Feb 25, 2016 - The dynamics of spherically symmetric vacuum bubbles has been studied in the thin wall limit by Berezin .
Black holes and the multiverse Jaume Garrigaa,b , Alexander Vilenkinb and Jun Zhangb a
Departament de Fisica Fonamental i Institut de Ciencies del Cosmos,
Universitat de Barcelona, Marti i Franques, 1, 08028, Barcelona, Spain b
Institute of Cosmology, Department of Physics and Astronomy,
arXiv:1512.01819v4 [hep-th] 25 Feb 2016
Tufts University, Medford, MA 02155, USA
Abstract Vacuum bubbles may nucleate and expand during the inflationary epoch in the early universe. After inflation ends, the bubbles quickly dissipate their kinetic energy; they come to rest with respect to the Hubble flow and eventually form black holes. The fate of the bubble itself depends on the resulting black hole mass. If the mass is smaller than a certain critical value, the bubble collapses to a singularity. Otherwise, the bubble interior inflates, forming a baby universe, which is connected to the exterior FRW region by a wormhole. A similar black hole formation mechanism operates for spherical domain walls nucleating during inflation. As an illustrative example, we studied the black hole mass spectrum in the domain wall scenario, assuming that domain walls interact with matter only gravitationally. Our results indicate that, depending on the model parameters, black holes produced in this scenario can have significant astrophysical effects and can even serve as dark matter or as seeds for supermassive black holes. The mechanism of black hole formation described in this paper is very generic and has important implications for the global structure of the universe. Baby universes inside super-critical black holes inflate eternally and nucleate bubbles of all vacua allowed by the underlying particle physics. The resulting multiverse has a very non-trivial spacetime structure, with a multitude of eternally inflating regions connected by wormholes. If a black hole population with the predicted mass spectrum is discovered, it could be regarded as evidence for inflation and for the existence of a multiverse.
1
I.
INTRODUCTION
A remarkable aspect of inflationary cosmology is that it attributes the origin of galaxies and large-scale structure to small quantum fluctuations in the early universe [1]. The fluctuations remain small through most of the cosmic history and become nonlinear only in relatively recent times. Here, we will explore the possibility that non-perturbative quantum effects during inflation could also lead to the formation of structure on astrophysical scales. Specifically, we will show that spontaneous nucleation of vacuum bubbles and spherical domain walls during the inflationary epoch can result in the formation of black holes with a wide spectrum of masses. The physical mechanism responsible for these phenomena is easy to understand. The inflationary expansion of the universe is driven by a false vacuum of energy density ρi ≈ const. Bubble nucleation in this vacuum can occur [2] if the underlying particle physics model includes vacuum states of a lower energy density, ρb < ρi .1 We will be interested in the case when ρb > 0. Once a bubble is formed, it immediately starts to expand. The difference in vacuum tension on the two sides of the bubble wall results in a force F = ρi − ρb per unit area of the wall, so the bubble expands with acceleration. This continues until the end of inflation (or until the energy density of the inflating vacuum drops below ρb during the slow roll). At later times, the bubble continues to expand but it is slowed down by momentum transfer to the surrounding matter, while it is also being pulled inwards by the negative pressure of vacuum in its interior. Eventually, this leads to gravitational collapse and the formation of a black hole. The nature of the collapse and the fate of the bubble interior depends on the bubble size. The positive-energy vacuum inside the bubble can support inflation at the rate Hb = > H −1 , its (8πGρb /3)1/2 , where G is Newton’s constant. If the bubble expands to a radius R ∼ b
interior begins to inflate2 . We will show that the black hole that is eventually formed contains
a ballooning inflating region in its interior, which is connected to the exterior region by a wormhole. On the other hand, if the maximum expansion radius is R Hb−1 , then internal
inflation does not occur, and the bubble interior shrinks and collapses to a singularity.3 1 2
3
Higher-energy bubbles can also be formed, but their nucleation rate is typically strongly suppressed [3]. This low energy internal inflation takes place inside the bubble, even though inflation in the exterior region has already ended. For simplicity, in this discussion we disregard the gravitational effect of the bubble wall. We shall see
2
Bubbles formed at earlier times during inflation expand to a larger size, so at the end of inflation we expect to have a wide spectrum of bubble sizes. They will form black holes with a wide spectrum of masses. We will show that black holes with masses above a certain critical mass have inflating universes inside. The situation with domain walls is similar to that with vacuum bubbles. It has been shown in Refs. [4] that spherical domain walls can spontaneously nucleate in the inflating false vacuum. The walls are then stretched by the expansion of the universe and form black holes when they come within the horizon. Furthermore, the gravitational field of domain walls is known to be repulsive [5, 6]. This causes the walls to inflate at the rate Hσ = 2πGσ, > H −1 develop where σ is the wall tension. We will show that domain walls having size R ∼ σ
a wormhole structure, while smaller walls collapse to a singularity soon after they enter the cosmological horizon. Once again, there is a critical mass above which black holes contain inflating domain walls connected to the exterior space by a wormhole. We now briefly comment on the earlier work on this subject. Inflating universes contained inside of black holes have been discussed by a number of authors [7–9]. Refs. [8, 9] focused on black holes in asymptotically flat or de Sitter spacetime, while Ref. [7] considered a different mechanism of cosmological wormhole formation. The possibility of wormhole formation in cosmological spacetimes has also been discussed in Ref. [14], but without suggesting a cosmological scenario where it can be realized. Cosmological black hole formation by vacuum bubbles was qualitatively discussed in Ref. [15], but no attempt was made to determine the resulting black hole masses. Black holes formed by collapsing domain walls have been discussed in Refs. [16–18], but the possibility of wormhole formation has been overlooked in these papers, so their estimate of black hole masses applies only to the subcritical case (when no wormhole is formed). In the present paper, we shall investigate cosmological black hole formation by domain walls and vacuum bubbles that nucleated during the inflationary epoch. We shall study the spacetime structure of such black holes and estimate their masses. We shall also find the black hole mass distribution in the present universe and derive observational constraints on the particle physics model parameters. We shall see that for some parameter values the black holes produced in this way can serve as dark matter or as seeds for supermassive black later that if the wall tension is sufficiently large, a wormhole can develop even when R Hb−1 .
3
holes observed in galactic centers. The paper is organized as follows. Sections II and III are devoted, respectively, to the gravitational collapse of vacuum bubbles and domain walls. Section IV deals with the mass distribution of black holes produced by the collapse of domain walls, and Section V with the observational bounds on such distribution. Our conclusions are summarized in Section VI. A numerical study of test domain walls is deferred to Appendix A, and some technical aspects of the spacetime structure describing the gravitational collapse of large domain walls are discussed in Appendix B.
II.
