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Introduction

\phi

\phi

V(\phi)

V(\phi)

Two characteristic time scales

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

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

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

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

1. Wall expansion

2. Cosmological expansion

t_{H}=\frac{8r_{i}^{3}}{27t_{i}^{2}}

t_{H}=\frac{8r_{i}^{3}}{27t_{i}^{2}}

(dust)

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

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

(radiation)

Hubble crossing

ar_i=\frac{1}{H}

ar_i=\frac{1}{H}

(\sigma)

(\sigma)

(r_i)

(r_i)

Two parameters:

\sigma,\ r_i

\sigma,\ r_i

where

Matching on the wall

\ddot{r}+(4-3a^{2}\dot{r}^{2})H\dot{r}+\frac{2}{a^{2}r}(1-a^{2}\dot{r}^{2})=6\pi\sigma\frac{(1-a^{2}\dot{r}^{2})^{\frac{3}{2}}}{a}

\ddot{r}+(4-3a^{2}\dot{r}^{2})H\dot{r}+\frac{2}{a^{2}r}(1-a^{2}\dot{r}^{2})=6\pi\sigma\frac{(1-a^{2}\dot{r}^{2})^{\frac{3}{2}}}{a}

EOM:

FRW +

ds^{2}=-dt^{2}+B^{2}(r,t)dr^{2}+R^{2}(r,t)d\Omega^{2}

ds^{2}=-dt^{2}+B^{2}(r,t)dr^{2}+R^{2}(r,t)d\Omega^{2}

Dust

Radiation

Simulation

Metric

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

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

Energy-momentum tensor

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

u^{\mu}=\left(\frac{1}{\sqrt{1-v^{2}}},\frac{v}{B\sqrt{1-v^{2}}},0,0\right)

u^{\mu}=\left(\frac{1}{\sqrt{1-v^{2}}},\frac{v}{B\sqrt{1-v^{2}}},0,0\right)

Variable transformations

r\to\frac{r}{H_{i}},\ t\to\frac{t}{H_{i}},\ \rho\to M_{Pl}^{2}H_{i}^{2}\rho,\ \phi\to M_{Pl}\phi,\ V\to M_{Pl}^{2}H_{i}^{2}V

r\to\frac{r}{H_{i}},\ t\to\frac{t}{H_{i}},\ \rho\to M_{Pl}^{2}H_{i}^{2}\rho,\ \phi\to M_{Pl}\phi,\ V\to M_{Pl}^{2}H_{i}^{2}V

Einstein equations + KG equation

B, R, \phi, \rho, v

B, R, \phi, \rho, v

Initial and boundary conditions

BH apparent horizon

\Theta_{out}\propto\frac{\dot{R}+\frac{R^{\prime}}{B}}{R},\ \Theta_{in}\propto\frac{\dot{R}-\frac{R^{\prime}}{B}}{R}

\Theta_{out}\propto\frac{\dot{R}+\frac{R^{\prime}}{B}}{R},\ \Theta_{in}\propto\frac{\dot{R}-\frac{R^{\prime}}{B}}{R}

\Theta_{out}=0,\ \Theta_{in}<0

\Theta_{out}=0,\ \Theta_{in}<0

Misner-Sharp mass

M=\frac{R}{2}(1-\frac{R^{\prime2}}{B^{2}}+\dot{R}^{2})

M=\frac{R}{2}(1-\frac{R^{\prime2}}{B^{2}}+\dot{R}^{2})

Expansions of radial null geodesics

Subcritical wall

Initial BH mass

M_{BHi}=4\pi\sigma CR_{H}^{2}

M_{BHi}=4\pi\sigma CR_{H}^{2}

where

R_{H}=\frac{1}{H(t_{H})}

R_{H}=\frac{1}{H(t_{H})}

C_{dust}\approx0.2,\ C_{radiation}\approx0.6

C_{dust}\approx0.2,\ C_{radiation}\approx0.6

Garriga, Vilenkin, Zhang (2016)

Final BH

\frac{dM_{BH}(t)}{dt}=4\pi FR^{2}_{BH}\rho(t)

\frac{dM_{BH}(t)}{dt}=4\pi FR^{2}_{BH}\rho(t)

M_{BH}(t)=\frac{1}{\frac{1}{M_{0}}+\frac{3}{2}F\left(\frac{1}{t}-\frac{1}{t_{0}}\right)}

M_{BH}(t)=\frac{1}{\frac{1}{M_{0}}+\frac{3}{2}F\left(\frac{1}{t}-\frac{1}{t_{0}}\right)}

M_{BHf}=\frac{1}{\frac{1}{M_{0}}-\frac{5.7}{t_{0}}}

M_{BHf}=\frac{1}{\frac{1}{M_{0}}-\frac{5.7}{t_{0}}}

F\approx 3.8

F\approx 3.8

Outer BH mass accretion

M_{BH2}^{(out)}(Kt)=KM_{BH1}^{(out)}(t)

M_{BH2}^{(out)}(Kt)=KM_{BH1}^{(out)}(t)

Self-similar process

K\approx\frac{M_{BHi2}}{M_{BHi1}}\approx\left(\frac{r_{i2}}{r_{i1}}\right)^{2}

K\approx\frac{M_{BHi2}}{M_{BHi1}}\approx\left(\frac{r_{i2}}{r_{i1}}\right)^{2}

where

\frac{M_{BHf2}^{(out)}}{M_{BHf1}^{(out)}}\approx\left(\frac{r_{i2}}{r_{i1}}\right)^{2}

\frac{M_{BHf2}^{(out)}}{M_{BHf1}^{(out)}}\approx\left(\frac{r_{i2}}{r_{i1}}\right)^{2}

M_{BHf}^{(out)}\approx2M_{BHi}

M_{BHf}^{(out)}\approx2M_{BHi}

We found that

Primordial Black Hole Formation by a Spherical Domain Wall

· Introduction
· Simulation
· Results

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Heling Deng

Primordial Black Hole Formation by a Spherical Domain Wall

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