black holes and baby universes from …...black holes of mass have inflating baby universes...

Stanford, March 2017

BLACK HOLES AND BABY UNIVERSES FROM INFLATION

Alex Vilenkin (Tufts University)

with Jaume Garriga, Jun Zhang and Heling Deng

Non-perturbative quantum effects during inflation may lead to the production of primordial black holes.

• Spherical domain walls or vacuum bubbles spontaneously nucleate during inflation.

• They are stretched to very large sizes and collapse to black holes after inflation ends.

Non-perturbative quantum effects during inflation may lead to the production of primordial black holes.

• Spherical domain walls or vacuum bubbles spontaneously nucleate during inflation.

• They are stretched to very large sizes and collapse to black holes after inflation ends.

Special features:

• A scaling distribution of PBH with a very wide spectrum of masses.

• BH larger than certain critical mass have inflating universes inside.

• For some parameter values, these BH can have significant observational effects.

First focus on vacuum bubbles.

• Particle physics models generally include a number of scalar fields.Then the inflaton field rolls in a multifield potential.

• As it rolls towards our vacuum, it can tunnel to another vacuum state.

Our vacuum

Bubble nucleation

ρi

ρb

ScalarField1

EnergyDensity

OurVacuum

ScalarField2

Bubbles of a lower-energy vacuum nucleate and expand during the slow roll: . ρb < ρi

ρiρb

ρb

Coleman & De Luccia (1980)Bubble nucleation

R ≈ Hi−1eHi (t−tn )

Time of nucleation t = tn:

At t > tn :

Scale-invariant size distribution:

dN ~ λ dRR4 dV

Hi = 8πGρi / 3( )1/2

Nucleation rate

Coleman & De Luccia (1980)Bubble nucleation

R

What happens to the bubbles when inflation ends?

1. Exterior view

Matter

• The bubble wall initially expands relativisticallyrelative to matter.

• It is quickly slowed down by particle scattering.

• An expanding relativistic shell of matter is formed.

Assume that matter particles are reflected from the bubble wall.

Similar to shocks in supernova explosions.

• The bubble eventually collapses to a black hole.

R


2. Interior view

Matter

Hb = (8πGρb / 3)1/2

For subcritical bubbles, with

an “ordinary” black hole forms.

R < Hb−1

Schwarzschild"singularity"

1

S

p

h

e

r

i

c

a

l

d

o

m

a

i

n

w

a

l

l

i

n

d

u

s

t

c

o

s

m

o

l

o

g

y

R=

2GMS

R=

0 R=

2GMbh , T �

�R

=2GM

bh , T ��

⇢ =�2GM

⇢ =const. r =

0 r =r0 ⇢ =

⇢0 t =

0t =

ti H �1=

2GMbh

Considera

spherical domain

wall embeddedin

aspatially

flatmatter

dominated

universe.Two

di↵erent time-scales are relevant to

the dynamics of such

awall. One is the cosm

ological scale t ⇠H �

1,

andthe

other is theacceleration

time-scale

dueto

therepulsive

gravitational fieldof the

domain

wall

t� ⇠(G�) �

1.

For R⌧

t� , therepulsive

fieldcan

beignored.

Inthis case, for t ⌧

R⌧

t� , thedom

ainwall is

conformally

stretchedby

cosmological expansion.

Eventually, whenthe wall falls within

the horizon, it

quicklyshrinks under its tension

andform

s ablack

hole of radius RS =

2GM⇠

t 2/t� ⌧

t.

Here, we will be primarily

interestedin

the opposite limit, where R �

t� . Inthis case, the wall repels

thematter around

it whileits size

grows faster thanthe

ambient expansionrate.

As weshall see, this

leads tothe form

ationof a

wormhole. The dust which

was originallyin

the interior of the wall goes into

ababy

universe, andin

theambient FRW

universewe

areleft with

ablack

holerem

nant cystedin

a

spherical regionof of vacuum

.

Beforewe

consider thee↵ect of the

domain

wall, let us first discuss thematching

of Schwarzschild

andadust cosm

ology.

1

.

1

M

a

t

c

h

i

n

g

S

c

h

w

a

r

z

s

c

h

i

l

d

t

o

a

d

u

s

t

c

o

s

m

o

l

o

g

y

Consider the Schwarzschildmetric

ds 2=

� ✓1� 2GM

R ◆dT 2

+ ✓1� 2GM

R ◆�1

dR 2+

R 2d� 2

.

(1)

InLem

aitre coordinates, this takes the formds 2=

�d� 2+ 2GM

Rd⇢ 2

+R 2

d� 2.

(2)

where �and

⇢are defined

bythe relations

d�=

±dT � s2GMR ✓

1� 2GMR ◆

�1

dR,

(3)

d⇢=

⌥dT+

s

R2GM✓

1� 2GMR ◆

�1

dR.

(4)

Subtracting(3) from

(4), we have

R=

32 (� +

⇢)�2/3

(2GM) 1/3

,

(5)

wherean

integrationconstant

hasbeen

absorbedby

ashift

inthe

originof the

⇢coordinate.

The

expressionfor T

as afunction

of �and

⇢can

be foundfrom

(3) as:

±T=

4GM�

� 13

✓R2GM

◆3/2

+s

R2GM �tanh �

1s

R2GM�

��

⇢,

(6)

withR

givenin

(5).A

secondintegration

constant has beenabsorbed

bya

shift inthe

Tcoordinate.

Sincethe

metric

(2) is synchronous, thelines of constant spatial coordinate

aregeodesics, and

�is the

1

S

p

h

e

r

i

c

a

l

d

o

m

a

i

n

w

a

l

l

i

n

d

u

s

t

c

o

s

m

o

l

o

g

y

R=

2GM

SR

=0R

=2G

Mbh

, T�

�R

=2G

Mbh

, T�

��⇢=

�2GM

⇢=

cons

t.r=

0r=

r 0⇢=

⇢ 0t =

0t =

t iH

�1 =2G

Mbh

Consid

era

sphe

rical

domain

wall

embe

dded

ina

spatially

flatmatterdo

minated

universe.

Two

di↵e

rent

time-scales

arerelev

ant to

thedy

namics

ofsu

chawa

ll.One

isth

eco

smolog

ical s

cale

t ⇠H

�1 ,

and

theothe

ris

theac

celer

ation

time-scaledu

eto

therepu

lsive

grav

itatio

nal fi

eldof

thedo

main

wall

t �⇠

(G�)

�1 .

