4. Estimation of PBH abundance

in peak theory

Chulmoon Yoo

1

Refs. CY, Harada, Garriga, Kohri arXiv:1805.03946

　 CY, Harada, Hirano, Kohri arXiv:2008.02425

PBH lecture

2

How can we count the number of PBHs?

◎Simplest conventional estimation(Carr’s formula, Press-Schecheter)

• Assumption 1: threshold is given by the amplitude of the density perturbation δ
• Assumption 2: Gaussian distribution of δ
• Assumption 3: PBH fraction β ～ production probability

[Carr ApJ 201(1975)1]

◎The exponential dependence on δ_th/σ

is well described

◎How can we refine this procedure

in a more accurate way?

→ peak theory

[Bardeen, Bond, Kaiser, Szalay ApJ 304 15 (1986)]

PBH lecture 4

Yoo, Chulmoon

3

How can we count the number of PBHs?

◎Strategy

• First, focus on the variable whose statistics are known (Gaussian curvature ζ)
• PBHs would be associated with a peak of the perturbation variable (peak of △ζ)
• Count the number of peaks which satisfy a PBH formation criterion (using C)

◎Peak theory [Bardeen, Bond, Kaiser, Szalay ApJ 304 15 (1986)]

• Basic assumption: Random Gaussian variable (ζ) with the power spectrum P(k)
• Number density of the peaks with the amplitude (μ₂) and the scale (1/k●)
• Typical profile of the Gaussian variable around the peak

Integrating above the threshold of μ₂

Relation between M,μ₂,k●: M=M(μ₂,k●)

with the typical profile

PBH lecture 4

Yoo, Chulmoon

Peak theory

ーPeak number densityー

4

5

Curvature perturbation

◎Spatial metric

◎ζ and density perturbation δ (w/ long wave-length approx. ＆ comoving slicing)

◎We assume ζ is a Gaussian variable with the power spectrum P (k)

◎Constant shift of ζ can be absorbed into the redefinition of the scale factor

◎Gradient moments

linear approx

PBH lecture 4

Yoo, Chulmoon

6

Random Gaussian variable

◎Random Gaussian variable ζ(xⁱ) with the power spectrum

Correlation matrix:

◎Probability distribution of linear combinations of ζ(xⁱ)

◎An example: the probability distribution of -ζ and △ζ ⇒

◎Gradient moments

PBH lecture 4

Yoo, Chulmoon

7

Probability distribution for the profile-1

◎Taylor expansion of ζ up through the second order: 10 independent variables

◎Non-zero correlations

◎variable transformation in 2nd order variables:

eigen values of the matrix ζ₂^ij with λ₁≥λ₂≥λ₃

Euler angles to take the principal direction

→factor 2π by integration

PBH lecture 4

Yoo, Chulmoon

8

Probability distribution for the profile-2

◎Remaining 7 independent variables:

◎Probability distribution:

PBH lecture 4

Yoo, Chulmoon

9

Number density of extrema

◎Number density distribution of extrema

in the space of

Values for a extremum labeled by p

Actually, integration over the space

Number of extrema N labeled by p

◎But, we don’t know the specific distribution of the extrema

◎What we need is the mean number density

Volume average

Average with respect to the parameters

PBH lecture 4

Yoo, Chulmoon

10

Mean number density of extrema

◎Extremum of ζ: ζⁱ=0 ⇔ η_i=0 at

◎Mean number density

Volume average

Average with respect to the parameters

Actually, the volume average trivially gives the mean number density N(ν, ξ₁ )ΔνΔξ₁ / V

But, we don’t know N(ν, ξ₁ ), and we need to calculate it in a different way based on the peak probability distribution for the parameters

PBH lecture 4

Yoo, Chulmoon

11

Peak number density

◎Mean number density of extrema

◎Peak(λ₁>λ₂>λ₃>0) number density

Step function

replace volume average by the ensemble average w.r.t.