GRAVITATIONAL COLLAPSE OF BUBBLES
In this Section we describe the gravitational collapse of bubbles after inflation, when they are embedded in the matter distribution of a FRW universe. Three different time scales will be relevant for the dynamics. These are the cosmological scale t ∼ H −1 = (3/8πGρm )1/2 ,
(1)
where ρm is the matter density, the scale associated with the vacuum energy inside the bubble, tb ≡ Hb−1 = (3/8πGρb )1/2 ,
(2)
and the acceleration time-scale due to the repulsive gravitational field of the domain wall tσ ≡ Hσ−1 = (2πGσ)−1 .
(3)
In what follows we assume that the the inflaton transfers its energy to matter almost instantaneously, and for definiteness we shall also assume a separation of scales, ti tb , tσ ,
(4)
where ti ∼ Hi−1 is the Hubble radius at the time when inflation ends. The relation (4) guarantees that the repulsive gravitational force due to the vacuum energy and wall tension are subdominant effects at the end of inflation, and can only become important much later. The interaction of bubbles with matter is highly model dependent. For the purposes of illustration, here we shall assume that matter is created only outside the bubble, and has reflecting boundary conditions at the bubble wall. For the case of domain walls, which will 4
be discussed in the following Section, we shall consider the case where matter is on both sides of the wall. As we shall see, this leads to a rather different dynamics. The dynamics of spherically symmetric vacuum bubbles has been studied in the thin wall limit by Berezin, Kuzmin and Tkachev [8].
4
The metric inside the bubble is the de Sitter
(dS) metric with radius Hb−1 . The metric outside the bubble is the Schwarzschild-de Sitter metric5 (SdS), with a mass parameter which can be expressed as 4 M = π(ρb − ρi )R3 + 4πσR2 [R˙ 2 + 1 − Hb2 R2 ]1/2 − 8π 2 Gσ 2 R3 . 3
(5)
Here, ρi is the vacuum energy outside the bubble, R is the radius of the bubble wall, and R˙ ≡ dR/dτ , where τ is the proper time on the bubble wall worldsheet, at fixed angular position. Eq. (5) can be interpreted as an energy conservation equation.
A.
6
Initial conditions
A bubble nucleating in de Sitter space has zero mass, MdS = 0, so the right hand side of (5) vanishes during inflation. Assuming that R, R˙ and the bubble wall tension are continuous in the transition from inflation to the matter dominated regime, we have 4 4 πρm (ti )Ri3 ≈ πρb Ri3 + 4πσRi2 [R˙ i2 + 1 − Hb2 Ri2 ]1/2 − 8π 2 Gσ 2 Ri3 . 3 3
(6)
Here, we have assumed that the transfer of energy from the inflaton to matter takes place almost instantaneously at the time ti , so that ρm (ti ) ≈ ρi , and that the vacuum energy ρ0 outside the bubble is completely negligible after inflation, ρ0 ρm .
(7)
4 Mi ≡ πρm (ti )Ri3 3
(8)
Note that the left hand side of Eq. (6),
4
5 6
A more detailed and pedagogical treatment in the asymptotically flat case was later given by Blau, Guendelman and Guth [9]. Here we are ignoring slow roll corrections to the vacuum energy during inflation. In principle, both signs for the square root are allowed for the second term at the right hand side. However, for ρb < ρi (which holds during inflation, when the energy density in the parent vacuum is higher than in the new vacuum) only the plus sign leads to a positive mass M , so the negative sign must be discarded.
5
is the “excluded” mass of matter in a cavity of radius Ri at time ti , which has been replaced by a bubble of a much lower vacuum energy density ρb . The difference in energy density goes into the kinetic energy of the bubble wall and its self-gravity corrections, corresponding to the second and third terms in Eq. (6). Let us now calculate the Lorentz factor of the bubble walls with respect to the Hubble flow at the time ti . Assuming the separation of scales (4), the right hand side of Eq. (6) is dominated by the kinetic term and we have 1 R˙ i ≈ Hi2 Ri tσ . 4
(9)
We wish to consider the motion of the bubble relative to matter. The metric outside the bubble is given by ds2 = −dt2 + a2 (t)(dr2 + r2 dΩ2 ).
(10)
Here, a(t) ∝ tβ is the scale factor, where β = 1/2 for radiation, or β = 2/3 for pressureless matter, as may be the case if the energy is in the form of a scalar field oscillating around the minimum of a quadratic potential. The proper time on the worldsheet is given by dτ =
√ 1 − a2 r02 dt,
(11)
where a prime indicates derivative with respect to t, and we have HR + ar0 dR =√ , dτ 1 − a2 r02
(12)
with H = a0 /a = β/t. The typical size of bubbles produced during inflation is at least of order ti , although it can also be much larger. Hence, at the end of inflation we have > 1. On the other hand, the second term in the numerator of (12) is bounded by Hi Ri ∼ unity, ar0 < 1. Combining (9) and (12) we have 1 tσ γi ≡ p ∼ , ti 1 − a2i r0 2i
(13)
and using (4), we find that the motion of the wall relative to matter is highly relativistic, γi 1. B.
Dissipation of kinetic energy
The kinetic energy of the bubble walls will be dissipated by momentum transfer to the surrounding matter. Let us now estimate the time-scale for this to happen. The force acting 6
on the wall due to momentum transfer per unit area in the radial direction can be estimated as ˆ
ˆ
T 0ˆr ∼ ρm U 0 U rˆ ∼ −ρm γ 2 ,
(14)
where U µˆ is the four-velocity of matter in an orthonormal frame in which the wall is at rest, hatted indices indicate tensor components in that frame, and we have used that the motion of matter is highly relativistic in the radial direction7 . The corresponding proper acceleration of the wall is given by α∼−
ρm γ 2 . σ
(15)
In the rest frame of ambient matter, we have dγ = γ 3 vv 0 = vα ≈ α, dt
(16)
where v = ar0 is the radial outward velocity of the wall. The second equality just uses the kinematical relation between acceleration and proper acceleration for the case when velocity and acceleration are parallel to each other. Combining (15) and (16) we have dγ dt ∼ −tσ 2 . 2 γ t
(17)
Solving (17) with the initial condition (13) at t = ti , it is straightforward to show that the wall loses most of its energy on a time scale ∆t ∼
t3i ti , t2σ
(18)
and that the wall slows down to γ ∼ 1 in a time of order ∆t0 ∼ t2i /tσ ti , which is still much less than the Hubble time at the end of inflation. Since the bubble motion will push matter into the unperturbed FRW region, this results in a highly overdense shell surounding the bubble, exploding at a highly relativistic speed. The total energy of the shell is Eshell ≈ Mi , with most of this energy contained in a thin
layer of mass Mshell ∼ (ti /G)(Ri /tσ )2 expanding with a Lorentz factor γshell ∼ γi2 ∼ (tσ /ti )2 7
Here we use a simplified description where the momentum transfer is modeled by superimposing an incoming fluid with four velocity U µˆ that hits the domain wall, and a reflected fluid which moves in the opposite direction. This description should be valid in the limit of a very weak fluid self-interaction. In a more realistic setup, shock waves will form [10, 11], but we expect a similar behaviour for the momentum transfer.