ForR

⌧t �, t

herepu

lsive

field

can

beigno

red.

Inth

isca

se, f

ort⌧

R⌧

t �, t

hedo

main

wall

is

conformally

stretche

dby

cosm

olog

ical e

xpan

sion.

Even

tually,

whenth

ewa

llfalls

within

theho

rizon

, it

quick

lysh

rinks

unde

r itstens

ionan

dform

s ablackho

leof

radius

R S=

2GM

⇠t2/t �

⌧t.

Here,

wewill

bepr

imarily

interested

inth

e oppo

site lim

it,whe

reR

�t �. I

nth

isca

se, t

hewa

llrepe

ls

thematterarou

ndit

while

itssiz

egrow

sfaster

than

theam

bien

tex

pans

ionrate.As

wesh

all s

ee, t

his

leads

toth

eform

ationof

awo

rmho

le.The

dust

which

was or

iginally

inth

einterio

r ofth

ewa

llgo

esinto

aba

byun

iverse, a

ndin

theam

bien

tFR

Wun

iverse

wearelef

twith

ablackho

leremna

ntcy

sted

ina

sphe

rical

region

ofof

vacu

um.

Before

weco

nsider

thee↵

ectof

thedo

main

wall,

letus

first

discus

sth

ematch

ingof

Schw

arzsch

ild

andadu

stco

smolog

y.

1

.

1

M

a

t

c

h

i

n

g

S

c

h

w

a

r

z

s

c

h

i

l

d

t

o

a

d

u

s

t

c

o

s

m

o

l

o

g

y

Consid

erth

eSc

hwarzsch

ildmetric

ds2 =

�✓ 1�

2GM

R◆ dT

2 +✓ 1�

2GM

R◆ �1 dR

2 +R

2 d�2 .

(1)

InLe

maitreco

ordina

tes,

this

take

s theform

ds2 =

�d�2 +

2GM

Rd⇢

2 +R

2 d�2 .

(2)

where

�an

d⇢arede

fined

byth

erelatio

ns

d�=

±dT�

s 2GM

R✓ 1�

2GM

R◆ �1 dR

,

(3)

d⇢=

⌥dT+

s R2G

M✓ 1�

2GM

R◆ �1 dR

.

(4)

Subt

racting(3) fro

m(4),

weha

ve R=

32

(�+

⇢)� 2/

3 (2GM

)1/

3 ,

(5)

where

anintegration

cons

tant

hasbe

enab

sorb

edby

ash

iftin

theorigin

ofth

e⇢

coordina

te.

The

expr

essio

nfor T

asafunc

tionof

�an

d⇢ca

nbe

foun

dfro

m(3) as

:

±T=

4GM

��

13

✓R

2GM

◆ 3/2 +

s R2G

M�

tanh

�1s R

2GM

��

⇢,

(6)

withR

give

nin

(5).

Aseco

ndintegration

cons

tant

hasbe

enab

sorb

edby

ash

iftin

theT

coordina

te.

Sinc

eth

emetric

(2)is

sync

hron

ous,

thelin

esof

cons

tant

spatial c

oord

inatearege

odesics

, and

�is

the

dust"FRW"1 Spherical domain wall in dust cosmology

R = 2GMS R = 0 R = 2GMbh, T � � R = 2GMbh, T � �� ⇢ = �2GM ⇢ = const. r = 0 r = r0 ⇢ =⇢0 t = 0 t = ti H�1 = 2GMbh

Consider a spherical domain wall embedded in a spatially flat matter dominated universe. Twodi↵erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale t ⇠ H�1,and the other is the acceleration time-scale due to the repulsive gravitational field of the domain wallt� ⇠ (G�)�1.

For R ⌧ t�, the repulsive field can be ignored. In this case, for t ⌧ R ⌧ t�, the domain wall isconformally stretched by cosmological expansion. Eventually, when the wall falls within the horizon, itquickly shrinks under its tension and forms a black hole of radius RS = 2GM ⇠ t2/t� ⌧ t.

Here, we will be primarily interested in the opposite limit, where R � t�. In this case, the wall repelsthe matter around it while its size grows faster than the ambient expansion rate. As we shall see, thisleads to the formation of a wormhole. The dust which was originally in the interior of the wall goes intoa baby universe, and in the ambient FRW universe we are left with a black hole remnant cysted in aspherical region of of vacuum.

Before we consider the e↵ect of the domain wall, let us first discuss the matching of Schwarzschildand a dust cosmology.

1.1 Matching Schwarzschild to a dust cosmology

Consider the Schwarzschild metric

ds2 = �✓

1 � 2GM

R

◆dT 2 +

✓1 � 2GM

R

◆�1

dR2 + R2d�2. (1)

In Lemaitre coordinates, this takes the form

ds2 = �d�2 +2GM

Rd⇢2 + R2d�2. (2)

where � and ⇢ are defined by the relations

d� = ±dT �

s2GM

R

✓1 � 2GM

R

◆�1

dR, (3)

d⇢ = ⌥dT +

sR

2GM

✓1 � 2GM

R

◆�1

dR. (4)

Subtracting (3) from (4), we have

R =3

2(� + ⇢)

�2/3

(2GM)1/3, (5)

where an integration constant has been absorbed by a shift in the origin of the ⇢ coordinate. Theexpression for T as a function of � and ⇢ can be found from (3) as:

±T = 4GM

�

�1

3

✓R

2GM

◆3/2

+

sR

2GM� tanh�1

sR

2GM

�

� � ⇢, (6)

with R given in (5). A second integration constant has been absorbed by a shift in the T coordinate.Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and � is the

Bubble"

1Sphericaldom

ain

wallin

dust

cosm

ology

R=

2GM

SR

=0

R=

2GM

bh,T

��

R=

2GM

bh,T

��

�⇢

=�

2GM

⇢=

const

.r

=0

r=

r 0⇢

=⇢ 0

t=

0t=

t iH

�1

=2G

Mbh

r=

0

V(z

)z

Con

sider

asp

her

ical

dom

ain

wal

lem

bed

ded

ina

spat

ially

flat

mat

ter

dom

inat

eduniv

erse

.T

wo

di↵

eren

tti

me-

scal

esar

ere

leva

ntto

the

dyn

amic

sof

such

aw

all.

One

isth

eco

smol

ogic

alsc

ale

t⇠

H�

1,

and

the

other

isth

eac

cele

rati

onti

me-

scal

edue

toth

ere

puls

ive

grav

itat

ional

fiel

dof

the

dom

ain

wal

lt �

⇠(G

�)�

1.

