# A possible mass distribution of primordial black holes implied by LIGO-Virgo

Heling Deng

Arizona State University

2101.11098

## Outline

• LIGO BHs
• Primordial?
• A simple model "fitting" LIGO
• A possible mechanism

## Outline

• LIGO BHs
• Primordial?
• A simple model "fitting" LIGO
• A possible mechanism

Mass distribution of LIGO BHs

BH mass

(BH number)

## Outline

• LIGO BHs
• Primordial?
• A simple model "fitting" LIGO
• A possible mechanism
M_\text{Pl} \newline 10^{-5}\ \rm g
M_\odot \newline 10^{33}\ \rm g
M_\text{PBH}

LIGO BHs

• Supermassive black holes
• LIGO black holes
• Dark matter
\mathcal O (10\text{-} 100)M_\odot

SMBH

\mathcal{O}(10^6 \text{-} 10^{10})M_\odot

...

Primordial black holes (PBHs)

...

Motivations

Observational constraints of PBHs as DM

PBH binary

PBH mass function

f_{PBH}\equiv\frac{\rho_{PBH}}{\rho_{CDM}} = \int \psi(m)\text{d}m
f(m)=m\psi(m)
\psi(m)\text{d}m=\frac{m}{\rho_{CDM}}\text{d}n

(Fraction of CDM in PBHs with $$\sim m$$)

(Fraction of CDM in PBHs within ($$m, m+\text{d}m)$$)

(Fraction of CDM in PBHs)

For example, $$f(30M_\odot)=10^{-3}$$ means PBHs of $$30M_\odot$$ constitute 0.1% DM

PBH mass function $$f(m)$$

Merger rate $$R(m_1,m_2,z)$$

Detection probability $$p_{det}(m_1,m_2,z)$$

Probability of each event $$p(m_1,m_2,z)$$

+

Likelihood of all LIGO events

\mathcal{L}\propto \prod_{i=1}^{44} p_i

Merger signals reaching the earth today

## Outline

• LIGO BHs
• Primordial?
• A simple model "fitting" LIGO
• A possible mechanism

Mass distribution of LIGO BHs

m^{\alpha_1},\ \ \ m< m_*
m^{\alpha_2},\ \ \ m> m_*
f(m)\propto
\{

A simple mass function

\log(m)
\log(f)
m^{\alpha_1}
m^{\alpha_2}
m_*
f_{PBH}
f_{PBH}
f_{PBH}\approx 10^{-3}

Maximizing $$\mathcal{L}$$ in a 4-parameter space

\alpha_2 = -4
\alpha_1 = 1.2
m_*=35M_\odot
\log(m)
\log(f)
m^{1.2}
m^{-4}
35M_\odot
10^{-3}

orange: broken power law (our model)

green: log-normal

blue: LIGO

## Outline

• LIGO BHs
• Primordial?
• A simple model "fitting" LIGO
• A possible mechanism

A possible mechanism

subcritical

supercritical

r

m\propto r^3
m\propto r^2

BH mass distribution $$f(m)$$

Parameters: $$\lambda$$, $$\gamma_i$$, $$\rho_i$$, $$\rho_b$$, $$\sigma$$

"initial speed"

"force"

"mass"

orange: broken power law

green: log-normal

black: our PBH model

Conclusions

PBH mass function

Merger rate

Detection probability

Expected number of observable events

Probability of each detected event

p_i
p_{Poisson}\propto N_e^{N_o}e^{-N_e}
\mathcal{L}\propto p_{Poisson}\prod_{i=1}^{N_o} p_i
N_e

+

+

Likelihood of all LIGO events

Intrinsic PBH merger rate

\text{d}R(m_1,m_2,z)\propto f(m_1) f(m_2)

(Number of event  $$(m_1,m_2,z)$$ per unit volume per unit time)

Signals reaching earth

\text{d}N(m_1,m_2,z)\propto \text{d}R(m_1,m_2,z)

(Number of event  $$(m_1,m_2,z)$$ reaching the earth per unit time)

Detection probability $$p_{det}(m_1,m_2,z)$$

\text{d}N_e(m_1,m_2,z)=p_{det}(m_1,m_2,z)\text{d}N\times T_o

Size distribution of bubbles after inflation

Constant bubble nucleation rate $$\lambda$$ during inflation

Evolution of bubbles in radiation background $$r$$($$\gamma_i$$, $$\rho_i$$, $$\rho_b$$, $$\sigma$$)

BH mass distribution $$f(m)$$

Parameters: $$\lambda$$, $$\gamma_i$$, $$\rho_i$$, $$\rho_b$$, $$\sigma$$

"initial speed"

"force"

"mass"

PBH mass function $$f(m)$$

Merger rate $$R(m_1,m_2,z) \propto f(m_1) f(m_2)$$

Detection probability $$p_{det}(m_1,m_2,z)$$

Probability of each event $$p_i(m_1,m_2,z)$$

+

Likelihood of all LIGO events

+

Expected number of detection $$N_e$$

\mathcal{L}\propto \prod_{i=1}^{N_o} p_i

Mergers reaching earth today

$$N(m_1,m_2,z) \propto R(m_1,m_2,z)$$

By Heling Deng

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