PBH lecture 4

Yoo, Chulmoon

12

1/peak length scale

peak amplitude

Transformation of variables

◎For convenience, we change the independent variables as

defined by

◎So far, we got the number density of the peaks characterized by

the peak amplitude μ and the length scale 1/k_*

◎We will see that these two parameter also characterize the peak profile

profile -ζ(r)

peak amplitude μ= -ζ(0)

peak scale 1/k_*, k_*²=△ζ(0)/μ

◎Peak number density

PBH lecture 4

Yoo, Chulmoon

Peak theory

ーTypical profileー

13

14

Conditional probability for ζ(r)

◎Let us consider a peak at r=0, and the following conditional probability

◎Some notations for a general multivariate Gaussian probability

The probability distribution is characterized by the matrix:

PBH lecture 4

Yoo, Chulmoon

15

Matrix decomposition and conditional probability

◎Matrix decomposition

Schur complement M/A of the block A:

P_X

P_X∩Y

◎Mean values of β is given by αA^-1B and the variance is (M/A)^-1

PBH lecture 4

Yoo, Chulmoon

16

Mean value of ζ(r) as a typical profile-1

◎Conditional probability for ζ(r)

P_X

P_X∩Y

◎Our case

PBH lecture 4

Yoo, Chulmoon

17

Mean value of ζ(r) as a typical profile-2

◎Covariance matrix

◎Mean for β₁=ν(r)=ζ(r)/σ₀

◎Variance for β₁=ν(r)=ζ(r)/σ₀

◎Correlations

PBH lecture 4

Yoo, Chulmoon

18

Mean value of ζ(r) as a typical profile-3

◎Mean profile

◎Variance

◎A typical profile is provided by the mean profile characterized by ζ(0)=σ₀ν and △ζ(0)=σ₂ξ₁ 　with the power spectrum P(k)

　through the two point correlation function

◎The variance is estimated as

◎The typical profile gives a good approximation as long as ζ(r)>>σ₀ for a rarely high peak

PBH lecture 4

Yoo, Chulmoon

19

Peak number density and profile with μ₀ , k_*

◎The typical profile

1/peak length scale

◎For convenience, we change the independent variables as

defined by

peak amplitude

◎Number density

where

PBH lecture 4

Yoo, Chulmoon

PBH abundance

20

21

Peak number density of △ζ

◎The typical profile

◎Let us consider peaks of △ζ and parameters:

◎With this modification ζ→△ζ，we need the replacements:

◎PBHs would be associated with peaks of △ζ ∵

◎Constant shift of ζ can be absorbed into the redef. of a ∵

◎Number density

PBH lecture 4

Yoo, Chulmoon

22

Typical profile of ζ associated with μ₂ and k_●

◎The typical profile

integration

integration constant

can be absorbed into

the scale factor

◎It can be shown(see Appendix in 2008.02425)

∴ζ_∞ can be regarded as a Gaussian probability variable

PBH lecture 4

Yoo, Chulmoon

23

Flow chart

PBH lecture 4

Yoo, Chulmoon

24

Flow chart

PBH lecture 4

Yoo, Chulmoon

25

Profile of compaction function and the threshold

◎The typical profile

◎Compaction function

◎Threshold

PBH lecture 4

Yoo, Chulmoon

26

Flow chart

PBH lecture 4

Yoo, Chulmoon

27

Estimation of PBH mass

◎Horizon entry

◎Including critical behavior

◎The PBH mass can be estimated from the typical profile (without critical behavior)

◎Radiation dominant

with ɤ≒0.36 and K(k_●) being a k_● dependent numerical factor which will be set to be 1 hereafter

