A possible mass distribution of primordial black holes implied by LIGO-Virgo
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}
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
\(N(m_1,m_2,z) \propto R(m_1,m_2,z)\)
DPF21: A possible mass distribution of primordial black holes implied by LIGO-Virgo
By Heling Deng