For

R⌧

t �,

the

repuls

ive

fiel

dca

nbe

ignor

ed.

Inth

isca

se,

for

t⌧

R⌧

t �,

the

dom

ain

wal

lis

confo

rmal

lyst

retc

hed

byco

smol

ogic

alex

pan

sion

.E

vent

ual

ly,w

hen

the

wal

lfa

lls

wit

hin

the

hor

izon

,it

quic

kly

shri

nks

under

its

tensi

onan

dfo

rms

abla

ckhol

eof

radiu

sR

S=

2GM

⇠t2

/t�

⌧t.

Her

e,w

ew

illbe

pri

mar

ily

inte

rest

edin

the

oppos

ite

lim

it,w

her

eR

�t �

.In

this

case

,th

ew

allre

pel

sth

em

atte

rar

ound

itw

hile

its

size

grow

sfa

ster

than

the

ambie

ntex

pan

sion

rate

.A

sw

esh

allse

e,th

isle

ads

toth

efo

rmat

ion

ofa

wor

mhol

e.T

he

dust

whic

hw

asor

igin

ally

inth

ein

teri

orof

the

wal

lgo

esin

toa

bab

yuniv

erse

,an

din

the

ambie

ntFRW

univ

erse

we

are

left

wit

ha

bla

ckhol

ere

mnan

tcy

sted

ina

spher

ical

regi

onof

ofva

cuum

.B

efor

ew

eco

nsi

der

the

e↵ec

tof

the

dom

ain

wal

l,le

tus

firs

tdis

cuss

the

mat

chin

gof

Sch

war

zsch

ild

and

adust

cosm

olog

y.

1.1

Matching

Schw

arzschild

to

adust

cosm

ology

Con

sider

the

Sch

war

zsch

ild

met

ric

ds2

=�

✓1

�2G

M R

◆dT

2+

✓1

�2G

M R

◆�

1

dR2+

R2d�

2.

(1)

InLem

aitr

eco

ordin

ates

,th

ista

kes

the

form

ds2

=�

d�2+

2GM R

d⇢2+

R2d�

2.

(2)

wher

e�

and

⇢ar

edefi

ned

byth

ere

lati

ons

d�=

±dT

�

s2G

M R

✓1

�2G

M R

◆�

1

dR,

(3)

d⇢=

⌥dT

+

sR

2GM

✓1

�2G

M R

◆�

1

dR.

(4)

Subtr

acti

ng

(3)

from

(4),

we

hav

e

R=

3 2(�

+⇢)

� 2/3

(2G

M)1/

3,

(5)

wher

ean

inte

grat

ion

const

ant

has

bee

nab

sorb

edby

ash

ift

inth

eor

igin

ofth

e⇢

coor

din

ate.

The

expre

ssio

nfo

rT

asa

funct

ion

of�

and

⇢ca

nbe

found

from

(3)

as:

±T

=4G

M

� �1 3

✓R

2GM

◆3/

2

+

sR

2GM

�ta

nh

�1

sR

2GM

� ��

⇢,(6

)

FIG. 2: Causal diagram showing the formation of a black hole by a small vacuum bubble, with mass

Mbh ⌧ Mcr, in a dust dominated spatially flat FRW. At the time ti when inflation ends, a small

bubble (to the left of the diagram) with positive energy density ⇢b > 0, initially expands with a

large Lorentz factor relative to the Hubble flow. The motion is slowed down by momentum transfer

to matter, turning around due to the internal negative pressure and the tension of the bubble wall,

and eventually collapsing into a Schwarzschild singularity. At fixed angular coordinates, the time-

like dashed line at the edge of the dust region represents a radial geodesic of the Schwarzschild

metric (see Appendix B). Such geodesic approaches time-like infinity with unit slope. The reason

is that a smaller slope would correspond to the trajectory of a particle at a finite radial coordinate

R, while the geodesic of a particle escaping to infinity has unbounded R. The thin relativistic

expanding shell of matter produced by the bubble is not represented in the figure. This shell

disturbs the homogeneity of dust near the boundary represented by the dashed line.

For �Hi/⇢b ⌧ 1, the first term in (20) dominates, and the turning point occurs after a small

fractional increase in the bubble radius

�R

Ri

⇡ �

⇢b

Hi ⌧ 1, (35)

which happens after a time-scale much shorter than the expansion time, �t ⇠ (�/⇢b) ⌧ ti.

For 1 ⌧ �Hi/⇢b ⌧ (RiHi)3/2 we have

Rmax

Ri

⇡✓3�Hi

⇢b

◆1/3

. (36)

11

Penrose diagram:

R


2. Interior view

Matter

Hb = (8πGρb / 3)1/2

a(t)∝ exp(Hbt)

R > Hb−1

For supercritical bubbles, with

the interior begins to inflate:

Inflatingbaby universe

ExteriorFRW region

The “wormhole” is seen as a black hole by exterior observers.

Black holes of mass have inflating baby universes inside.M >Mcr ~ ρbHb−3

The wormhole formation here does NOT violate the NEC or any singularity theorem.

• The wormhole closes in about a light crossing time.

• Some matter may follow the bubble into the wormhole. . Need numerical simulation to determine Mbh.

(work in progress)

Penrose diagram

Futureinfinity

r =∞

r = 0Light propagatesat 45o.

FutureInfinity

BLACK HOLES FROM DOMAIN WALLS

Domain walls Form when a discrete symmetry is broken.

V

ϕ−η ηTwo degenerate vacua: ϕ = ±η

ϕ = +ηϕ = −η

Domainwall

Mass per unit area: . σ ~η3

Tension = σ

Domain walls Form when a discrete symmetry is broken.