7
relative to the Hubble flow. Possible astrophysical effects of this propagating shell are left as a subject for further research. After the momentum has been transferred to matter, the bubble starts receding relative to the Hubble flow. As we shall see, for large bubbles with Ri ti the relative shrinking speed becomes relativistic in a time-scale which is much smaller than the Hubble scale. To illustrate this point, let us first consider the case of pressureless matter.
C.
Bubbles surrounded by dust
Once the bubble wall has stopped with respect to the Hubble flow, it is surrounded by an infinitesimal layer devoid of matter, with a negligible vacuum energy ρ0 which we shall take to be zero. Since the matter outside of this layer cannot affect the motion of the wall, from that time on we can use Eq. (5) with ρi replaced by the vanishing vacuum energy after inflation, ρ0 = 0, 4 Mbh = πρb R3 + 4πσR2 [R˙ 2 + 1 − Hb2 R2 ]1/2 − 8π 2 Gσ 2 R3 . 3
(19)
The mass parameter Mbh for the Schwarzschild metric within the empty layer can be estimated from the initial condition R˙ i ≈ Hi Ri , at the time t = ti + ∆t ≈ ti when the bubble is at rest with respect to the Hubble flow. Using the separation of scales (4), we have 4 πρb + 4πσHi Ri3 . Mbh ≈ (20) 3 The motion of the bubble is such that it eventually forms a black hole of mass Mbh , nested in an empty cavity of co-moving radius rc ≈
Ri . a(ti )
(21)
A remarkable feature of (20) is that Mbh is proportional to the initial volume occupied by the bubble, with a universal proportionality factor which depends only on microphysical parameters. Under the condition (4), this factor is much smaller than the initial matter density ρm (ti ), and we have Mbh ∼
t2i ti + 2 tb tσ
Mi Mi ,
(22)
where Mi is the excluded mass given in (8). On dimensional grounds, the first term in parenthesis in (20) is of order ηb4 , where ηb is the energy scale in the vacuum inside the 8
bubble, whereas the second term is of order ηw3 ηi2 /Mp , where ηw is the energy scale of the bubble wall and ηi is the inflationary energy scale. The second term is Planck suppressed, and so in the absence of a large hierarchy between ηw , ηi and ηb , the first term will be dominant. In this case we have Mbh ≈ (ti /tb )2 Mi . However, we may also consider the limit when the vacuum energy inside the bubble is completely negligible, and then we have Mbh ∼ (ti /tσ )Mi . It was shown by Blau, Guendelman and Guth [9] that the equation of motion (19) can be written in the form
dz d˜ τ
2
+ V (z) = E,
(23)
where the rescaled variables z and τ˜ are defined by H+2 R3 , z = 2GMbh H+2 τ˜ = τ, 4Hσ 3
(24) (25)
with τ the proper time on the bubble wall trajectory, and H+ defined by H+2 = Hb2 + 4Hσ2 .
(26)
The potential and the conserved energy in Eq. (23) are given by 2 g2 1 − z3 − , V (z) = − z2 z 2 −16Hσ E = , 8/3 (2GMbh )2/3 H+
(27) (28)
where g=
4Hσ . H+
(29)
The shape of the potential is plotted in Fig. 1. The maximum of the potential is at 3
z =
3 zm
i 1 hp 2 2 2 ≡ 8 + (1 − g /2) − (1 − g /2) . 2
(30)
Note that 0 ≤ g ≤ 2, and 1 ≤ zm ∼ 1 in the whole parameter range.
The qualitative behaviour of the bubble motion depends on whether the Schwarzshild mass Mbh is larger or smaller than a critical mass defined by 3 6 2 1/2 ¯ g z√m (1 − g /4) , Mcr = M 6 − 1)3/2 3 3(zm
9
(31)
FIG. 1: curve).
where
R = 2GMS R = 0 R = 2GMbh , T ! ⇢0 t = 0 t = ti H 1 = 2GMbh V (z) z 1 4 Consider a spherical domain w 1 Spherical domain wall2 in dust3 cosmology -5 di↵erent time-scales are relevant to -10 and the other is the acceleration ti R = 2GMS R = 0 R = 2GMbh , T ! 1 R = 2GM bh , T ! 1 ⇢ = 2GM ⇢ = const. r = 0 r 1 ⇢0 t = 0 t -15 = ti H = 2GMbh t ⇠ (G ) 1 . V (z) z For R ⌧ t , the repulsive field -20 Consider a spherical domain wall embedded in a spatially flat matter dominated univer conformally by cosmologic di↵erent time-scales are relevant to the dynamics of such a wall.stretched One is the cosmological scale -25 and the other is the acceleration time-scale duequickly to the repulsive of the dom shrinksgravitational under itsfield tension an t ⇠ (G )-301 . Here, we will be primarily intere For R ⌧ t , the repulsive field can be ignored. In this case, for t ⌧ R ⌧ t , the domai theEventually, matter when around it while its size conformally stretched by cosmological expansion. the wall falls within the ho 2 quickly shrinks underforitsg tension andcurve), forms ga = black holetoofcurve) radius RSg = ⇠ ta/twormhol ⌧ t. leads the formation of The potential in Eq. (27), = 0 (upper 1 (middle and = 2GM 2 (lower Here, we will be primarily interested in the opposite limit, where R t . In this case, the w a baby universe, and in the ambien the matter around it while its size grows faster than the ambient expansion rate. As we shall region in ofthe of interior vacuum. leads to the formation of a wormhole. The dustspherical which was originally of the wall a baby universe, and in the ambient FRW universeBefore we are left with a black hole remnant cy we consider the e↵ect of spherical region of of vacuum. and a dust cosmology. matching of Schw ¯ ≡ tb . M Before we consider the e↵ect of the domain wall, let us first discuss the(32)
2G and a dust cosmology. Although the expression for the critical mass is somewhat cumbersome, we can estimate it
1.1
as
1.1
Matching Schwarzschild
Matching Schwarzschild to a dust cosmology
GMcrmetric ∼ Min{tσ , tb }. Consider the Schwarzschild ✓
Consider the Schwarzschild metric (33)
◆
✓
◆
1 2 2GM 2 3 .2 In the absence 2GM of a 2hierarchy On dimensional grounds tb ∼ Mp /η σ + 1 dsb2, =while1tσ ∼ Mp /ηdT dR + R2 d⌦2 . 2 ds = R R between ηb and ησ we may expect GMcr ∼ tb , but the situation where GMcr ∼ tσ is of In Lemaitre coordinates, this takes the form course also possible. In2GM Lemaitre coordinates, this 2 2 ds = d⌧ + d⇢2 + R2 d⌦2 . R
1.