NOTE: the mass estimation is not valid for Type II PBH formation

PBH lecture 4

Yoo, Chulmoon

28

Flow chart

PBH lecture 4

Yoo, Chulmoon

29

PBH mass spectrum

◎Threshold

by eliminating k_●

may be bounded below for a fixed M by

◎PBH formation with mass M for

◎Peak number density for the variables μ₂ and M instead of k_●

◎PBH number density and the mass spectrum at equality time

PBH lecture 4

Yoo, Chulmoon

Specific cases

ーmonochromatic power spectrumー

30

31

Monochromatic power spectrum-1

◎Power spectrum

◎Gradient moments

◎Profile

◎PBH mass

without critical behavior(ɤ=0) → the minimum value of M is given by

with critical behavior(ɤ=0.36) → zero mass PBH is possible

from numerical simulation

PBH lecture 4

Yoo, Chulmoon

32

Monochromatic power spectrum-2

◎Probability distribution P₁ in

k_●integration

◎PBH number density

PBH lecture 4

Yoo, Chulmoon

33

Monochromatic power spectrum-3

◎PBH number density

◎PBH fraction

with

defined by the inverse function of

PBH lecture 4

Yoo, Chulmoon

34

Monochromatic power spectrum:results

◎PBH fraction

with critical behavior

without

critical behavior

◎

◎Results with k₀=10⁵k_eq

PBH lecture 4

Yoo, Chulmoon

Specific cases

ーsingle scale extend power spectrumー

35

36

Flow chart

PBH lecture 4

Yoo, Chulmoon

37

An example of for a narrow power spectrum

◎Single scale narrow power spectrum

New: CY, Harada, Hirano, Kohri arXiv:2008.02425

Old : CY, Harada, Garriga, Kohri arXiv:1805.03946

PBH lecture 4

Yoo, Chulmoon

Implementing a window function

38

39

Procedure for a broad power spectrum

◎A window function is needed for controlling UV contribution

◎Our guiding principle: UV cut-off without too much reduction in the relevant scale

◎Procedure to calculate PBH mass spectrum from a broad power spectrum

2. Calculate the mass spectrum for a value of k_W
3. Take the envelope curve of the mass spectrum for all values of k_W

⇒For a fixed scale k₀, k_W=k_W^max maximizing the abundance is relevant

Our procedure minimizes the extra-reduction due to the window function.

PBH lecture 4

Yoo, Chulmoon

Flat power spectrum case

40

◎Power spectrum with a window function

• k-space top-hat gives the largest abundance
• The final mass spectrum is of course flat

​

PBH lecture 4

Yoo, Chulmoon

Window function and peak number density

41

• For k₀≪k_W , every peak has much larger k_●

because of smaller scale perturbations

on top of the perturbation w/ the scale k₀

⇒ essentially no peak with the scale k₀

​

• In both limits, the number of peaks

with the scale ~ k₀ decreases.

• For k₀≫k_W no peak with the scale k₀

◎We expect k_W^max is comparable to (and slightly larger than) k₀

◎For a fixed scale k₀, k_W=k_W^max maximizing the abundance is relevant

PBH lecture 4

Yoo, Chulmoon

Implementing a window function

42

◎If the window function reduces the amplitude of the power spectrum

in the region of k much smaller than k_W

(like the yellow curve in the figure),

the number density of peaks with k₀

also decreases due to the window function

​

⇒ sharp cut-off (like the light-blue curve)

would provide more peak numbers

PBH lecture 4

Yoo, Chulmoon

Flat power spectrum case

43

◎Power spectrum with a window function

• k-space top-hat gives the largest abundance
• The final mass spectrum is of course flat

​

PBH lecture 4

Yoo, Chulmoon

44

Summary and advertisements

◎We provide a procedure to calculate the PBH mass spectrum

• for an arbitrary power spectrum with a window function
• by using plausible PBH formation criterion
• with nonlinear relation taken into account

◎Some advertisements about our recent works

• Application to local type non-Gaussianity

CY, Gong, Yokoyama arXiv:1906.06790

Kitajima, Tada, Yokoyama, CY arXiv:2109.00791

Escrivà, Tada, Yokoyama, CY arXiv:2202.01028

• Simulation of non-spherical PBH formation
• CY, Harada, Okawa arXiv:2004.01042
• PBH formation from massless scalar isocurvature

CY, Harada, Hirano, Okawa, Sasaki arXiv:2112.12335

Thank you for your attention

PBH lecture 4

Yoo, Chulmoon