V

ϕ−η ηTwo degenerate vacua: ϕ = ±η

Mass per unit area: . σ ~η3ϕ = +ηϕ = −η

Domainwall

Tension = σ

Wall gravity is repulsiveA.V. (1983)Ipser & Sikivie (1984)

Spherical walls of radius inflate: R

1 Spherical domain wall in dust cosmology

H�

= t�1

�

⌘ 2⇡G�R > t

�

fM

1

⇠ Mcr

⇠ 1/G2�M

2

⇠ f4/3M1

<⇠ M1

R = 0 R = 2GM,T ! 1 R = 2GM,T ! �1 ⇢ = �2GM ⇢ = const. r = 0 r = r0

⇢ = ⇢0

t = 0

[gij

,�] ⇠ e�W

QFT

[g

ij

,�]

V (�) �1 �2

[a2gij

,�] = e�W

QFT

[g

ij

,�; µ]

k ⌧ kc

⌘ aH

Consider a spherical domain wall embedded in a spatially flat matter dominated universe. Two di↵erenttime-scales are relevant to the dynamics of such a wall. One is the cosmological scale t ⇠ H�1, and theother is the acceleration time-scale due to the repulsive gravitational field of the domain wall t

�

⇠ (G�)�1.

gij

= a2e2⇣ [�ij

+ hij

]

SR = 4(k2/a2)⇣

h00ij

+ 2a0

ah0ij

+ k2hij

= 0

⇣ 00 + 2z0

z⇣ 0 + k2⇣ = 0

⇣, hij

�horizon

= H�1

[gij

,�] = e�W

QFT

[�; µ]

[a, hij

] ⇠ e�W

CFT

[a,h

ij

]

| |2 ⇠��e�W

CFT

��2

⇠ eiS0+iS2[a,hij

] ⇠ e�W

CFT

[a,h

ij

]

S0

[a] = �2M2

P

VHa3 = �c Vk3c

S2

[a, hij

] =c

2

X

k

[�kc

k2 + ik3 +O(k4/kc

)]|hk

|2

tσ = (2πGσ )−1

R(τ )∝ exp(τ / tσ )

Basu, Guth & A.V. (1991)

Scale-invariant size distribution: dN ~ λ dRR4 dV

Spherical domain walls of initial radius spontaneously nucleate during inflation.

Nucleation rate:


R(tn) = H�1

i

H� = t�1

� ⌘ 2⇡G�R > t�R(⌧) / eH�

⌧

fM

1

⇠ Mcr ⇠ 1/G2�M

2

⇠ f4/3M1

<⇠ M1

R = 0 R = 2GM,T ! 1 R = 2GM,T ! �1 ⇢ = �2GM ⇢ = const. r = 0 r = r0

⇢ = ⇢0

t = 0

[gij ,�] ⇠ e�WQFT

[gij

,�]

V (�) �1 �2

[a2gij ,�] = e�WQFT

[gij

,�; µ]

k ⌧ kc ⌘ aH

Consider a spherical domain wall embedded in a spatially flat matter dominated universe. Two di↵erenttime-scales are relevant to the dynamics of such a wall. One is the cosmological scale t ⇠ H�1, and theother is the acceleration time-scale due to the repulsive gravitational field of the domain wall t� ⇠ (G�)�1.

gij = a2e2⇣ [�ij + hij ]

SR = 4(k2/a2)⇣

h00ij + 2a0

ah0ij + k2hij = 0

⇣ 00 + 2z0

z⇣ 0 + k2⇣ = 0

⇣, hij

�horizon

= H�1

[gij ,�] = e�WQFT

[�; µ]

[a, hij ] ⇠ e�WCFT

[a,hij

]

| |2 ⇠��e�W

CFT

��2

⇠ eiS0+iS2[a,hij

] ⇠ e�WCFT

[a,hij

]

S0

[a] = �2M2

PVHa3 = �c Vk3c

R ≈ Hi−1eHi (t−tn )

λ ~ exp −2π 2σHi

3

"

#$

%

&'

RH < tσ

After inflationSubcritical walls come within the horizon having radius

They collapse to BH of mass . Mbh ~ 4πRH2σ

RH < tσ

After inflationSubcritical walls come within the horizon having radius

They collapse to BH of mass . Mbh ~ 4πRH2σ

A supercritical wall reaches radius before horizon crossing.

It inflates in a baby universe. .

R > tσ

Expandingballofmatter

Schwarzschildsphere

Inflatingdomainwall

Transition between the two regimes is at

Mbh =Mcr ~1/G2σ

Figure 5: These embedding diagrams show how a wormhole develops with time outside the wall.

The flat-looking region at the top and the bottom area represent the interior and exterior regions

respectively. The ring that encircles the cap is where the wall is located. The distance from the cap

center along a longitude line is the proper radius´Bdr, while the radius of a latitude circle is the

area radius R. The throat will be pinched off when the black hole singularity is encountered. These

diagrams are not shown with the same scale. The radius of the ring should grow exponentially.

(M (in)BHf and M

(out)BHf ) that are not identical because more fluid falls in from outside due to the

initial wall thickness. In a radiation-dominated universe, there’s also mass accretion as that

in sub-critical cases. As we’ll see, for dust background M(in)BHf ⇡ M

(out)BHf , while for radiation

M(out)BHf ⇡ 2M

(in)BHf .

A. Spacetime Structure

Applying adaptive mesh refinement, we are able to evolve the thin wall. However, as more

and more grid points are assigned to the wall, the code becomes computationally expensive.

Fortunately, since the wall grows exponentially away from both the interior and the exterior

regions, it can be removed when it gets almost detached and is only connected to theses

regions via two almost empty layers, where the energy density of either the scalar field or

the fluid is negligible compared to the FRW density (Fig. 6). Since we are more interested

in the black holes formed outside the wall, we may also remove the interior region and the

inner layer.

Now let’s take a look at the evolution of a constant time hypersurface near the wormhole.

We illustrate this with the help of Fig. 7, which shows how the expansions evolve, and

Fig. 8, which shows the spacetime outside the wall. These two graphs are obtained from a

simulation of a super-critical wall in radiation background with ri = 6.

15

Figure 8: Depicted under simulation coordinates, this graph shows the radiation energy density

distribution outside the wall. The wall is located at r ⇠ 6. The energy density has been rescaled so

that ⇢FRW = 1. Two apparent horizons and the cosmological horizon are shown with black dots.

Five constant time slices respectively correspond to five different stages discussed in text.

where the outer white hole and cosmological horizons arise, and the slice begins to cross a

normal region outside the throat. (4) Then the intersection of ⇥out and ⇥in reaches zero

(purple), which corresponds to a bifurcating marginal surface, and the white hole begins to

turn into a black hole. (5) The slice then leaves the WH region and two apparent horizons

appear with ⇥out = 0 (⇥in < 0) and ⇥in = 0 (⇥out < 0) respectively (brown).

As in the sub-critical case, we can excise the BH region to avoid simulation breakdown.

Now we need to cut at two apparent horizons instead of one (Fig. 9). However, we are

still able to evolve this region for a while to see how the singularity arises. After apparent

horizons appear, the area radius of the throat begins to decrease and approach zero (Fig. 10

and 11). Correspondingly, fluid energy density at the throat starts to increase and a spike

is formed that eventually goes to infinity (Fig. 11).