1
takes
Small bubbles surrounded by dust
d
where ⌧ and ⇢ are defined by the relations s
✓
◆
1 point. In For Mbh < Mcr , we have V (zm ) > E, and the bubble trajectory a turning 2GM has 2GM d⌧ = ±dT 1⌧ and ⇢ are dR,defined where R R is confined to this case it is easy to show that with the condition (4), the bubble trajectory s ✓ ◆ 1 R of z which 2GM corresponds the left of the potential barrier, zi < 1 ≤ zm , where z is the value to i + d⇢ = ⌥dT 1 dR. 2GM R the initial radius Ri . The bubbles grow from small size up to a maximum radius Rmax and Subtracting (3) from (4), we have then recollapse. The turning point is determined from (19) with R˙ = 0. For Mbh Mcr we
have
✓
by the re
d⌧
=
2/3 3 d⇢ = R = (⌧ + ⇢) (2GM )1/3 , 4 2 3 2 Mbh ≈ πρb Rmax + 4πσRmax . (34) 3 where an integration constant has been absorbed by a shift in the origin of the ⇢ coordina Subtracting (3) from (4), we have expression for T as a function of ⌧ and ⇢ can be found from (3) as:
0
1 ±T = 4GM10@ 3
✓
R 2GM
◆3/2
+
s
R 2GM
tanh
1
s
1
R A 2GM
⇢,
M
bh
in
bh
,T
!
w al
li
n
1
du st
Schwarzschild"singularity"
co vi th r do hi iv hi n t lim f r y, w se, m t a e do ns ds 2 ad h f o er ch h ld m e t c i e t. m t s ai i o os t, iu en r an ric Schwarzschild e w t te 1.1 =Matching a i to a dust cosmology n n r s n w o t m d w m a ✓ a s t a = w R l fi ol te e s o bi he ,t he ⌧ ⇢ a a 0 S o a r e ar h d l r e d e l g r R n w Consider the Schwarzschild metric = 1 i i , l e gi t s e us d i ca u n R al l 2 ⌧ e n t 2 l d e e✓ a x ◆ 1 G l f of l ak t◆ t ✓ G i efi Su al 2 t 2 M ◆ 2GM us 2 ft w ll2GM es co th sca vers y pan t dRM ne 2 2 bt l R , ds = 1 dT + 1 + R d⌦ . s l i e fi s t i ⇠ . d n si o I ra th R he m rs R d o e t e. by t h n n t 2 wi t t h e ct t d o a f m ⇠ / T2 in or di lo ds 2 th b l e i n r at t h i s t h i n d o ai H g m sc ✓ g In Lemaitre coordinates, this takes the form m e + a n ⌧ e t w (3 th a y us ck er . ca = re h wa ) i e s i t A s ex er n l 1 o . a h e f h d⌧ ll ro 2GM th 2 tio pr e wa d⌧ 2 2 2 2 ol 2 r o s w , t o 2 e r m ds = d⌧ + d⇢ + R d⌦ . e a G es n f t e he iz ll ns m r = R M ◆ o + 2 si o i (4 h e n, i s at m e sha wa R ), d⇢ n nte ch na w l l l l ± where ⌧ and ⇢ areGMdefined by the relations it 1 we fo gr a d r i n R ng T r s t l l g see ep at = h w d T d s cy o , el s av ⇢2 o R2 ✓ ◆ as ion ⌥ Si i t h st es t h i e 2G 2GM 2GM f S1c + dT + ed i n s nc R a co 1 s hdR, R 2 d⌧ = ±dTR 2 M e n f wa in to + u n st R R ✓ R th giv d d ⌦2 r s c t an a ⌦ z e en 2 ✓ ◆ sc io t . m ± . R 2GM 1 hi 2G R ✓ 1 2 n R h et i n T d⇢ = ⌥dT + 1 dR. ld of as G = ric (5 M = 2GM R M ◆ 3 ⌧ be 4G 1 (2 ). R a 2 (⌧ ) A M 0 n d en 2G (3) from (4), 1 i s se Subtracting we have + @ 1 ✓ ⇢ ab M ◆ sy co ⇢ d s c 2 R ) R o a / nc nd 3 3 n rb 2/3 , 1 hr 3 2G R ◆ b e e d R = (⌧ + ⇢) (2GM ()1/3 , on int ( 2 d fo by 2 G 1) ou eg R M 3 un /2 M . s, rat a d 1 ) s i th on /3 an integration constant has been absorbed by a shift in the origin of the ⇢ coordinate. where f r o sh + e , m if c
n
w V (z) z qui for r R ) 1 er cal her = 2 2G a ck m is es ic G M . ⌧ a a Consider domain wall embedded inb a spatiallyll flat matter dominated universe. ly t M th Ha spherical a l l h h, in is the cosmological scale t ⇠ e rdynamics t d e e s ly srelevant T a wall. One le time-scales di↵erent ac e re om bhof such ad ma re, hrare tr , t to the i du ! n a l c a w acceleration el evdue s t ttis the k e h time-scale and the gravitational field of the domain 1 er an into the repulsive sp bab other er e w s u tch e r st o 1 at t wa t ⇠he(G y) .th a il nd ed ep R io to ll r i c u e r ou l b e r b u l s = for t ⌧coR ⌧ t , the domain w n an BFor y can R ⌧ field be ignored. Ine this case, ni t f,o thenrepulsive t e i a 2G s h i v t m v e l c d im Eventually, e e os expansion. er rm dby pcosmological ri ts M the wall m conformally falls a fore reg stretched e- dy bed when ol 2within the horizo 1. du w io se, atio it w ma ten mo field bh scholenaof radius d quickly under its tension and forms a black R = 2GM ⇠ to/t ⌧ t. s , r l 1 n S a e h st shrinks i a o n ily on g c T e g l e mi d of bendprimarily C of ile interested ic theanopposite co will co we ! t . In this ycase, the wall re in limit, where R c i on i M Here, d n a n a s o i i n s a t n u n l t f b m si d o si a d faster at mattero around e than e v it whilewits ssize egrows the thee ambient 1 rate. As we shall see, to f su expansion sp er lo der acu the or size rest for xpa ig ch m g n a c leads to the formation of a wormhole. The dust which was originally in the interior of the wall goes e ⇢ th h ti th y. th um am h g d ms ns or in = In alblack hole e with e a baby io we ol ro FRW a e e i g b n . a universe, and in the ambient universe are left a remnant cysted d l Sc re w y e↵ i e e . ws t h b n . . Le 2G pu al fl nt T f hw Scregion l ec I E a spherical of of vacuum. e m l h n l s i . O at t M ar FR he ast op ck ve ai w ve the er pwall, tr zs Before we discuss of m ar consider theof e↵ect of ho letnt usthfirst ne matching Wthedudomain i ⇢ Schwarzs o e s u c a t zs gr i tt s t t h s i t l e al c hi cosmology. he = co and a dust u s a a ld a l e o c e n w
ai
2GM ⇢ = const. r = 0 r = r0
G
do m
w
he
re
⌧
or
di
na
= 1
sp
s
e-
th
on Fo
S
Spherical domainR =wall inphdust er cosmology ⇢0 2 ic t G al di C = 0 MS ↵ o 1
