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
\rho_b
\rho_i
\rho_b< \rho_i
radiation
H_b^{-1}
H_b^{-1}
BH
subcritical
H_b^{-1}
subcritical spacetime
subcritical BH mass
M \sim \left(\frac{4}{3} \pi\rho_b+4\pi H_i\sigma \right) R_i^3
R_i
\rho_b
\sigma
1512.01819 by Jaume Garriga, Alex Vilenkin and Jun Zhang (2016)
- bubble mass doesn't change much after bubble decouples from Hubble flow
\equiv \kappa R_i^3/G
- bubble doesn't grow much before it decouples from Hubble flow
H_b^{-1}
supercritical
supercritical spacetime
supercritical BH mass
GM<2.8H_iR_i^2
upper bound:
1612.03753 with Jaume Garriga and Alex Vilenkin (2016)
- results
- simulation setup
- observations
- dynamics & spacetime
metric
ds^2=-A^2dt^2+B^2dr^2+R^2d\Omega^2
fluid
T^{\mu\nu}=(p+\rho)u^\mu u^\nu+pg^{\mu\nu}
u^1=0
p=\frac{1}{3}\rho
radiation
comoving gauge
BH apparent horizon
\Theta_\text{out} \propto \dot{R}/A + R^\prime /B =0
Misner-Sharp mass
M_\text{bh}=R_\text{bh}/2G
M=\frac{R}{2G}\left[1+(\dot{R}/A)^2-(R^\prime/B)^2\right]
(M^\prime = 4\pi\rho R^2 R^\prime)
boundary condition
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
\rho=\frac{3}{32\pi Gt_i^2}
A=1
...
- results
- simulation setup
- observations
- dynamics & spacetime
\rho/\rho_\text{FRW}
r
early stage of evolution of \(\rho \)
evolution of \(\rho \) after BH formation (subcritical)
\rho/\rho_\text{FRW}
r
evolution of \(\rho \) after BH formation (supercritical)
\rho/\rho_\text{FRW}
r
supercritical
M<2.8H_iR_i^2/G
M\approx\kappa R_i^3
subcritical
\kappa R_i^3,\ \ \ M< M_*
H_iR_i^2,\ \ \ M> M_*
GM\sim
\{
mass of BH formed by vacuum bubbles
where
M_*\sim \frac{H_i^3}{G\kappa}
bubble size distribution
number density
dn= \lambda\frac{dR}{(R+H_i^{-1})^4}
PBH mass function (fraction of CDM in PBH with mass ~ \(M\) )
f(M)\equiv \frac{M^2}{\rho_\text{CDM}} \frac{dn}{dM}\propto
M_*^{-1/2},\ \ \ M< M_*
M^{-1/2},\ \ \ M> M_*
\{
BH mass distribution
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 \(\to\) our \(f_\text{PBH} \)
1705.05567 by Carr et al. (2017)
LIGO PBH?
\(f(M\sim 30\ M_\odot) \sim 10^{-3} \ \to \) merger rate for LIGO events
1603.08338 by Sasaki et al. (2016)
M_* \sim 30\ M_\odot
M_\text{min} \sim 10^{-5}\ M_\odot
f_\text{PBH}\sim0.01
\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
Copy of Primordial Black Hole Formation by Vacuum Bubbles
By Heling Deng
