Primordial Black Hole Formation by Vacuum Bubbles
Heling Deng
1710.02865 with Alex Vilenkin
Tufts University
- dynamics & spacetime
- results
- simulation setup
- observations
- results
- simulation setup
- observations
- dynamics & spacetime
inflaton
ρb
\rho_b
ρi
\rho_i
ρb<ρi
\rho_b< \rho_i
radiation
Hb−1
H_b^{-1}
Hb−1
H_b^{-1}
BH
subcritical
Hb−1
H_b^{-1}
subcritical spacetime
subcritical BH mass
M∼(34πρb+4πHiσ)Ri3
M \sim \left(\frac{4}{3} \pi\rho_b+4\pi H_i\sigma \right) R_i^3
Ri
R_i
ρb
\rho_b
σ
\sigma
1512.01819 by Jaume Garriga, Alex Vilenkin and Jun Zhang (2016)
- bubble mass doesn't change much after bubble decouples from Hubble flow
≡κRi3/G
\equiv \kappa R_i^3/G
- bubble doesn't grow much before it decouples from Hubble flow
Hb−1
H_b^{-1}
supercritical
supercritical spacetime
supercritical BH mass
GM<2.8HiRi2
GM<2.8H_iR_i^2
upper bound:
1612.03753 with Jaume Garriga and Alex Vilenkin (2016)
- results
- simulation setup
- observations
- dynamics & spacetime
metric
ds2=−A2dt2+B2dr2+R2dΩ2
ds^2=-A^2dt^2+B^2dr^2+R^2d\Omega^2
fluid
Tμν=(p+ρ)uμuν+pgμν
T^{\mu\nu}=(p+\rho)u^\mu u^\nu+pg^{\mu\nu}
u1=0
u^1=0
p=31ρ
p=\frac{1}{3}\rho
radiation
comoving gauge
BH apparent horizon
Θout∝R˙/A+R′/B=0
\Theta_\text{out} \propto \dot{R}/A + R^\prime /B =0
Misner-Sharp mass
Mbh=Rbh/2G
M_\text{bh}=R_\text{bh}/2G
M=2GR[1+(R˙/A)2−(R′/B)2]
M=\frac{R}{2G}\left[1+(\dot{R}/A)^2-(R^\prime/B)^2\right]
(M′=4πρR2R′)
(M^\prime = 4\pi\rho R^2 R^\prime)
boundary condition
A′=−AB(σρ/3+ρb+BR2R′+6πσ)
A^\prime = -AB\left( \frac{\rho/3+\rho_b}{\sigma}+\frac{2R^\prime}{BR} + 6\pi \sigma \right)
outer: FRW solutions
inner: junction conditions
initial condition
ρ=32πGti23
\rho=\frac{3}{32\pi Gt_i^2}
A=1
A=1
...
- results
- simulation setup
- observations
- dynamics & spacetime
ρ/ρFRW
\rho/\rho_\text{FRW}
r
r
early stage of evolution of ρ
evolution of ρ after BH formation (subcritical)
ρ/ρFRW
\rho/\rho_\text{FRW}
r
r
evolution of ρ after BH formation (supercritical)
ρ/ρFRW
\rho/\rho_\text{FRW}
r
r
supercritical
M<2.8HiRi2/G
M<2.8H_iR_i^2/G
M≈κRi3
M\approx\kappa R_i^3
subcritical
κRi3, M<M∗
\kappa R_i^3,\ \ \ M< M_*
HiRi2, M>M∗
H_iR_i^2,\ \ \ M> M_*
GM∼
GM\sim
{
\{
mass of BH formed by vacuum bubbles
where
M∗∼GκHi3
M_*\sim \frac{H_i^3}{G\kappa}
bubble size distribution
number density
dn=λ(R+Hi−1)4dR
dn= \lambda\frac{dR}{(R+H_i^{-1})^4}
PBH mass function (fraction of CDM in PBH with mass ~ M )
f(M)≡ρCDMM2dMdn∝
f(M)\equiv \frac{M^2}{\rho_\text{CDM}} \frac{dn}{dM}\propto
M∗−1/2, M<M∗
M_*^{-1/2},\ \ \ M< M_*
M−1/2, M>M∗
M^{-1/2},\ \ \ M> M_*
{
\{
BH mass distribution
fPBH≡ρCDMρPBH=∫MdMf(M)
f_\text{PBH}\equiv\frac{\rho_\text{PBH}}{\rho_\text{CDM}} = \int\frac{dM}{M}f(M)
total fraction of CDM in PBH
- results
- simulation setup
- observations
- dynamics & spacetime
observational constraints on monochromatic PBH
observational constraints → our fPBH
1705.05567 by Carr et al. (2017)
LIGO PBH?
f(M∼30 M⊙)∼10−3 → merger rate for LIGO events
1603.08338 by Sasaki et al. (2016)
M∗∼30 M⊙
M_* \sim 30\ M_\odot
Mmin∼10−5 M⊙
M_\text{min} \sim 10^{-5}\ M_\odot
fPBH∼0.01
f_\text{PBH}\sim0.01
→ f(M∼30 M⊙)∼10−3
\to \ f(M\sim30\ M_\odot) \sim 10^{-3}
Conclusions
- a subcritical bubble collapses into an ordinary BH
- PBH formed by vacuum bubbles can have a distinct, wide range of mass spectrum
- a supercritical bubble has an inflating baby universe connected to our universe by a wormhole
- vacuum bubbles may nucleate during inflation, leading to the formation of BHs
- can make up 10% of DM and account for LIGO BHs
Primordial Black Hole Formation by Vacuum Bubbles Heling Deng 1710.02865 with Alex Vilenkin Tufts University
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By Heling Deng
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