17

RD universe

Deng, Garriga & A.V. (2016)

A wormhole starts developingwhen .R ~ tσ

Numerical simulation of supercritical collapse

Black hole mass:

Mbh ~ RH / 2G

Comoving radius of the wall at horizon crossing

Mass distribution of black holes

Upper bound on f(M)

f (M ) = M 2

ρDM

dndM

Mass density of black holes per logarithmic mass interval in units of dark matter density:

MeqMcr

M-1/2M1/4

logM/M⊙

logf(M)

Meq ~1017M . – mass within the

horizon at teq.

Mcr ~1/G2σ

Effect of quantum fluctuations on subcritical wallsGarriga & A.V. (1992)

δRR~ B−1/2 B <100 – tunneling action

δR ≈ const during the wall collapse. €

~

BH forms only if δR < 2GM M >Mcr

B

€

~

€

~

B = 2π2σ

Hi3

Quantum fluctuations have little effect on supercritical walls.

Widrow (1989)

Mass distribution of black holes

Upper bound on f(M)

MeqMcr

M-1/2M1/4

logM/M⊙

logf(M)

Adiscoveryofblackholeswiththepredictedmassdistributionwouldprovideevidenceforinflation– andforbabyuniverses.

Two free parameters: σ , Hi

The critical mass:

Mcr ~1/G2σ

10 kg <Mcr <1015M

For

we have

100 GeV <η <1016GeV

.

Normalization depends on wall nucleation rate.

Observational bounds1-Gamma-ray background from BH evaporation with

€

~

Mbh ~1015g

.2-CMB spectral distortions by radiation emitted by gas accretion onto BH with : Mbh >103M

3-Overdensity bound : f (M )<1

Supermassive black hole seeds:

Mbh >103M

4- Atleast one seed pergalaxy.

.

Nucleation rate:

Mcr = 103 M⊙ Mcr = 1024 g Mcr = 1015 g

-20 -18 -16 -14 -12 -10 -8 -61.0

1.5

2.0

2.5

3.0

3.5

4.0

Log10Hi

MPl

ξ

Meq

Mcr

M-1ê2

M1ê4

logMêM ü

logfHML

t s<t eq

FIG

.9:

Mas

sdis

trib

uti

onfu

nct

ion

for

t �<

t eq.

V.

OB

SERVA

TIO

NA

LB

OU

ND

S

Observational

bou

ndson

themassspectrum

ofprimordialblack

holes

havebeenexten-

sively

studied;foran

upto

datereview

,see,

e.g.,[20].

For

smallblack

holes,themost

stringent

bou

ndcomes

from

the�-ray

backg

roundresultingfrom

black

holeevap

oration:

f(M

⇠10

15g)< ⇠

10�

8.

(98)

For

massive

black

holes

withM

>10

3M

�,thestrongest

bou

ndisdueto

distortionsof

the

CMBspectrum

producedby

theradiation

emittedby

gasaccreted

onto

theblack

holes

[21]:

f(M

>10

3M

�)< ⇠

10�

6.

(99)

Ofcourse,thetotalm

assdensity

ofblack

holes

cannot

exceed

thedensity

ofthedarkmatter.

Since

themassdistribution

inFig.9ispeakedat

Mbh

=M

cr,thisim

plies

f(M

cr)<

1.(100)

Thesebou

ndscannow

beusedto

imposeconstraints

onthedom

ainwallmod

elthat

we

analyzed

inSection

IV.Asbefore,

weshallproceed

under

theassumption

that

themass

bou

nd(87)

issaturated.

Themod

elis

fullycharacterizedby

theparam

eters⇠=

�/H

3 ian

dH

i/M

pl,whereH

iis

theexpan

sion

rate

duringinflation.Thenu

cleation

rate

ofdom

ainwalls�dep

endson

lyon

31

FIG. 10: Observational constraints on the distribution of black holes produced by domain walls.

Red and purple regions mark the parameter values excluded, respectively, by small black hole

evaporation and by gas accretion onto large black holes. The green region indicates the parameter

values allowing the formation of superheavy black hole seeds.The solid straight line marks the

parameter values where f(Mcr) = 1, so the parameter space below this line is excluded by Eq. (100).

⇠,

� ⇠ ⇠2e�2⇡2⇠ (101)

(see Eqs. (76), (77), (79)). The parameter space {⇠, Hi/Mpl} is shown in Fig. 10, with red

and purple regions indicating parameter values excluded by the constraints (98) and (99),

respectively. We show only the range ⇠ >⇠ 1, where the semiclassical tunneling calculation

is justified. Also, for ⇠ � 1 the nucleation rate � is too small to be interesting, so we only

show the values ⇠ ⇠ few.

The dotted lines in the figure indicate the values Mcr ⇠ 1015 g ⌘ Mevap and Mcr ⇠

103M� ⌘ Maccr. These lines, which mark the transitions between the subcritical and super-

critical regimes in the excluded red and purple regions, are nearly vertical. This is because

⇠ ⇠ 1 in the entire range shown in the figure, and therefore Mcr ⇠ ⇠�1(Mpl/Hi)3Mpl depends

32

1

2

Parameterspace

4


€

~

Mbh ~1015g




Mbh >103M


.

Nucleation rate:

Mcr = 103 M⊙ Mcr = 1024 g Mcr = 1015 g

-20 -18 -16 -14 -12 -10 -8 -61.0

1.5

2.0

2.5

3.0

3.5

4.0

Log10Hi

MPl

ξ

Meq

Mcr

M-1ê2

M1ê4

logMêM ü

logfHML

t s<t eq

FIG

.9:

Mas

sdis

trib

uti

onfu

nct

ion

for

t �<

t eq.

V.

OB

SERVA

TIO

NA

LB

OU

ND

S

Observational

bou

ndson

themassspectrum

ofprimordialblack

holes

havebeenexten-

sively

studied;foran

upto

datereview

,see,

e.g.,[20].

For

smallblack

holes,themost

stringent

bou

ndcomes

from

the�-ray

backg

roundresultingfrom

black

holeevap

oration:

f(M

⇠10

15g)< ⇠

10�

8.

(98)

For

massive

black

holes

withM

>10

3M

�,thestrongest

bou

ndisdueto

distortionsof

the

CMBspectrum

producedby

theradiation

emittedby

gasaccreted

onto

theblack

holes

[21]:

f(M

>10

3M

�)< ⇠

10�

6.

