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
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