sized regions. We consider (numerically and analytically) the eﬀect

of non-Gaussianities on the threshold for overdensities to collapse into a PBH. Since typical

high peaks have some dispersion in their shape or proﬁle, we also consider the eﬀect of such

dispersion on the corresponding threshold for collapse. With these results we estimate the

most likely channel for PBH production as a function of the non-Gaussianity parameter

f

NL

. We also compare the threshold for collapse coming from the perturbative versus the

non perturbative template for the non-Gaussianity arising in this model. We show that i)

for f

NL

& 3.5, the population of PBH coming from false vacuum regions dominates over

that which comes from the collapse of large adiabatic overdensities, ii) the non-perturbative

template of the non-Gaussianities is important to get accurate results. iii) the eﬀect of the

dispersion is small in determining the threshold for the compaction function, although it can

be appreciable in determining the threshold amplitude for the curvature perturbation at low

f

NL

. We also conﬁrm that the volume averaged compaction function provides a very accurate

universal estimator for the threshold.

Contents

1 Introduction 1

2 Large and rare peaks from single ﬁeld inﬂation 3

2.1 The typical high peak proﬁles 3

2.2 Non-Gaussianity 5

3 The formation of PBH 6

3.1 The Misner-Sharp equations 7

3.2 The long wavelength approximation and initial conditions 8

3.3 The criterion for BH production 8

3.4 A universal threshold for collapse 10

4 Results 10

4.1 Case A: Perturbative template 11

4.2 Case B: Non-perturbative template 14

5 Summary and conclusions 16

1 Introduction

Primordial Black Holes (PBH) may have formed during the radiation dominated era due to

unusually high peaks in the distribution of cosmological density perturbations [1–4]. There

are strong observational constraints on the abundance of PBH over a wide range of mass scales

[5]. Nonetheless, these still allow for several phenomenologically interesting possibilities. For

instance, PBH of sublunar [6] or stellar mass [7, 8] may constitute a sizable fraction of all

dark matter in the universe

1

. Also, the origin of supermassive black holes at the center of

galaxies is not very well understood at present, and one possibility is that they may have

formed by accretion from a smaller intermediate mass PBH seed (for a recent review, see

[13]).

In order to make accurate predictions on the statistical properties of PBHs, it is nec-

essary to be speciﬁc about their formation process. One of the simplest mechanisms is

the collapse of large adiabatic perturbations seeded during a period of single-ﬁeld inﬂation

[14–23]. While ﬂuctuations must be predominantly Gaussian at the cosmic microwave back-

ground scales, those leading to PBH formation at smaller scales are typically non-Gaussian

[24–27]

2

. Suﬃciently large ampliﬁcation of the perturbations are induced while the inﬂation

overshoots a small barrier on its way down the potential, undergoing a periond of “constant

1

In the case of stellar masses, the strongest constraint on the fraction f of dark matter in the form of

PBHs may come from the observed rate of merger events by the LIGO/Virgo collaboration. Nonetheless,

such constraint can be substantially relaxed, or even voided, due to various environmental eﬀects which may

contribute to the eccentricity of PBH binaries at the time of their formation, or shortly after. Such eﬀects

may include the infall of PBH onto binaries and the collision of binaries with compact N body systems [9–11],

as well as the torque exerted by an enhanced power spectrum of cosmological perturbations at small scales

[12].

2

We expect this to be the case also in other scenarios leading to PBH formation, as variants of multiﬁeld

inﬂation [28–30] and non-canonical inﬂation [31–33].

– 1 –

roll” (see Fig. 1). In this context, PBH can be formed not only from the collapse of a large

adiabatic overdensity, but also from false vacuum bubbles which continue inﬂating in the

ambient radiation dominated universe, and eventually pinch oﬀ from it. This results in a

black hole which separates the ambient universe from an inﬂating baby universe [25, 34, 35].

3

A question of practical interest is to determine the abundance of PBHs. Several works

have already treated the inﬂuence of non-Gaussianities in the abundance of PBHs [25, 36–

45]. Since this turns to be large, it is important to i) predict the amplitude and shape of

the non-Gaussianities for a given model of PBH formation, and ii) consider their inﬂuence

beyond perturbation theory.

