Primordial Black Holes from Primordial Bubbles

Heling Deng

with Alex Vilenkin, Jaume Garriga and Masaki Yamada

1804.10059

1612.03753

1710.02865

  • Background
  • Simulations
  • Observational constraints
  • Analytic estimates
  • Background
  • Simulations
  • Observational constraints
  • Analytic estimates
M_\text{Pl} \newline \sim10^{-5}\ \rm g
M_\text{evp} \newline \sim10^{15}\ \rm g
M_\odot \newline \sim10^{33}\ \rm g
M_\text{PBH}

LIGO BH

Primordial Black Hole (PBH)

Supermassive black holes

LIGO black holes

Dark matter

\sim \mathcal O (10) \ M_\odot

SMBH

\sim 10^6 \text{-} 10^{10}\ M_\odot

...

...

Stellar BH

Observational constraints on PBH as DM

  • Background
  • Simulations
  • Observational constraints
  • Analytic estimates
\rho_i
\rho_i
\rho_b

Inflation

\rho_i
\rho_b
\rho_b< \rho_i

Bubble interior

\rho_i

Inflation

\rho_b
\rho_b< \rho_i

Bubble interior

\rho_b
\rho_i

Radiation

\rho_b
\rho_b
\rho_b
H_b^{-1}
H_b^{-1}

Subcritical

BH

H_b^{-1}
H_b^{-1}

Subcritical

BH

H_b^{-1}
H_b^{-1}

Subcritical

Supercritical

BH

Spacetime and BH

Subcritical

Spacetime and BH

R_i
\rho_b
\sigma

Subcritical

M \sim \kappa(\rho_i,\rho_b,\sigma) R_i^3

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

Supercritical

Spacetime and BH

M \sim \kappa(\rho_i,\rho_b,\sigma) R_i^3
GM <\mathcal O (1)H_iR_i^2

upper bound:

Subcritical

Supercritical

R_i
\rho_b
\sigma
  • Background
  • Simulations
  • Observational constraints
  • Analytic estimates

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}
p=\frac{1}{3}\rho

Radiation

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)
\rho/\rho_\text{FRW}
r

Early stage of evolution of \(\rho \)

r
\rho/\rho_\text{FRW}

Evolution of \(\rho \) after BH formation (subcritical)

r
\rho/\rho_\text{FRW}

Evolution of \(\rho \) at late time (subcritical)

\rho/\rho_\text{FRW}
r

Evolution of \(\rho \) after BH formation (supercritical)

Estimates

Sub:

M< \mathcal O(1) M_\text{Pl}^2 H_iR_i^2

Super:

Simulations

BH mass

M \sim \kappa R_i^3
\kappa R_i^3,\ \ \ M< M_*
H_iR_i^2,\ \ \ M> M_*
GM\sim
\{

BH mass

where

M_*\sim \frac{H_i^3}{G\kappa}
  • Background
  • Simulations
  • Observational constraints
  • Analytic estimates

BH mass distribution

PBH mass function (fraction of DM in PBH with mass \(\sim M\))

f(M)\equiv \frac{\rho_t(M)}{\rho_\text{CDM}}

Mass density after inflation (\(\sim M\))

\rho_t(M)=Mn_t(M)

Number density after inflation (\(\sim R_i\))

n_t(R_i)\approx\frac{\Gamma}{R_i^3}\frac{a(t_i)^3}{a(t)^3}
M_*^{-1/2},\ \ \ M< M_*
M^{-1/2},\ \ \ M> M_*
f(M)\propto
\{

Lower cutoff

Bubble wall fluctuations

Small bubbles

BH mass distribution

Observational constraints on monochromatic PBH

Observational constraints on our PBH

LIGO PBH?

\(f(M\sim 30\ M_\odot) \sim 10^{-3} \ \to \)  merger rate for LIGO events

1603.08338  Misao Sasaki, Teruaki Suyama, Takahiro Tanaka, and Shuichiro Yokoyama (2016)

SMBH?

\(M \gtrsim 10^3\ M_\odot \ \to \)  primordial seeds of SMBH

astro-ph/0406260 Norbert Duechting (2004)

Galaxy number density:

\sim 0.1\ \text{Mpc}^{-3}

All DM

SMBH

LIGO

(10\%)

CMB distortion?

CMB: black-body spectrum (no chemical potential)

\(\mu \)-era:

photon number changing processes are ineffective

Photon number not conserved

\(\mu \)-distortion: spectrum with chemical potential

Photon diffusion

Silk damping: wiping off inhomogeniety

CMB distortion?

BH

BH

BH

BH

BH

BH

Photon diffusion length at recombination

10\ \text{Mpc}

BH

BH

BH

BH

BH

BH

BH

BH

10\ \text{Mpc}

BH

BH

Photon diffusion length at recombination

BH

BH

BH

BH

BH

BH

10\ \text{Mpc}

BH

BH

Photon diffusion length at recombination

\(\bar \mu \) dominated by

10\ \text{Mpc}

Photon diffusion length at recombination

Observational constraints on \(\mu \)-distortion

Possible large BHs in some rare regions on LSS \(\to\) spiky \(\mu\)

\mu_\text{max} \sim 10^{10} \bar{\mu}

Observations:

\bar{\mu}< 10^{-4}

Observations:

\mu_\text{max} \lesssim10^{-4}

Conclusions

  • This model can be naturally implemented in landscape models of the kind suggested by string theory
  • Depending on their sizes after inflation ends, bubbles can be classified into two categories: subcritical and supercritical, both leading to black hole formation
  • It predicts distinctive PBH mass spectra, ranging over many orders of magnitude
  • With some parameter choices, these PBH can account for SMBHs, for the black hole mergers observed by LIGO, and/or for the dark matter
  • It predicts a spiky distribution of \(\mu \)-distortion in CMB

Bubble size distribution

R(t-t_n)\approx H_i^{-1}\left[e^{H_i(t-t_n)}-1\right]

Bubble radius during inflation

Number of bubbles

dN=\lambda H_i^{} e^{3H_i t_n}d^3xdt_n

Number density during inflation

Number density after inflation

dn = \frac{dN}{H_i^{-3}e^{3H_i t}d^3x} = \lambda\frac{dR}{(R+H_i^{-1})^4}
dn(t)= \lambda\frac{dR_i}{(R_i+H_i^{-1})^4}\frac{a(t_i)^3}{a(t)^3}

CMB distortion?