1
do, T ! 1 ⇢ = R = 2GMS R = 0 R a= 1 R = 2GMbh er bhn,s T t! n 2GM en i d = t R = m ⇢0 t = 0 t = ti H t1 = d2GM bh t e t ⇠ he t i r i H 0 R a r = 0 a in m ( o c G
R w co = to all sm 2G t h em M ol m e d be e- y n d d bh og ,T sc a e al m d y ! e i i c n du s be 1Bubble" e of a s dust"FRW" s t p o uc a ig 1 Spherical domain wall in dust cosmology ⇢ th h tia = ns nor a e lly i o ed w r 2G= 2GMS R = 0 R = 2GMbh, T ! 1 R = 2GMbh, T ! 1 ⇢ = 2GM ⇢ = const. r = 0 r = r0 n e R pu all fla bl . E . I 1 = 2GM M ⇢ . 0 t = 0 t = ti H bh ac l si t ve n t O Consider a spherical domain wall embedded in a spatially flat matter dominated universe. k m v ⇢ n pp h nt hi e time-scales are relevant to the dynamics of such a wall. One is the cosmological scale t ⇠ = gr e is attedi↵erent os ole ua s c and the other is the time-scale due to the repulsive gravitational field of the domain c a lly as r ⇠ formation onof aacceleration th tthe v 1. an ite of FIG. (G ) i 2: Causal diagram showing black hole by a small vacuum bubble, with mass e d , ta e For field can be ignored. In this case, for t ⌧ R ⌧ t , the domain w tio co omR ⌧ t , thestrepulsive . th limi rad wh , fo r cosmological in stretched by expansion. falls within the horizo spatially flat FRW. At the time ti whenEventually, inflationwhen ends,thea wall small sm h e t Mbhi Me cr , inr a dust dominated na conformally = a wa am , w us R n t t ⌧ quickly shrinks under its tension and forms a black hole of radius R = 2GM ⇠ t2 /t ⌧ t. S o t l 0 e l h h fie Here, ogwepositive with energyinterested density ρbthe > opposite 0, initially willdbe primarily in limit,expands where R with t . Inathis case, the wall r s b bubble r e (to thee left of the diagram) theld mattericarounduit while its size=grows faster than the ambient expansion rate. As we shall see ar ori ien re R S = wa R ni ofmotion al The Lorentz2factorllrelative Hubble is slowed down bywas momentum e gin tlarge ⌧ to theleads r0The dust ofthe flow. to formation which originally intransfer the interior of the wall goes vea wormhole. e s G l ef a x c f a baby universe, and in the ambient FRW universe we are left with a black hole remnant cysted ⇢ t r t a M a p he negative spressure t llyto matter, l due to the internal around and = the tension of the bubble wall, e. t region of oflevacuum. wi in ans turning ⇠ ls w , spherical t . d t Before we consider the e↵ect of the domain wall, let us first discuss the matching of Schwarzs h osingularity. fir io collapsing th and teventually Twfixed angular ⇠ At In t 2 intoit a and e Schwarzschild coordinates, the timem a dust cosmology. n st h he d H a th /t a o i o r in n dustmregion represents di is edge⌧of the bl dashed in lineateat the like a1 radial geodesic of the Schwarzschild t sc a w 1.1 Matching Schwarzschild , to a dust cosmology a h t c . i e c as t. e a us n r k l A i l infinity with unit slope. The reason e, geodesic or sB). Such Considerhthe Schwarzschild metric s metric (see wa time-like ho Appendix oapproaches th r t ✓ ◆ ✓ ◆ 1 w o i ll 2 of he le slope zthe 2GM 2GM f t would eis that a smaller 2 a finite 2 e o correspond to trajectory a particle at radial coordinate ds = 1 dT + 1 dR + R2 d⌦2 . i r e h sh w n, s m R R m e w a al at the geodesic t this takes R, while escaping to iinfinity has unbounded R. The thin relativistic l r coordinates, ll In Lemaitre na of aa particle the form ch s e nt ll in e p 2GM 2figure. els bubble is not represented go e, by the expanding gshell of matter produced shell ds2 = d⌧ 2 + in the d⇢ + R2 d⌦2This . c y t R of e h s snear is boundary disturbs the homogeneity oftedust by the dashed line. Sc in ⌧ the where and ⇢ are definedrepresented by the relations d t hw s o i ✓ ◆ n 2GM 2GM 1 ar a d⌧ = ±dT 1 dR, For σHi /ρb 1, the first zs term in (20) dominates, and the turning point R occursR after a small ⌦2 s ch ✓ ◆ R 2GM 1 ild radius fractional increase in the bubble . d⇢ = ⌥dT + 1 dR. 2GM R ∆R Ri
σ ρb
Subtracting (3)≈ from H (4), we have 1, i
(35)
3
2/3
1/3
R = (⌧ + ⇢) , which happens after a time-scale much shorter than the expansion time, (2GM ∆t ∼) (σ/ρ 2 b ) ti .
where an integration constant has been absorbed by a shift in the origin of the For 1 σHi /ρb (Ri Hi )3/2 we have expression for T as a function of ⌧ and ⇢ can be found from (3) as: 1/30 (1 1 s ✓ ◆3/2 s Rmax 3σHi ) R R(36) 1 ≈ ±T = 4GM @. 1 R A ⇢, + tanh Ri ρb 3 2GM 2GM 2GM
(2
)
⇢ coordinate.
with R given in (5). A second integration constant has been absorbed by a shift in the T coordi 11 Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and ⌧ i
Finally, for σHi /ρb (Ri Hi )3/2 we have Rmax ≈ (Hi Ri )1/2 . Ri
(37)
< R Rmax the In the last two cases, Rmax Ri . From Eq. (19), we find that for Ri ∼ bubble radius behaves as R ∝ τ 1/3 . This is in contrast with the behaviour of the scale factor
in the matter dominated universe, a(t) ∝ t2/3 , and therefore the bubble wall decouples from
the Hubble flow in a time-scale which is at most of order ti . Once the bubble radius reaches its turning point at R = Rmax , the bubble collapses into a Schwarzschild singularity. This is represented in Fig. 2. Denoting by tH the time it takes for the empty cavity of co-moving radius rc = Ri /a(ti ) to cross the horizon, we have tH ∼ (Hi Ri )2 Ri .