When PBHs are formed from rare overdensities, their abundance will depend on the

threshold for the amplitude of the overdensity to collapse once it reenters the horizon. This

threshold notoriously depends on the shape (or proﬁle) for the overdensity [46–50]. For

a Gaussian random ﬁeld, the typical shape of high peaks is determined from the power

spectrum, but if the distribution is non-Gaussian, the shape will also depend on the nature of

the non-Gaussianity [25, 43, 44]. Furthermore, since ﬂuctuations are drawn from a statistical

distribution, the shapes of perturbations susceptible of collapsing will inherit a dispersion.

While the mean proﬁle is usually taken to be representative of the typical shape, it seems

important to consider how the threshold may vary due to the dispersion of shapes. This

point is particularly relevant when a mean proﬁle for the perturbations cannot be deﬁned,

as it is the case for large overdensities coming from the model of single-ﬁeld inﬂation with a

barrier

4

[25].

In this work we study the dependence of the threshold on the dispersion of the proﬁles,

including the non-Gaussianity resulting from the physics of single-ﬁeld inﬂation. The non-

Gaussianity is entirely due to the non-linear relation between the Gaussian variable

ζ

g

≡ − H

δφ

˙

φ

sr

, (1.1)

and the non-Gaussian gauge-invariant curvature perturbation ζ. Here δφ is the inﬂaton ﬁeld

perturbation in the ﬂat slicing, evaluated at the onset of the slow roll attractor behaviour

past the top of the barrier, and H is the expansion rate during inﬂation

5

. For the non-

linear relation between ζ and ζ

g

, we will compare the non-perturbative expression which

follows from the single ﬁeld model where the inﬂaton overshoots a small barrier [25], with

the more widely used perturbative Taylor expansion of ζ to second order in ζ

g

(parametrized

by the standard coeﬃcient f

NL

). These non-perturbative and perturbative versions of local

non-Gaussianity are given, respectively, in Eqs. (2.13) and (2.15) below.

We will ﬁnd the thresholds for collapse into a PBH under the assumption of spherical

symmetry, by using a recently developed numerical code [54]. This solves the Misner-Sharp

(MS) partial diﬀerential equations by using spectral methods. We will also compare the

3

These are sometimes refered to as black holes with a baby universe inside. Note, however, that the baby

universe is not in the trapped region, or “interior” of the black hole. Rather, the trapped region separates

two normal regions, one in the parent ambient universe and the other in the baby universe, which were once

causally connected but are not anymore, after the trapped region forms.

4

In a nutshell, the problems is that ζ diverges when the amplitude of ζ

g

reaches a critical value µ

∗

, and it

is not even deﬁned for larger amplitude of ζ

g

, for which there is a ﬁnite probability.

5

Refs. [44, 51–53] consider the non-Gaussianity in the density perturbation δ due to the non-linear relation

between δ and ζ. Note that such discussion would be redundant in our approach, where the initial conditions

for numerical evolution, as well as the threshold estimators for gravitational collapse, are expressed directly

in terms of ζ.

– 2 –

results obtained by numerical evolution with the results which can be obtained from a recently

proposed universal estimator for the strength of a perturbation [55]. This is given by a

suitable spatial average

¯

C of the so-called initial compaction function C[ζ(r)] [46], out to a

certain optimal radius r

m

. The threshold value for

¯

C which triggers gravitational collapse

turns out to be extremely robust, in the sense that it is nearly independent of the radial

proﬁle of the perturbation ζ(r).

The plan of the paper is the following: In section 2, we consider the typical shapes of a

high peak in the curvature perturbation proﬁle, within one standard deviation of the median

proﬁle, and we introduce the non-perturbative relation between the curvature perturbation

ζ and the Gaussian variable ζ

g

. In Section 3, we present the Misner-Sharp equations and we

review the criteria for the formation of PBHs. The results of the numerical simulation and

the analytical estimates, together with their interpretation are presented in section 4.

