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)

...

BHs formed during radiation era

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}\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

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

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