(99)

Ofcourse,thetotalm

assdensity

ofblack

holes

cannot

exceed

thedensity

ofthedarkmatter.

Since

themassdistribution

inFig.9ispeakedat

Mbh

=M

cr,thisim

plies

f(M

cr)<

1.(100)

Thesebou

ndscannow

beusedto

imposeconstraints

onthedom

ainwallmod

elthat

we

analyzed

inSection

IV.Asbefore,

weshallproceed

under

theassumption

that

themass

bou

nd(87)

issaturated.

Themod

elis


theparam

eters⇠=

�/H

3 ian

dH

i/M

pl,whereH

iis

theexpan

sion

rate


cleation

rate

ofdom

ainwalls�dep

endson

lyon

31






⇠,

� ⇠ ⇠2e�2⇡2⇠ (101)










32

1

2

Parameterspace

4

Mcr ~1020 gAccounts for DM and SMBH.


€

~

Mbh ~1015g




Mbh >103M


.

Nucleation rate:

Mcr = 103 M⊙ Mcr = 1024 g Mcr = 1015 g

-20 -18 -16 -14 -12 -10 -8 -61.0

1.5

2.0

2.5

3.0

3.5

4.0

Log10Hi

MPl

ξ

Meq

Mcr

M-1ê2

M1ê4

logMêM ü

logfHML

t s<t eq

FIG

.9:

Mas

sdis

trib

uti

onfu

nct

ion

for

t �<

t eq.

V.

OB

SERVA

TIO

NA

LB

OU

ND

S

Observational

bou

ndson

themassspectrum

ofprimordialblack

holes

havebeenexten-

sively

studied;foran

upto

datereview

,see,

e.g.,[20].

For

smallblack

holes,themost

stringent

bou

ndcomes

from

the�-ray

backg

roundresultingfrom

black

holeevap

oration:

f(M

⇠10

15g)< ⇠

10�

8.

(98)

For

massive

black

holes

withM

>10

3M

�,thestrongest

bou

ndisdueto

distortionsof

the

CMBspectrum

producedby

theradiation

emittedby

gasaccreted

onto

theblack

holes

[21]:

f(M

>10

3M

�)< ⇠

10�

6.

(99)

Ofcourse,thetotalm

assdensity

ofblack

holes

cannot

exceed

thedensity

ofthedarkmatter.

Since

themassdistribution

inFig.9ispeakedat

Mbh

=M

cr,thisim

plies

f(M

cr)<

1.(100)

Thesebou

ndscannow

beusedto

imposeconstraints

onthedom

ainwallmod

elthat

we

analyzed

inSection

IV.Asbefore,

weshallproceed

under

theassumption

that

themass

bou

nd(87)

issaturated.

Themod

elis


theparam

eters⇠=

�/H

3 ian

dH

i/M

pl,whereH

iis

theexpan

sion

rate


cleation

rate

ofdom

ainwalls�dep

endson

lyon

31






⇠,

� ⇠ ⇠2e�2⇡2⇠ (101)










32

1

2

Parameterspace

4

Mcr ~1020 gAccounts for DM and SMBH.

Merger rate observed by LIGO requiresfor Mbh ~ 30M .f ~10−3

.Mcr ~100M

Accounts for LIGO and SMBH.

Sasaki et al (2016)

The mechanism is generic, but requires a high nucleation rate of bubbles or domain walls.

String theory suggests a huge landscape of vacua. Some of the bubbles are likely to have a high nucleation rate.

Similarly, string theory predicts an “axiverse” with hundreds of axionsand associated domain walls.

A discovery of black holes with the predicted distribution of masses would provide evidence for inflation – and for nontrivial spacetime structure.

Concluding remarks


€

~

Mbh ~1015g


Nucleation rate:


Mcr = 103 M⊙ Mcr = 1024 g Mcr = 1015 g

-20 -18 -16 -14 -12 -10 -8 -61.0

1.5

2.0

2.5

3.0

3.5

4.0

Log10Hi

MPl

ξ

Meq

Mcr

M-1ê2

M1ê4

logMêM ü

logfHML

t s<t eq

FIG

.9:

Mas

sdis

trib

uti

onfu

nct

ion

for

t �<

t eq.

V.

OB

SERVA

TIO

NA

LB

OU

ND

S

Observational

bou

ndson

themassspectrum

ofprimordialblack

holes

havebeenexten-

sively

studied;foran

upto

datereview

,see,

e.g.,[20].

For

smallblack

holes,themost

stringent

bou

ndcomes

from

the�-ray

backg

roundresultingfrom

black

holeevap

oration:

f(M

⇠10

15g)< ⇠

10�

8.

(98)

For

massive

black

holes

withM

>10

3M

�,thestrongest

bou

ndisdueto

distortionsof

the

CMBspectrum

producedby

theradiation

emittedby

gasaccreted

onto

theblack

holes

[21]:

f(M

>10

3M

�)< ⇠

10�

6.

(99)

Ofcourse,thetotalm

assdensity

ofblack

holes

cannot

exceed

thedensity

ofthedarkmatter.

Since

themassdistribution

inFig.9ispeakedat

Mbh

=M

cr,thisim

plies

f(M

cr)<

1.(100)

Thesebou

ndscannow

beusedto

imposeconstraints

onthedom

ainwallmod

elthat

we

analyzed

inSection

IV.Asbefore,

weshallproceed

under

theassumption

that

themass

bou

nd(87)

issaturated.

Themod

elis


theparam

eters⇠=

�/H

3 ian

dH

i/M

pl,whereH

iis

theexpan

sion

rate


cleation

rate

ofdom

ainwalls�dep

endson

lyon

31






⇠,

� ⇠ ⇠2e�2⇡2⇠ (101)










32

1

2

Parameterspace

4

(Inomata etal.’17)

Ligo

Merger rate observed by LIGO requiresfor Mbh ~ 30M .f ~10−3

.Mcr ~100M

Accounts for LIGO and SMBH.

Sasaki et al (2016)

Penrose diagram for a supercritical wallGarriga, A.V. and Zhang (2015)

dust"FRW"

end"of"infla=on"

iden=fy"

I"

II"

III"

IV"

1

S

p

h

e

r

i

c

a

l

d

o

m

a

i

n

w

a

l

l

i

n

d

u

s

t

c

o

s

m

o

l

o

g

y

R=0R=2G

M,T

!1

R=2G

M,T

!�1

⇢=�2

GM

⇢=2G

Mr=0

Considerasphericaldomainwallem

bedded

inaspatially

flatmatterdominated

universe.