2 Large and rare peaks from single ﬁeld inﬂation

At cosmological scales, the power spectrum of primordial perturbations must be of the order

of 10

−9

, in accordance with observations of the cosmic microwave background. However, in

order for PBH formation to be signiﬁcant, the power must be of the order of 10

−3

− 10

−2

at the PBH scale. This jump in the amplitude can be achieved if the inﬂaton ﬁeld passes

trough a transient period with

¨

φ/H

˙

φ ≈ const. < −3. Throughout this paper, we shall refer

to this as “constant-roll” (CR).

6

Parametrically, the fraction of dark matter in PBH is Ω

P BH

∼ 10

9

(M

/M

P BH

)

1/2

β

0

,

where the probability of PBH formation at the time when a large perturbation crosses the

horizon can be roughly estimated as β

0

∼ exp[−ζ

2

th

/2σ

2

0

], for some threshold value ζ

th

∼ 1.

The remaining factors in the estimate of Ω

P BH

account for the dilution of radiation relative

to PBH, from the time of their formation until the time t

eq

. For M

P BH

in the broad range

10

−13

− 10

2

M

, the threshold for the perturbations to undergo gravitational collapse must

be in the range ζ

th

∼ (6 −8)σ

0

, sizably larger than the standard deviation, in order to obtain

a signiﬁcant Ω

P BH

∼ 1. Because these perturbations are very rare, we can use the theory of

high peaks to describe them.

2.1 The typical high peak proﬁles

Since the non-Gaussian curvature perturbation ζ is a local function of the Gaussian ﬁeld ζ

g

,

let us start by reviewing the latter [57]. This will be the basis to describe the non-Gaussian

realisations. Fluctuations of ζ

g

are characterized by the power spectrum P

ζ

(k), representing

the variance of the random ﬁeld per logarithmic interval in k,

hζ

2

g

i ≡ σ

2

0

=

Z

dk

k

P

ζ

(k). (2.1)

Introducing the normalized two point correlation function of ζ

g

(~x) as

ψ(r) ≡

1

σ

2

0

hζ

g

(r)ζ

g

(0)i =

1

σ

2

0

Z

P

ζ

(k) sinc kr

dk

k

, (2.2)

6

In its original deﬁnition [56], constant-roll refers to any period where

¨

φ = −(3 + α)H

˙

φ, with any constant

value of α. Ultra slow-roll (USR) corresponds to α = 0, and can also enhance the amplitude of the power

spectrum. However, to our knowledge, there is no concrete model of transient USR where the ampliﬁcation

is suﬃcient to provide a signiﬁcant abundance of PBH [24]. Hence, here we consider a transient period with

α > 0. This corresponds to the presence of a small barrier in the potential which slows down the motion of

the inﬂaton for a short period of time (see Fig. 1).

– 3 –

peaks of the Gaussian random ﬁeld of given amplitude µ = νσ

0

at the origin, have a mean

proﬁle given by

hζ

g

(r)|ν, peaki = σ

0

[νψ(r) + O(ν

−1

)], (2.3)

where the last term can be neglected in the limit of high peaks ν 1. Note that ψ(0) = 1.

The above expectation is calculated by using the number density distribution of peaks. This

distribution is almost Gaussian, except for a Jacobian prefactor which relates the condition

of being an extremum with the condition for the peak to be at a certain location. If we

simply condition the ﬁeld value to be at a certain height, the distribution is Gaussian, an

leads to the simpler expression [57]

hζ

g

(r)|νi = σ

0

νψ(r), (2.4)

which coincides with the large ν limit of (2.3). For a Gaussian distribution, the mean and

the median coincide, and therefore for the rest of this paper we shall refer to (2.4) as the

median Gaussian proﬁle.