(38)
> Rmax , so the black hole forms on a time-scale much shorter Note that tH (Hi Ri )1/2 Ri ∼ than tH .
2.
Large bubbles surrounded by dust
For Mbh > Mcr we have V (zm ) > E, so the trajectory is unbounded and the bubble wall grows monotonically towards infinite size in the asymptotic future. The unbounded growth of the bubble is exponential, and starts at the time t ∼ Min{tb , tσ }. This leads to the formation of a wormhole which eventually “pinches off”, leading to a baby universe (see Figs. 3 and 4). The bubble inflates at the rate Hb , and transitions to the higher energy inflatinary vacuum with expansion rate Hi will eventually occur, leading to a multiverse structure [3, 12]. Initially, a geodesic observer at the edge of the matter dominated region can send signals that will end up in the baby universe. However, after a time t ∼ GMbh , the wormhole closes and any signals which are sent radially inwards end up at the Schwarzschild singularity.
D.
Bubbles surrounded by radiation
Due to the effects of pressure, the dynamics of bubbles surrounded by radiation is somewhat more involved than in the case of pressureless dust. Here, we will only present a 12
ph e
= ic t = 2GM al di C 0 S do an ↵er ons t = R m t d t ent ide ti = ai ⇠ he t i m r a H 0 R n ( co F G oth e-s sp 1 = w qu nfo or R ) 1 er cal her = 2 2G al is es ica G M ick rm li . M ⌧ bh t a a l l h n th H y s lly t e r e d o bh , T e e du r h a m ! l ea m r e r i st , t e c l ce ev ai 1 st a ds att , we nks retc he n l a e ra nt w R sp bab to er w un hed rep co t i o t o al = he y th ar ill de b u l n sm l 2G e o b r tim the emb an B ica univ for und e p r it y co sive M ol st s d d ef o l r er m r e fi e i y d b m -s c n d e e og h, a a it m e s am ed T du re w gio e, tio wh ari nsi olo ld 1. a y le ! 1 st e n o and n o ile ly on gic can i i c n du s co co f o i f a it int an al Co o a b e 1 M n n f s s e d ex e ns m si d f v t o su sp fo p i g at ol id a t wo si re ⇢ th ch atia og er cuu he a rm ze sted rm ans no ch er = r g e a t y. l in th he m mb hol ro in s a ion ed l y re wa 2G . w . e e g In pu ll fla i en . e↵ s th b . Sc M l si . O t ec Le t Th fas e o lack Eve In t hw Sch ve n m FR e t m te pp h nt hi ⇢ w ar o a e d ai r s o = u W f g t o ar us t zs r a i s t er sit le all ca tr th h co ch t z e t v o u a y se el f e h sc w wh ns i n , ild n co d t do i v hi at e c om hi th im rad wh , fo t. or er e c m i m o i on sm in rs h ld e r r e e ds 2 di t i u e n , a e = w a t tr ⌧ na a s a i w n to l fi ol te we as mb h R th ⌧ ic an = te o w d ✓ e e S e g i o d re s, a al e R a l w i r = u d n c re ig ⇢ l, th du R t 2G all ⌧ of al s niv 1 ar l et is l ef i n a ex st e 2G t h ca er s ta t l l y p a t M f al l t , u de e le wi i n s . co ke s M ◆ ⇠ s w th fin do t ⇠ n fi Su s t i s In t 2 i t e R o h r t m e t m s n bt he d t a he th /t hin do ai o r ra d d by a b l i i n m fo T i s n o t 2 l s ⌧ ct t e ac te . ds 2 c rm a g h c w th ✓ u in i a + y k r i o A s e t. e h n ss e g = h w s , r r o (3 t o e 1 wh a t r h w o la le h i zo l l )f e 2G d⌧ d⌧ 2 tio ex er e r e f t h e sh e w m ro n, i s ns M ◆ m e pr at al m a + it a = w n l 2 l ch R es n l G a ( r a 4) si o i n nt ll se ep in 1 M d⇢ ± e ,w g g e n te , R cy o e t l s dT of s dR fo gr e d⇢ 2 st s h i = 2 r T at ha Sc ed i n s io + v + wi h ⌥ 2 e w in to G n a dT R2 R2 sa c ar Si t h s M a ✓ z d o d nc R + R sc ⌦2 ⌦2 fu ns e hi nc t a . . th giv l 1 R n d tio t e en 2G 2G R ± ✓ n m h T M = M ◆ et i n of as = ( ric 5 3 R 1 ⌧ b ) 4 e 1 . a 2 ( e 0 G (2 2 ⌧ nd n G M ) A + M ◆ d i s se ⇢ abs @ 1 ✓ R ⇢ R ca o ) 2/3 sy co , 1 n rb nc nd 3 (1 ed b R hr ) ( d e 2G 2G R ◆ on inte fo by . 3 M M un ou gr /2 )1 a s d a
R
ri ca t = 2GM ld di C 0 S om an ↵er ons t = R t d t ent ide ti = 0 ai ⇠ he t i m r a H R n ( 1 o = co Fo G w t h e -s s p h = 2 n c e al G ) q u fo r R 2G M 1 r i ale eri li c ick rm s . s a al M b ⌧ t n h e r e d bh h , T th H ly s ally t o du ac re m ! Schwarzschild"singularity" l ea e m er e, h r i st r , t l a c e st 1 ele va in a ds att we nks etc he n ra R co sp bab to er wi un hed rep tio t to wall = he y th ar ll de b ul sm n 2G e t ric un e f ou be r i y siv h m tim e M be an B al iv or nd p ts co e ol d e- y n d d bh d efo re er ma it rim te sm fie og ,T sc se ld a n r a o e g t a e a w d m s l y i i , du w on a on h ril io og c l ! e i .1 st e nd o ile y i n a ica an d u cs i n o f c f n o a co o be 1 i n l e M sm ns of v in t a w ts s tere d f ex Infla=ng"" t o f su sp ig ⇢ ch ati at ol ide ac he orm ize ste orm pbubble" a n t = de h ns or og r uu a ch a all e d g rt s h i e t dust"FRW" y m o r y w r i m he d. " in o 2G a . o n n e he l b a w pu Spherical g ll. fla i en e. e↵ . s t h e b l a . E In 1 domain wall in dust cosmology M Sc l t T S si v O f c v t ec ch hw ⇢ FR he aste opp k h ent thi t e n e ma = w ar d s o t o g r u W us t os le a ft ra is te ar c zs c he ch un t w han ite of lly, Ras=e, 2GM vitR =the0 Rr d= 2GMon,sT ! 1 R = 2GM , T ! 