Primordial Black Holes from Bubbles

Heling Deng

July, 2019

Tufts University

Dissertation Defense

Advisor: Alexander Vilenkin

1804.10059 with Alex Vilenkin and Masaki Yamada (2018)

"Primordial black hole formation by vacuum bubbles"

1612.03753 with Jaume Garriga and Alex Vilenkin (2016)

1512.01819 by Jaume Garriga, Alex Vilenkin and Jun Zhang (2016)

"Black holes and the multiverse"

"Primordial black hole and wormhole formation by domain walls"

"CMB spectral distortions from black holes formed by vacuum bubbles"

1710.02865 with Alex Vilenkin (2017)

Background

Spherical domain walls

Conclusions

Vacuum bubbles

Background

Spherical domain walls

Conclusions

Vacuum bubbles

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

Spherical domain walls

Conclusions

Vacuum bubbles

\rho_i

\rho_b

\rho_i

R

M_i=\frac{4}{3} \pi R^3 \rho_i

Inflation

\rho_i

\rho_b< \rho_i

M_b=\frac{4}{3} \pi R^3 \rho_b+4\pi R^2 \sigma

Bubble interior

\rho_b

\sigma

R

M_i=\frac{4}{3} \pi R^3 \rho_i

Inflation

\rho_b

\rho_i

\rho_b< \rho_i

M_b-M_i

\sigma

Bubble interior

R

=4\pi R^2 \sigma -\frac{4}{3} \pi R^3 (\rho_i-\rho_b)

=0

\to R=3\sigma/(\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

H_b^{-1}

Subcritical

BH

H_b^{-1}

Subcritical

BH

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 Black Hole

Supercritical

Spacetime and Black Hole

Heling Deng

Department of Physics and Astronomy

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

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

\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}

BH mass distribution

PBH mass function (Fraction of CDM 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

CMB distortion?

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}

Background

Spherical domain walls

Conclusions

Vacuum bubbles

\rho_i

Inflation

\rho_i

H_b^{-1}

\sigma

\rho_i

Inflation

\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

Background

Spherical domain walls

Conclusions

Vacuum bubbles

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

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)

BH

Silk patch

10\ \text{Mpc}

BH

Silk patch

10\ \text{Mpc}

BH

Silk patch

10\ \text{Mpc}

BH

Silk patch

\(\bar \mu \) dominated by

10\ \text{Mpc}

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

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

early stage of evolution of bubble mass


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)

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}

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