Two

di↵erent

time-scales

arerelevant

tothedynamicsofsuch

awall.One

isthecosm

ologicalscalet⇠H

�1 ,

andtheotheristheacceleration

time-scaledueto

therepulsivegravitationalfield

ofthedomainwall

t �⇠(G

�)�

1 .ForR

⌧t �,therepulsivefield

canbe

ignored.

Inthiscase,fort⌧

R⌧

t �,thedomainwallis

conformally

stretchedby

cosm

ologicalexpansion.

Eventually,whenthewallfalls

withinthehorizon,

it

quicklyshrinksunderitstensionandform

sablackholeofradius

RS=2G

M⇠t2/t�⌧

t.

Here,wewillbe

primarily

interested

intheoppositelim

it, w

hereR�

t �.In

thiscase, the

wall repels

thematteraround

itwhileitssize

grow

sfaster

than

theam

bientexpansionrate.Asweshallsee,this

leadsto

theform

ationofawormhole.The

dustwhich

was

originallyin

theinterior

ofthewall goesinto

ababy

universe,andin

theam

bientFRW

universe

weareleftwithablackholeremnant

cysted

ina

spherical regionofofvacuum

.

Beforeweconsider

thee↵ectof

thedomainwall,letus

first

discussthematchingof

Schw

arzschild

andadustcosm

ology.

1

.

1

M

a

t

c

h

i

n

g

S

c

h

w

a

r

z

s

c

h

i

l

d

t

o

a

d

u

s

t

c

o

s

m

o

l

o

g

y

ConsidertheSchw

arzschild

metric

ds2=�

✓ 1�

2GM

R◆ dT

2 +✓ 1�

2GM

R◆ �

1 dR2 +

R2 d⌦

2 .

(1)

InLem

aitrecoordinates,thistakestheform

ds2=�d

⌧2 +

2GM

Rd⇢

2 +R

2 d⌦

2 .

(2)

where

⌧and⇢aredefined

bytherelations

d⌧=

±dT�

s 2GM

R✓ 1�

2GM

R◆ �

1 dR,

(3)

d⇢=

⌥dT+

s

R2G

M✓ 1�

2GM

R◆ �

1 dR.

(4)

Subtracting(3)from

(4),wehave

R=

32(⌧

+⇢)

� 2/3 (2

GM)1/3 ,

(5)

where

anintegrationconstant

hasbeen

absorbed

byashift

intheorigin

ofthe⇢coordinate.

The

expression

forTas

afunction

of⌧and⇢canbe

foundfrom

(3)as:

±T=4G

M0

@1

3

✓R

2GM

◆ 3/2 +

s

R2G

M�tanh

�1

s

R2G

M

1A�⇢,

(6)

withR

givenin

(5).

Asecond

integrationconstant

hasbeen

absorbed

byashift

intheTcoordinate.

Sincethemetric(2)issynchronous,thelines

ofconstant

spatialcoordinate

aregeodesics,and⌧isthe


R = 2GMS R = 0 R = 2GM,T ! 1 R = 2GM,T ! �1 ⇢ = �2GM ⇢ = const. r = 0 r = r0 ⇢ =⇢0 t = 0 t = ti







ds2 = �✓1� 2GM

R

◆dT 2 +

✓1� 2GM

R

◆�1

dR2 +R2d⌦2. (1)


ds2 = �d⌧2 +2GM

Rd⇢2 +R2d⌦2. (2)

where ⌧ and ⇢ are defined by the relations

d⌧ = ±dT �

s2GM

R

✓1� 2GM

R

◆�1

dR, (3)

d⇢ = ⌥dT +

sR

2GM

✓1� 2GM

R

◆�1

dR. (4)


R =3

2(⌧ + ⇢)

�2/3(2GM)1/3, (5)

where an integration constant has been absorbed by a shift in the origin of the ⇢ coordinate. Theexpression for T as a function of ⌧ and ⇢ can be found from (3) as:

±T = 4GM

0

@1

3

✓R

2GM

◆3/2

+

sR

2GM� tanh�1

sR

2GM

1

A� ⇢, (6)

with R given in (5). A second integration constant has been absorbed by a shift in the T coordinate.Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and ⌧ is the


R = 2GMS R = 0 R = 2GM,T ! 1 R = 2GM,T ! �1 ⇢ = �2GM ⇢ = const. r = 0 r = r0 ⇢ =⇢0 t = 0 t = ti







ds2 = �✓1� 2GM

R

◆dT 2 +

✓1� 2GM

R

◆�1

dR2 +R2d⌦2. (1)


ds2 = �d⌧2 +2GM

Rd⇢2 +R2d⌦2. (2)


d⌧ = ±dT �

s2GM

R

✓1� 2GM

R

◆�1

dR, (3)

d⇢ = ⌥dT +

sR

2GM

✓1� 2GM

R

◆�1

dR. (4)


R =3

2(⌧ + ⇢)

�2/3(2GM)1/3, (5)


±T = 4GM

0

@1

3

✓R

2GM

◆3/2

+

sR

2GM� tanh�1

sR

2GM

1

A� ⇢, (6)


Schwarzschild"singularity"

1

S

p

h

e

r

i

c

a

l

d

o

m

a

i

n

w

a

l

l

i

n

d

u

s

t

c

o

s

m

o

l

o

g

y

R=2GMSR=0R=2GMbh , T

!1R=2GMbh , T

!�1

⇢=�2GM

⇢=const. r

=0r=r0 ⇢

=

⇢0 t =

0t =

tiH �

1=2GMbh

Consideraspherical

domain

wallembedded

inaspatially

flatmatter

dominated

universe.Two

di↵erent time-scales are

relevant tothe

dynamics of such

awall. O

neis the

cosmological scale

t ⇠H �

1,

andthe

otheristhe

accelerationtim

e-scaledue

tothe

repulsivegravitational field

of thedom

ainwall

t� ⇠(G�) �

1.

ForR⌧t� , the

repulsivefield

canbe

ignored.Inthis

case, fort ⌧

R⌧t� , the

domain

wall is

conformally

stretchedby

cosmological expansion.

Eventually, whenthe

wall fallswithin

thehorizon, it

quicklyshrinks under its tension

andform

s ablack

holeof radius

RS =

2GM⇠t 2/t� ⌧

t.

Here, we will be primarily

interestedinthe opposite lim

it, whereR�t� . In

this case, the wall repels

thematter

arounditwhile

itssize

growsfaster

thanthe

ambient

expansionrate.

