Heling Deng
with Alex Vilenkin, Jaume Garriga and Masaki Yamada
- 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?
\(f(M\sim 30\ M_\odot) \sim 10^{-3} \ \to \)Â merger rate for LIGO events
1603.08338Â Misao Sasaki, Teruaki Suyama, Takahiro Tanaka, and Shuichiro Yokoyama (2016)
SMBH?
\(M \gtrsim 10^3\ M_\odot \ \to \)Â primordial seeds of SMBH
astro-ph/0406260Â 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
Copy of PBHs from Bubbles
By Heling Deng