Still, there will be some deviations around the median, so that the typical proﬁle will

be of the form

ζ

g

(r) = µψ(r) ± ∆ζ, (2.5)

where the variance of the shape is given by [57]

(∆ζ(r))

2

σ

2

0

= 1−

ψ

2

1 − γ

2

−

1

γ

2

(1 − γ

2

)

2γ

2

ψ +

R

2

s

∇

2

ψ

3

R

2

s

∇

2

ψ

3

−

5R

4

s

γ

2

ψ

0

r

−

∇

2

ψ

3

2

−R

2

s

ψ

02

γ

2

.

(2.6)

Here γ ≡ σ

2

1

/(σ

2

σ

0

), and R

s

≡

√

3σ

1

/σ

2

, where the gradient moments of the power spectrum

are given by

σ

2

n

=

Z

k

2n

P

ζ

(k)d ln k. (2.7)

In what follows, we are going to consider two diﬀerent forms for the enhancement of the

power spectrum at the PBH scale.

Monochromatic power spectrum: This is simply an idealized a delta function enhance-

ment, such that the power spectrum is given by

P

δ

ζ

(k) = σ

2

0

k

0

δ(k − k

0

). (2.8)

In this case the median shape in (2.5) is given by

ψ(r) = sinc(k

0

r), (2.9)

while the dispersion takes the following form

(∆ζ(r))

2

σ

2

0

= 1 − ψ

2

− 5

R

2

s

ψ

0

r

+ ψ

2

− R

2

s

(ψ

0

)

2

. (2.10)

Note that in this case, we have γ = 1, and the general expression (2.6) contains indeterminate

ratios. In order to obtain (2.10) we have regularized the delta function by using a normalized

distribution which is constant in an interval of radius ε around k = k

0

, and vanishes outside

of this interval, taking the limit ε → 0 at the end.

Sharply peaked power spectrum: We are also going to consider a more realistic case, in

which the enhancement follows a power law growth k

n

. Models of the type considered here

– 4 –

##### Citations

More filters

••

TL;DR: In this article, it was shown that a small bump-like feature behaves like a local speed-breaker and lowers the speed of the scalar field, thereby locally enhancing the power spectrum.

Abstract: Scalar perturbations during inflation can be substantially amplified by tiny features in the inflaton potential. A bump-like feature behaves like a local speed-breaker and lowers the speed of the scalar field, thereby locally enhancing the scalar power spectrum. A bump-like feature emerges naturally if the base inflaton potential Vb(φ) contains a local correction term such as Vb(φ)[1+v(φ)] at φ=φ0. The presence of such a localised correction term at φ0 leads to a large peak in the curvature power spectrum and to an enhanced probability of black hole formation. Remarkably this does not significantly affect the scalar spectral index nS and tensor to scalar ratio r on CMB scales. Consequently such models can produce higher mass primordial black holes (MPBHg 1 Mo) in contrast to models with `near inflection-point potentials' in which generating higher mass black holes severely affects nS and r. With a suitable choice of the base potential—such as the string theory based (KKLT) inflation or the α-attractor models—the amplification of primordial scalar power spectrum can be as large as 107 which leads to a significant contribution of primordial black holes (PBHs) to the dark matter density today, fPBH = Ω0, PBH/Ω0,DM ∼ O(1). Interestingly, our results remain valid if the bump is replaced by a dip. In this case the base inflaton potential Vb(φ) contains a negative local correction term such as Vb(φ)[1−v(φ)] at φ=φ0 which leads to an enhanced probability of PBH formation. We conclude that primordial black holes in the mass range 10−17 Mo l MPBH l 100 Mo can easily form in single field inflation in the presence of small bump-like and dip-like features in the inflaton potential.

88 citations

••

TL;DR: In this article, it was shown that the acceleration of the scalar field can be substantially amplified by small bump-like and dip-like features in the inflaton potential, leading to a large peak in the curvature power spectrum and an enhanced probability of black hole formation.