1 ⇢ = 2GM ⇢ = const. r = 0 r = r ⇢ = zs 0 S a l bh t bh ild om r w ch i i v hi do c m t a t f i h o o 1 h d e c or in bh . r = m ild m rs h e it, iu ⇢0ent = r0 tt = ti oH 2GM na s= m ds 2 di a a e e a w s te in tr l fi olo domain th ⌧ a spherical na 0 we as mb whe R Consider to in a spatially flat matter dominated universe. Two i = wa eld gic d u wall rembedded te ✓ c or ien re S = e w R = a a s, di↵erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale t ⇠ H 1 , n l a r i a l, e gin t e R ⌧ r o i l d l 2 th v l f t sc time-scale 0 G l et us 1 lef a x and the e fa is tthe acceleration ⇢due to the repulsive gravitational field of the domain wall is he ale rse t lly pa t Mother 2G lls , t ta us . M = M , in a dust dominated wi inbytnsa⇠large ⇠1vacuum co t (G ) . . d t ke M ◆3: Formation w FIG. of a black hole bubble, with cr om ⇠ T bh fir i o In t 2 i t h e th sm st R n Rt ⌧ /t , the st do H w field beo ignored. In this case, for t ⌧ R ⌧ t , the domain wall is ai can a the For he ra hi t hin repulsive d o n d m 1 b d i l s by nt te ⌧ thby cosmological la conformally T 2 flat FRW. fo spatially og In thisiscase, expansion. when thesingularity. wall falls within the horizon, it wa into , cadoes cu the the Eventually, Schwarzschild in er . A stretched ck bubble rm e acollapse ds 2 t. not th y seunder ll + ✓ ss io shrinks quickly itshtension and forms a black hole of radius RS = 2GM ⇠ t2 /t ⌧ t. w e h , s o r = th ol riz all th re we be 1 e ofwe Here, in the opposite limit, where R this case, the wall repels e primarily la oninterested istarts 2G time tcr ∼ Min{tbe ,mtσ } the th will re size s d⌧ 2Instead, at the wa bubble tio ofshathe growing exponentially tin. In a baby ⌧ e m a M the matter around it while its, size grows faster than the ambient expansion rate. As we shall see, this i l ns t t w ◆ l l n + c r R 2G an formation = a l hi to the leads ng 1 t ll g see,ofeapewormhole. The dust which was originally in the interior of the wall goes into M ± ls ambient c universe, which is connected by a wormhole to the parent dust dominated universe. Initially, o y t R o e a baby universe, and in FRW universe FRW we are left with a black hole remnant cysted in a dT dR st s hthe fS s d⇢ 2 2 ed of ivacuum. nt is c spherical region of h + o + incan 2G ⌥ a geodesic edge of the Before dustwwe region signals through wormhole the of Schwarzschild ar consider R 2 observer atR the 2 e↵ect of the domain wall, letthe us first discuss theinto matching dT a thesend M ✓ s zs d d ⌦ c + R ⌦2 2 and a dust cosmology. h ild . . 1 baby universe. This is represented by the blue arrow in the Figure. However, after a proper time 2G R ✓ 2G 1.1 Matching Schwarzschild to a dust cosmology M M ◆ R 1 1 bh , the wormhole “closes” and any signals which are sent radially inwards end up at the 2Gt ∼ 2GM Consider the Schwarzschild metric M ◆ + dR ✓ ◆ ✓ ◆ R ⇢) 2/ , 2GM 2GM 1 2 1 3 Schwarzschild singularity. (1 ds2 = 1 dT 2 + 1 dR + R2 d⌦2 . (1) ) R R dR (2 G . y M In Lemaitre coordinates, this takes the form a ) 1/ 3 s h (2 ro if , 2GM 2 ) m t ds2 = d⌧ 2 + d⇢ + R2 d⌦2 . (2) (3 in R ) a th s: e where ⌧ and ⇢ are defined by the relations or (3 ig an s i ) n ✓ ◆ h 1 s of 2GM 2GM 1 d⌧ = ±dT 1 dR, (3) th ( 4 R R e ) s 2G R 1 ⇢ ✓ ◆ co be M A R 2GM 1 or d d⇢ = ⌥dT + 1 dR. (4) d b 2GM R in ⇢, in y a a ( te 5) at . e sh i Subtracting (3) from (4), we have ar ft Th e e ge in 2/3 3 o d th R = (⌧ + ⇢) (2GM )1/3 , (5) es e T 2 i cs (6 , a co o ) nd rd where an integration constant has been absorbed by a shift in the origin of the ⇢ coordinate. The ⌧ i na expression for T as a function of ⌧ and ⇢ can be found from (3) as: i s t e. th 0 1 s e ✓ ◆3/2 s =
⇢0
R
1
Sp he r
⇢0
±T = 4GM @
1 3
R 2GM
+
R 2GM
tanh
1
R A 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
FIG. 4: A space-like slice of an inflating bubble connected by a wormhole to a dust dominated flat FRW universe.
13
qualitative analysis of this case, since a detailed description will require further numerical studies which are beyond the scope of the present work. Initially, as mentioned in Subsection II A, the Lorentz factor of the bubble wall relative to the Hubble flow at the end of inflation is of order γi ∼ tσ /ti 1. As the wall advances into ambient radiation with energy density ρm , a shock wave forms. The shocked fluid is initially at rest in the frame of the wall, and its proper density is of order ρs ∼ γ 2 ρm , where γ is the Lorentz factor of the shocked fluid relative to the ambient [11]. In the frame of shocked radiation, the shock front moves outward at a subsonic speed β ≈ 1/3, and therefore the pressure of the shocked gas has time to equilibrate while the wall is pumping energy into it. Consider a small region of size L ti
(39)
near the wall. At such scales, the wall looks approximately flat, and gravitational effects (including the expansion of the universe) can be neglected. In the frame of ambient radiation, energy conservation per unit surface of the bubble wall can be expressed as γi σ ∼ γ 2 ρm ∆t + γσ.