Aswe

shall see, this

leads tothe

formation

of aworm

hole. Thedust which

was originallyinthe

interior of thewall goes into

ababy

universe, andinthe

ambient

FRWuniverse

weare

leftwith

ablack

holerem

nantcysted

ina

spherical regionof of vacuum

.

Beforewe

considerthe

e↵ectof the

domain

wall, letusfirst

discussthe

matching

of Schwarzschild

andadust cosm

ology.

1

.

1

M

a

t

c

h

i

n

g

S

c

h

w

a

r

z

s

c

h

i

l

d

t

o

a

d

u

s

t

c

o

s

m

o

l

o

g

y

Consider theSchwarzschild

metric

ds 2=� ✓

1� 2GM

R ◆dT 2

+ ✓1� 2G

MR ◆

�1

dR 2+R 2d⌦ 2

.

(1)

InLem

aitrecoordinates, this takes the

formds 2=�d⌧ 2

+ 2GM

Rd⇢ 2

+R 2d⌦ 2

.

(2)

where⌧and

⇢are

definedbythe

relations

d⌧=

±dT � s

2GM

R ✓1� 2G

MR ◆

�1

dR,

(3)

d⇢=

⌥dT

+s

R2GM ✓

1� 2GM

R ◆�1

dR.

(4)

Subtracting(3) from

(4), wehave

R=

32 (⌧

+⇢)

�2/3

(2GM) 1/3

,

(5)

wherean

integrationconstant

hasbeen

absorbedby

ashift

inthe

originofthe

⇢coordinate.

The

expressionfor

Tas a

functionof ⌧

and⇢can

befound

from(3) as:

±T=4GM 0

@ 13

✓

R2GM

◆3/2

+s

R2GM �

tanh �1

s

R2GM

1A�⇢,

(6)

withRgiven

in(5).

Asecond

integrationconstant

hasbeen

absorbedby

ashift

inthe

Tcoordinate.

Sincethe

metric

(2)issynchronous, the

linesof constant

spatial coordinateare

geodesics, and⌧isthe

1

S

p

h

e

r

i

c

a

l

d

o

m

a

i

n

w

a

l

l

i

n

d

u

s

t

c

o

s

m

o

l

o

g

y

R=2GMSR=0R=2GMbh, T

!1R=2GMbh, T

!�1

⇢=�2GM

⇢=const.r=0r=r 0⇢=

⇢ 0t =

0t =

t iH

�1=2GMbh

Considerasphericaldomainwallembedded

inaspatially

flatmatterdominated

universe.Tw

o

di↵erenttim

e-scalesarerelevant tothedynamics

ofsuchawall.Oneisthecosmological scalet ⇠

H�1 ,

andtheotheristheaccelerationtim

e-scaleduetotherepulsivegravitational field

ofthedomainwall

t �⇠(G�)

�1 .

ForR⌧

t �, the

repulsivefieldcanbe

ignored.

Inthiscase, fort⌧

R⌧

t �, the

domainwallis

conformallystretchedby

cosmological expansion.Eventually,whenthewallfallswithinthehorizon, it

quicklyshrinksunder itstensionandforms a

blackholeofradiusR S

=2GM⇠t2 /t�⌧t.

Here,we

willbeprimarilyinterestedinthe opposite

limit,where R

�t �. Inthiscase, thewallrepels

thematteraround

itwhileits

sizegrowsfasterthan

theambientexpansionrate.Aswe

shall see, this

leadstotheformationofawormhole.

Thedustwhich

was originallyintheinterior of thewallgoesinto

ababy

universe, and

intheambientFRW

universewe

areleftwithablackholeremnantcysted

ina

sphericalregionofofvacuum.

Beforewe

considerthee↵ectofthedomainwall,let

usfirstdiscussthematchingofSchwarzschild

andadustcosmology.

1

.

1

M

a

t

c

h

i

n

g

S

c

h

w

a

r

z

s

c

h

i

l

d

t

o

a

d

u

s

t

c

o

s

m

o

l

o

g

y

ConsidertheSchwarzschild

metric

ds2 =�

✓ 1�2GM

R◆ dT

2 +✓ 1�

2GM

R◆ �1 dR

2 +R

2 d⌦2 .

(1)

InLemaitrecoordinates,thistakes theform

ds2 =�d⌧2 +

2GM

Rd⇢

2 +R

2 d⌦2 .

(2)

where⌧and⇢aredefinedbytherelations

d⌧=

±dT�

s 2GM

R✓ 1�

2GM

R◆ �1 dR

,

(3)

d⇢=

⌥dT+

s R2GM

✓ 1�2GM

R◆ �1 dR

.

(4)

Subtracting(3) from(4),we

have R=

32(⌧+⇢)

� 2/3 (2GM)1/3 ,

(5)

wherean

integrationconstanthasbeen

absorbed

byashift

intheoriginofthe⇢coordinate.The

expressionfor T

asafunctionof⌧and⇢canbefoundfrom(3) as:

±T=4GM

0@1

3✓R

2GM

◆ 3/2 +

s R2GM

�tanh

�1s R

2GM

1A �

⇢,

(6)

withRgivenin(5).Asecond

integrationconstanthasbeenabsorbedby

ashiftintheTcoordinate.

Sincethemetric

(2)issynchronous,thelinesofconstantspatial coordinatearegeodesics, and

⌧isthe

Baby"universe" Parent"universe"


R = 2GMS R = 0 R = 2GMbh, T ! 1 R = 2GMbh, T ! �1 ⇢ = �2GM ⇢ = const. r = 0 r = r0 ⇢ =⇢0 t = 0 t = ti H�1 = 2GMbh







ds2 = �✓1� 2GM

R

◆dT 2 +

✓1� 2GM

R

◆�1

dR2 +R2d⌦2. (1)


ds2 = �d⌧2 +2GM

Rd⇢2 +R2d⌦2. (2)


d⌧ = ±dT �

s2GM

R

✓1� 2GM

R

◆�1

dR, (3)

d⇢ = ⌥dT +

sR

2GM

✓1� 2GM

R

◆�1

dR. (4)


R =3

2(⌧ + ⇢)

�2/3(2GM)1/3, (5)


±T = 4GM

0

@1

3

✓R

2GM

◆3/2

+

sR

2GM� tanh�1

sR

2GM

1

A� ⇢, (6)


For pressureless matter,analytic solution canbeobtainedby gluing exact solutions

black holes and baby universes from …...black holes of mass have inflating baby universes...

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