Abstract: Scalar perturbations during inflation can be substantially amplified by tiny features in the inflaton potential. A bump-like feature behaves like a local speed-breaker and lowers the speed of the scalar field, thereby locally enhancing the scalar power spectrum. A bump-like feature emerges naturally if the base inflaton potential $V_b(\phi)$ contains a local correction term such as $V_b(\phi)\left[1+\varepsilon(\phi)\right]$ at $\phi=\phi_0$. The presence of such a localised correction term at $\phi_0$ leads to a large peak in the curvature power spectrum and to an enhanced probability of black hole formation. Remarkably this does not significantly affect the scalar spectral index $n_{_S}$ and tensor to scalar ratio $r$ on CMB scales. Consequently such models can produce higher mass primordial black holes ($M_{\rm PBH}\geq 1 M_{\odot}$) in contrast to models with `near inflection-point potentials' in which generating higher mass black holes severely affects $n_{_S}$ and $r$. With a suitable choice of the base potential - such as the string theory based (KKLT) inflation or the $\alpha$-attractor models - the amplification of primordial scalar power spectrum can be as large as $10^7$ which leads to a significant contribution of primordial black holes (PBHs) to the dark matter density today, $f_{\rm PBH} = \Omega_{0,\rm PBH}/\Omega_{0,\rm DM} \sim O(1)$. Interestingly, our results remain valid if the bump is replaced by a dip. In this case the base inflaton potential $V_b(\phi)$ contains a negative local correction term such as $V_b(\phi)\left[1-\varepsilon(\phi)\right]$ at $\phi=\phi_0$ which leads to an enhanced probability of PBH formation. We conclude that primordial black holes in the mass range $10^{-17} M_{\odot} \leq M_{\rm PBH} \leq 100\, M_{\odot}$ can easily form in single field inflation in the presence of small bump-like and dip-like features in the inflaton potential.

86 citations

••

TL;DR: In this article, the authors considered the constant-roll inflation with small positive value of the constant roll parameter and investigated how a stage of constant roll inflation may realize the growth in the primordial curvature power spectrum necessary to produce a peaked spectrum of primordial black hole abundance.

Abstract: The constant-roll inflation with small positive value of the constant-roll parameter $\beta\equiv \frac{\ddot\phi}{H\dot\phi}={\rm const.}$ has been known to produce a slightly red-tilted curvature power spectrum compatible with the current observational constraints. In this work, we shed light on the constant-roll inflation with negative $\beta$ and investigate how a stage of constant-roll inflation may realize the growth in the primordial curvature power spectrum necessary to produce a peaked spectrum of primordial black hole abundance. We first review the behavior of constant-roll models in the range of parameters $-\frac32<\beta<0$, which allows for a constant-roll attractor stage generating a blue-tilted curvature power spectrum without superhorizon growth. As a concrete realization, we consider a potential with two slow-roll stages, separated by the constant-roll stage, in a way that satisfies the current constraints on the power spectrum and the primordial black hole abundance. The model can produce primordial black holes as all dark matter, LIGO-Virgo events, or OGLE microlensing events. Due to the range of different scalar tilts allowed by the constant-roll potential, this construction is particularly robust and testable by future observations.

40 citations

••

TL;DR: In this paper , an analytically solvable simplified model for primordial black hole (PBH) production in most models of single-field inflation is presented. But the model is not suitable for probing PBHs with scalar-induced gravitational wave backgrounds, since the COBE/Firas μ-distortion constraints exclude the production of a PBH heavier than 104 M ⊙ in single field inflation.

Abstract: We construct an analytically solvable simplified model that captures the essential features for primordial black hole (PBH) production in most models of single-field inflation. The construction makes use of the Wands duality between the constant-roll (or slow-roll) and the preceding ultra-slow-roll phases and can be realized by a simple inflaton potential of two joined parabolas. Within this framework, it is possible to formulate explicit inflationary scenarios consistent with the CMB observations and copious production of PBHs of arbitrary mass. We quantify the variability of the shape of the peak in the curvature power spectrum in different inflationary scenarios and discuss its implications for probing PBHs with scalar-induced gravitational wave backgrounds. We find that the COBE/Firas μ-distortion constraints exclude the production of PBHs heavier than 104 M ⊙ in single-field inflation.

34 citations

••

TL;DR: In this paper , an extension of the ultra-slow-roll inflation that incorporates a transition process is presented, where the inflaton climbs up a tiny potential step at the end of the non-attractor stage before it converges to the slow-roll attractor.

