Merger Rate Distribution of Primordial-Black-HoleBinaries
Zu-Cheng ChenDone with Fan Huang and Prof. Qing-Guo HuangBased on arXiv:1801.10327 and a work in progress
Institute of Theoretical Physics
May 25, 2018
.................................
.......
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Outline
1 Introduction
2 Merger Rate Distribution of PBH Binaries
3 Stochastic Gravitational-Wave Background
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Outline
1 Introduction
2 Merger Rate Distribution of PBH Binaries
3 Stochastic Gravitational-Wave Background
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
https://www.ligo.caltech.edu
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
What we know after LIGO/Virgo
There are many BH binaries.
They do have mass distribution.
They can merge within Hubble time,
The merger rate is 12 ∼ 213Gpc−1yr−1. PRL 118, 221101 (2017)
What we don’t know after LIGO/Virgo
Where do these BHs come from?
What is the mechanism of the binary formation?
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
What we know after LIGO/Virgo
There are many BH binaries.
They do have mass distribution.
They can merge within Hubble time,
The merger rate is 12 ∼ 213Gpc−1yr−1. PRL 118, 221101 (2017)
What we don’t know after LIGO/Virgo
Where do these BHs come from?
What is the mechanism of the binary formation?
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Maybe, primordial BHs!
All PBHs with mass 30M⊙ and the torque from the nearestthird BH. PRL 117, 061101 (2016)
General mass function, but the torque from the nearest thirdBH. PRD 96, 123523 (2017)
A monochromatic mass function, and the torques from allother BHs. PRD 96, 123523 (2017)
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Outline
1 Introduction
2 Merger Rate Distribution of PBH Binaries
3 Stochastic Gravitational-Wave Background
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Merger Rate Distribution of PBH Binaries
PBHs have a general mass function.
PBHs distributed randomly in the early Universe.
Torques from all other BHs and linear density perturbations.
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Merger Rate Density
R(t,mi,mj) ≈ 3.9 · 106 ×(
t
t0
)− 3437
f2(f2 + σ2eq)
− 2174
×min
(P (mi)
mi,P (mj)
mj
)(P (mi)
mi+
P (mj)
mj
)×(mimj)
337 (mi +mj)
3637
The fraction of PBHs in DM is fpbh ≡ Ωpbh/ΩDM ≈ f/0.85.
P (m) is the mass function (PDF)∫ ∞
0P (m)dm = 1.
σeq is the variance of density perturbations of the rest of DM.
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Power-law: α = 1.5, M = 1M⊙
P (m) ≈ α− 1
M
(m
M
)−α
Lognormal: σ = 1, mc = 40M⊙
P (m) =1√
2πσme−
log2(m/mc)
2σ2
Power-law
Lognormal
20 40 60 80
0.01
0.02
0.03
0.04
m [M⊙]
PD
F[M
⊙-
1]
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Constrains on fpbh: 5M⊙ ≤ m1,m2 & m1 +m2 ≤ 100M⊙
Power-law
Lognormal
0.001 0.010 0.100 10.01
1
100
104
fpbh
Ev
en
tra
te[G
pc-
3y
r-1]
Power-law: 0.0036 ≲ fpbh ≲ 0.021
Lognormal: 0.0021 ≲ fpbh ≲ 0.011
Consistent with other observations.Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
1D merger rate distribution
RH(t0,mH) =
∫ mH
mmin
R(t0,m,mH)dm
Power-law
Lognormal
5 10 50 10010-4
0.001
0.010
0.100
1
10
mH [M⊙]
Ev
en
tra
te[G
pc-
3y
r-1]
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
2D merger rate distribution
GW150914
GW151226
LVT151012
GW170104
GW170608
GW170814
6 8 11 15 19 26 34 45 60 80
80
60
45
34
26
19
15
11
8
6
m1 [M⊙]
m2[M
⊙]
Power-law
1
2
3
4
5
6
GW150914
GW151226
LVT151012
GW170104
GW170608
GW170814
6 8 11 15 19 26 34 45 60 80
80
60
45
34
26
19
15
11
8
6
m1 [M⊙]
m2[M
⊙]
Lognormal
1
2
3
4
5
6
7
Merger rate distributions are quite sensitive to P (m).
P (m) could be reconstructed from merger rate distributions.
Understand the origin of BHs detected by LIGO/Virgo.
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Outline
1 Introduction
2 Merger Rate Distribution of PBH Binaries
3 Stochastic Gravitational-Wave Background
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Events observed by LIGO/Virgo
Events Primary mass Secondary mass Source Redshift
GW150914 36.2+5.2−3.8M⊙ 29.1+3.7
−4.4M⊙ 0.09+0.03−0.04
LVT151012 23+18−6 M⊙ 13+4
−5M⊙ 0.20+0.09−0.09
GW151226 14.2+8.3−3.7M⊙ 7.5+2.3
−2.3M⊙ 0.09+0.03−0.04
GW170104 31.2+8.4−6.0M⊙ 19.4+5.3
−5.9M⊙ 0.18+0.08−0.07
GW170608 12+7−2M⊙ 7+2
−2M⊙ 0.07+0.03−0.03
GW170814 30.5+5.7−3.0M⊙ 25.3+2.8
−4.2M⊙ 0.11+0.03−0.04
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Constraints for lognormal PDF
P (m) =1√
2πσmexp
(− log2(m/mc)
2σ2
)with (mc, σ) = (20.82M⊙, 0.58)
-1 0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
log10(mc/M⊙)
σ
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Fractional energy density ΩGW(ν)
The energy-density spectrum of a GW background is characterizedby the dimensionless quantity
ΩGW(ν) =ν
ρc
dρGW
dν,
which represents the contribution of GW to the critical energydensity of the Universe, ρc = 3H2
0/(8πG), and can be expressed as
ΩGW(ν) =ν
ρc
∫dz dmi dmj
R(z,mi,mj)
(1 + z)H(z)
dEGW
dνs
∣∣∣∣νs=(1+z)ν
.
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
O2
O3
Design
LISA
10-5 10-4 10-3 10-2 10-1 100 101 102 103 10410-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
Frequency [Hz]
ΩG
W
Several times larger than that from astrophysical BBHs as inPRL 120, 091101 (2018)
Can be detected by LIGO/Virgo and LISA.
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Signal-to-Noise Ratio (SNR) for LISA
1σ
3σ
5σ
0.5 1 5 10 50 100 5000.1
0.5
1
5
10
50
100
Observation time (hours)
SN
R
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Introduction Merger Rate Distribution of PBH Binaries Stochastic Gravitational-Wave Background
Conclusions and Discussions
We derive the merger rate distribution of PBH binaries with ageneral mass function by taking into account the torques byall primordial black holes and linear density perturbations.
It will help us to settle down the formation mechanism ofBBHs observed by LIGO/Virgo.
The SGWB can be detected by LIGO/Virgo and LISA if BBHsare of primordial orgin.
Zu-Cheng Chen Merger Rate Distribution of PBH Binaries May 25, 2018