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

M_\text{Pl} \newline \sim10^{-5}\ \rm g

M_\text{Pl} \newline \sim10^{-5}\ \rm g

M_\text{evp} \newline \sim10^{15}\ \rm g

M_\text{evp} \newline \sim10^{15}\ \rm g

M_\odot \newline \sim10^{33}\ \rm g

M_\odot \newline \sim10^{33}\ \rm g

M_\text{PBH}

M_\text{PBH}

LIGO BH

Primordial Black Hole (PBH)

Supermassive black holes

LIGO black holes

Dark matter

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

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

SMBH

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

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

...

Stellar BH

Background

Simulations

Observational constraints

Analytic estimates

BH apparent horizon

\Theta_\text{out} \propto \dot{R}/A + R^\prime /B =0

\Theta_\text{out} \propto \dot{R}/A + R^\prime /B =0

Misner-Sharp mass

M_\text{bh}=R_\text{bh}/2G

M_\text{bh}=R_\text{bh}/2G

M=\frac{R}{2G}\left[1+(\dot{R}/A)^2-(R^\prime/B)^2\right]

M=\frac{R}{2G}\left[1+(\dot{R}/A)^2-(R^\prime/B)^2\right]

(M^\prime = 4\pi\rho R^2 R^\prime)

(M^\prime = 4\pi\rho R^2 R^\prime)

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}}

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

Mass density after inflation ( $\sim M$ )

\rho_t(M)=Mn_t(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}

n_t(R_i)\approx\frac{\Gamma}{R_i^3}\frac{a(t_i)^3}{a(t)^3}

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}

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

Observations:

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

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

Observations:

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

\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]

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

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 = \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}

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(t) \propto t^{5/4}

\lambda_D^{(c)}(t_\text{rec}) \sim 10\ \text{Mpc}

\lambda_D^{(c)}(t_\text{rec}) \sim 10\ \text{Mpc}

$\mu$ -era:

Recombination:

sound horizon (shell comoving radius)

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

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

(Silk patch)

Inflating wall

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

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

where

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

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

Hubble crossing

aR_i=\frac{1}{H}

aR_i=\frac{1}{H}

Gravity vs. Tension

t_{\sigma}\gg t_{H}

t_{\sigma}\gg t_{H}

Subcritical:

t_{\sigma}\ll t_{H}

t_{\sigma}\ll t_{H}

Supercritical:

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

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

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

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

where

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

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

Hubble crossing

aR_i=\frac{1}{H}

aR_i=\frac{1}{H}

Gravity vs. Tension

t_{\sigma}\gg t_{H}

t_{\sigma}\gg t_{H}

Subcritical:

t_{\sigma}\ll t_{H}

t_{\sigma}\ll t_{H}

Supercritical:

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

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

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

\propto R_i^4

\propto R_i^4

Inflating wall

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

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

where

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

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

Hubble crossing

aR_i=\frac{1}{H}

aR_i=\frac{1}{H}

Gravity vs. Tension

t_{\sigma}\gg t_{H}

t_{\sigma}\gg t_{H}

Subcritical:

t_{\sigma}\ll t_{H}

t_{\sigma}\ll t_{H}

Supercritical:

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

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

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

\propto R_i^4

\propto R_i^4

GM <\mathcal O (1)H_iR_i^2

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_*

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

H_iR_i^2,\ \ \ M> M_*

H_iR_i^2,\ \ \ M> M_*

GM\sim

GM\sim

\{

\{

BH mass

where

M_*\sim \mathcal O(0.1)(G\sigma)^{-1}

M_*\sim \mathcal O(0.1)(G\sigma)^{-1}

M_\text{min}\sim \mathcal O(10)\frac{\sigma}{GH_i^2}

M_\text{min}\sim \mathcal O(10)\frac{\sigma}{GH_i^2}

Metric

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

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

Fluid

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

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}^{(\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

T_{\nu}^{(f)\mu}=\left(\rho+p\right)u^{\mu}u_{\nu}-\delta_{\nu}^{\mu}p


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})

R_i(H_i^{-1})

M_\text{est}(M_\text{Pl}^2H_i^{-1})

M_\text{est}(M_\text{Pl}^2H_i^{-1})

M_\text{bh}(M_\text{Pl}^2H_i^{-1})

M_\text{bh}(M_\text{Pl}^2H_i^{-1})

H_b=0.05,

H_b=0.05,

H_\sigma=0.03

H_\sigma=0.03

H_b=0.25,

H_b=0.25,

H_b=0.5,

H_b=0.5,

H_b=0.75,

H_b=0.75,

H_\sigma=0.01

H_\sigma=0.01

H_\sigma=0.02

H_\sigma=0.02

H_\sigma=0.03

H_\sigma=0.03

M_\text{est}\approx\kappa R_i^3

M_\text{est}\approx\kappa R_i^3

subcritical BH mass

(H_i)

(H_i)

Primordial Black Holes from Primordial Bubbles

Primordial Black Hole (PBH)

BH mass

BH mass

BH mass distribution

BH mass distribution

Observational constraints on monochromatic PBH

Observational constraints on our PBH

LIGO PBH?

SMBH?

CMB distortion?

CMB distortion?

Observational constraints on $\mu$ -distortion

Conclusions

Bubble size distribution

CMB distortion?

BH mass distribution

Observational constraints on our PBH

BH mass

Copy of PBHs from Bubbles

Copy of PBHs from Bubbles

Heling Deng

Primordial Black Holes from Primordial Bubbles

Copy of PBHs from Bubbles

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