Abstract: For primordial perturbations, deviations from Gaussian statistics on the tail of the probability distribution can be associated with non-perturbative effects of inflation. In this paper, we present some particular examples in which the tail of the distribution becomes highly non-Gaussian although the statistics remains almost Gaussian in the perturbative regime. We begin with an extension of the ultra-slow-roll inflation that incorporates a transition process, where the inflaton climbs up a tiny potential step at the end of the non-attractor stage before it converges to the slow-roll attractor. Through this example, we identify the key role of the off-attractor behaviour for the upward-step transition, and then extend the analysis to another type of the transition with two slow-roll stages connected by a tiny step. We perform both the perturbative and non-perturbative analyses of primordial fluctuations generated around the step in detail, and show that the tiny but nontrivial transition may affect large perturbations in the tail of the distribution, while the perturbative non-Gaussianity remains small. Our result indicates that the non-Gaussian tails can have rich phenomenology which has been overlooked in conventional analyses. We also study the implications of this non-Gaussian tail for the formation of primordial black holes, and find that their mass fraction can be parametrically amplified by several orders of magnitudes in comparison with the case of the Gaussian distribution. Additionally, we also discuss a mechanism of primordial black holes formation for this upward step inflation model by trapping the inflaton in the bottom of the step.

30 citations

##### References

More filters

••

TL;DR: In this paper, a set of new mathematical results on the theory of Gaussian random fields is presented, and the application of such calculations in cosmology to treat questions of structure formation from small-amplitude initial density fluctuations is addressed.

Abstract: A set of new mathematical results on the theory of Gaussian random fields is presented, and the application of such calculations in cosmology to treat questions of structure formation from small-amplitude initial density fluctuations is addressed. The point process equation is discussed, giving the general formula for the average number density of peaks. The problem of the proper conditional probability constraints appropriate to maxima are examined using a one-dimensional illustration. The average density of maxima of a general three-dimensional Gaussian field is calculated as a function of heights of the maxima, and the average density of 'upcrossing' points on density contour surfaces is computed. The number density of peaks subject to the constraint that the large-scale density field be fixed is determined and used to discuss the segregation of high peaks from the underlying mass distribution. The machinery to calculate n-point peak-peak correlation functions is determined, as are the shapes of the profiles about maxima.

3,098 citations

••

1,827 citations

••

TL;DR: Relativistic gravitational collapse equations assuming spherical symmetry, adiabatic flow and pressure gradient forces were proposed in this paper, where spherical symmetry was assumed to be a function of the density.

Abstract: Relativistic gravitational collapse equations assuming spherical symmetry, adiabatic flow and pressure gradient forces

1,442 citations

••

1,363 citations

••

TL;DR: The existence of galaxies indicates that the early universe must have been inhomogeneous and might have been highly chaotic as discussed by the authors, which could have lead to regions of the size of the particle horizon undergoing gravitational collapse to produce black holes with initial masses from 10-5 g upwards.

Abstract: The existence of galaxies indicates that the early universe must have been inhomogeneous and might have been highly chaotic. This could have lead to regions of the size of the particle horizon undergoing gravitational collapse to produce black holes with initial masses from 10-5 g upwards. Radiation pressure in the early Universe would cause these black holes to grow by accretion. However, despite previous expectations, this accretion would not be very much unless the initial conditions of the Universe were arranged in a special and a causal manner. Observations indicate that, at the most, only a small fraction of the matter in the early Universe can have undergone gravitational collapse.

1,334 citations

##### Related Papers (5)

##### Frequently Asked Questions (2)

###### Q2. What are the future works mentioned in the paper "Pbh in single field inflation: the effect of shape dispersion and non-gaussianities" ?

The authors leave a more detailed consideration of such effect for future work. Prospects for observational discrimination of these two possibilities remain an interesting direction for further reseach. Another possible phenomenological application of their results may be in the study of gravitational waves induced by non-Gaussian scalar perturbations [ 67–69 ].