Photon diffusion

Silk damping: wiping off inhomogeniety

Damping scale:

\lambda_D(t) \propto t^{5/4}
\lambda_D^{(c)}(t_\text{rec}) \sim 10\ \text{Mpc}

\(\mu \)-era:

Recombination:

sound horizon (shell comoving radius)

0.1 \ \text{-}\ 2\ \text{Mpc}

(Silk patch)

  • Background
  • Spherical domain walls
  • Conclusions
  • Vacuum bubbles
\rho_i
\rho_i
\rho_i

Inflation

\rho_i
\rho_i
H_b^{-1}
\sigma
\rho_i

Inflation

\rho_i
\rho_i
\rho_i

Radiation

M^{1/4}M_*^{-3/4},\ \ \ M_\text{min} < M < M_*
M^{-1/2},\ \ \ M> M_*
f(M)\propto
\{

BH mass distribution

Observational constraints on our PBH

Repulsive planar wall

  • Constant acceleration away from the wall
  • Exponential expansion on the wall

Evolution of \(\rho \)

Inflating wall

t_{\sigma}=\frac{1}{2\pi\sigma}

where

t_{H}=\frac{R_{i}^{2}}{4t_{i}}

Hubble crossing

aR_i=\frac{1}{H}

Gravity vs. Tension

t_{\sigma}\gg t_{H}

Subcritical:

t_{\sigma}\ll t_{H}

Supercritical:

ds^{2}=-dt^{2}+e^{2t/t_{\sigma}}\left(dy^{2}+dz^{2}\right)
t_{\sigma}=\frac{1}{2\pi\sigma}

where

t_{H}=\frac{R_{i}^{2}}{4t_{i}}

Hubble crossing

aR_i=\frac{1}{H}

Gravity vs. Tension

t_{\sigma}\gg t_{H}

Subcritical:

t_{\sigma}\ll t_{H}

Supercritical:

ds^{2}=-dt^{2}+e^{2t/t_{\sigma}}\left(dy^{2}+dz^{2}\right)

Subcritical

BH

M \sim 4\pi\sigma\mathcal O(1)R_H^2
\propto R_i^4

Inflating wall

t_{\sigma}=\frac{1}{2\pi\sigma}

where

t_{H}=\frac{R_{i}^{2}}{4t_{i}}

Hubble crossing

aR_i=\frac{1}{H}

Gravity vs. Tension

t_{\sigma}\gg t_{H}

Subcritical:

t_{\sigma}\ll t_{H}

Supercritical:

ds^{2}=-dt^{2}+e^{2t/t_{\sigma}}\left(dy^{2}+dz^{2}\right)

Supercritical

Subcritical

BH

M \sim 4\pi\sigma\mathcal O(1)R_H^2
\propto R_i^4
GM <\mathcal O (1)H_iR_i^2

upper bound:

Inflating wall

\mathcal O(10) \sigma H_i^2 R_i^4,\ \ \ M_\text{min} < M < M_*
H_iR_i^2,\ \ \ M> M_*
GM\sim
\{

BH mass

where

M_*\sim \mathcal O(0.1)(G\sigma)^{-1}
M_\text{min}\sim \mathcal O(10)\frac{\sigma}{GH_i^2}
  • results
  • simulation setup
  • observations
  • dynamics & spacetime

Metric

ds^2=-dt^2+B^2dr^2+R^2d\Omega^2

Fluid

p=\frac{1}{3}\rho

radiation

T_{\nu}^{(\phi)\mu}=\partial^{\mu}\phi\partial_{\nu}\phi-\delta_{\nu}^{\mu}\left[\frac{1}{2}(\partial \phi)^{2}-V(\phi)\right]
T_{\nu}^{(f)\mu}=\left(\rho+p\right)u^{\mu}u_{\nu}-\delta_{\nu}^{\mu}p

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

early stage of evolution of bubble mass

parameters
2 0.5 0.3
3 1.7 1.0
4 3.9 2.1
5 7.9 3.7
2 0.8 0.7
1.8 0.8 0.9
1 0.3 0.3
R_i(H_i^{-1})
M_\text{est}(M_\text{Pl}^2H_i^{-1})
M_\text{bh}(M_\text{Pl}^2H_i^{-1})
H_b=0.05,
H_\sigma=0.03
H_b=0.25,
H_b=0.5,
H_b=0.75,
H_\sigma=0.01
H_\sigma=0.02
H_\sigma=0.03
M_\text{est}\approx\kappa R_i^3

subcritical BH mass

(H_i)
f_\text{PBH}\equiv\frac{\rho_\text{PBH}}{\rho_\text{CDM}} = \int \frac{M n_t (M)}{\rho_\text{CDM}} dM= \int\frac{dM}{M}f(M)

Total fraction of CDM in PBH

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