(40)
The left hand side of this relation represents the initial energy of the wall. The first term in the right hand side is the energy of the layer of shocked fluid. This is of order γρs ∆τ ∼
γ 2 ρm ∆t, where ∆τ is the proper time ellapsed since the shock starts forming, and ∆t = (t − ti ) = γ∆τ is the corresponding interval in the ambient frame. The second term in the right hand side is just the energy of the wall at time t. Here, we have neglected the rest energy of the ambient fluid, which is of order γ −2 relative to the shocked fluid. Eq. (40) shows that the bubble wall looses most of its energy on a time scale ∆t ∼
σ/(γi ρm ) ∼ (ti /tσ )2 ti ti . This is the characteristic time which is required for the pressure of the shocked fluid to accelerate the wall away from it, and is actually in agreement with the rough estimate (18), which is based on a simplified fluid model. The recoil velocity of the wall cannot grow too large, though, because the backward pressure wave can only travel at the speed of sound. Hence, the relative Lorentz factor γw between the wall and the shocked fluid will be at most of order one. The initial energy of the wall per unit surface is is γi σi ∼ G/ti . After time of order ti , this energy will be distributed within a layer of width ti near the position of the wall, and its density will be comparable to the ambient energy
14
density ρm ∼ G/t2i . By that time, the shock will have dissipated, and the wall will have, at most, a mildly relativistic speed with respect to the ambient Hubble flow. > ti , the expansion of the universe, and of the bubble, start playing a role. The For t ∼ wall’s motion becomes then dominated by the effect of its tension, and of the vacuum energy density in its interior. Again, the fate of gravitational collapse depends on whether the bubble is larger or smaller than a critical size.
1.
Small bubbles surrounded by radiation
Shortly after t ∼ ti , the bubble wall will be recoiling from ambient radiation at mildly relativistic speed, due to the pressure wave which proceeds inward at the speed of sound, in > ti , we expect that the wall will only be in contact with the aftermath of the shock. For t ∼
rarified radiation dripping ahead of the pressure wave. Neglecting the possible effect of such radiation, the dynamics proceeds as if the bubble were in an empty cavity, along the lines of our discussion in Subsection II C. In that case, the bubble has a conserved energy Mbh given approximately by Eq. (20). For Mbh Mcr , the bubble initially grows until it reaches a turning point, at some maximum radius Rmax , after which it collapses to form a black hole < R Rmax , the radius grows as of mass Mbh . As mentioned after Eq. (37), for Ri ∼ R ∝ τ 1/3 ,
(41)
where τ is proper time on the wall. Note that the ambient radiation of the FRW universe expands faster than that, as a function of its own proper time, since a(t) ∝ t1/2 . This means that even in the absence of external pressure, the bubble motion decouples from the Hubble flow on the timescale ti , with a tendency for the gap between the bubble and the ambient Hubble flow to grow in time. In what follows, and awaiting confirmation from numerical studies, we assume that the influence of radiation on the wall’s motion is negligible thereafter. Eventually the wall will collapse, driven by the negative pressure in its interior and by the wall tension. We can estimate the mass of the resulting black hole as the sum of bulk and bubble wall contributions, as in Eq. (20): 4 πρb + 4πσHi Ri3 . Mbh ∼ 3 15
(42)
In a radiation dominated universe, the co-moving region of initial size Ri crosses the horizon at the time tH ∼ and we can write (42) as GMbh ∼
Ri2 , ti
ti Ri Ri + t2b tσ
(43)
tH tH ,
(44)
< Ri tb , tσ . Hence, where in the last relation we have used that for Mbh Mcr , we have ti ∼ the black hole radius is much smaller than the co-moving region affected by the bubble at the time of horizon crossing.
2.
Large bubbles surrounded by radiation
For Mbh Mcr , the situation is more complicated. A wormhole starts developing at t ∼ GMcr , when the growth of the bubble becomes exponential. On the other hand, the wormhole stays open during a time-scale of order ∆t ∼ GMbh GMcr . During that time, some radiation can flow together with the bubble into the baby universe, thus changing the estimate (42) for Mbh . On general grounds, we expect that the radius of the black hole is at most of the size of the cosmological horizon at the time tH when the co-moving scale corresponding to Ri crosses the horizon < tH . GMbh ∼
(45)
The study of modifications to Eq. (42) for Mbh Mcr requires numerical analysis and is left for further research.
III.
GRAVITATIONAL COLLAPSE OF DOMAIN WALLS
In the previous sections we considered bubbles with ρb ρi . This condition led to a highly boosted bubble wall at the time when inflation ends. Here we shall concentrate instead on domain walls, which correspond to the limit ρb = ρi . Domain walls will be essentially at rest with respect to the Hubble flow at the end of inflation. Another difference is that for the case of bubbles we assumed that the inflaton field lives only outside the bubble, so its energy is transfered to matter that lives also outside. Here, we will explore the alternative possibility where matter is created both inside and outside the domain wall. This seems 16
to be a natural choice, since domain walls are supposed to be the result of spontaneous symmetry breaking, and we may expect the physics of both domains to be essentially the same. As we shall see, the presence of matter inside the domain wall has a dramatic effect on its dynamics. For t tσ , the graviational field of the walls is completely negligible. Walls of superhorizon size R ti are initially at rest with respect to the Hubble flow, and are conformally stretched by the expansion of the universe, R ∝ a. Depending on their initial size, their fate will be different. As we shall see, small walls enter the horizon at a time tH tσ and then they begin to decouple from the Hubble flow. Depending on the interaction of matter with the domain wall, they may either collapse to a black hole singularity, or they may develop long lived remnants in the form of pressure supported bags of matter, bounded by the wall. Larger walls with tH tσ start creating wormholes at the time t ∼ tσ , after which the walls undergo exponential expansion in baby universes.
A.
Domain walls surrounded by dust
The case of walls surrounded by dust is particularly simple, since the effect of matter outside the wall can be completely ignored. As we did for bubbles, here we may distinguish between small walls, for which self-gravity is negligible, and large walls, for which it is important.
1.
Small walls
Small walls are frozen in with the expansion until they cross the horizon, at a time tH tσ . The fate of the wall after horizon crossing depends crucially on the nature of its interaction with matter, which is highly model dependent [13]. To illustrate this point we may consider two contrasting limits. Let us first consider a situation where matter is reflected off the wall with very high probability. In this case, even if the wall tension exerts an inward pull, matter inside of it keeps pushing outwards. Initially, this matter has the escape velocity. In a Newtonian description, its positive kinetic energy exactly balances the negative Newtonian potential V (R). However, part of the kinetic energy will be invested in increasing the surface of the
17
domain wall, up to a maximum size Rmax , which can be estimated through the relation 2 ∼ −V (Rmax ) ∼ σRmax
2 GMbag t2H ∼ . Rmax GRmax
(46)
Here Mbag ∼ tH /G is the mass of the mater contained inside the wall, and we have neglected a small initial contribution of the domain wall to the energy budget. From (46) we have Rmax ∼ (t2H tσ )1/3 tH .
(47)
At that radius the gravitational potential of the system Φ ∼ V /M is very small − Φ ∼ (tH /tσ )1/3 1.
(48)
After reaching the radius Rmax , the ball of matter inside the wall starts collapsing under its own weight, helped also by the force exerted by the wall tension. As a result, matter develops a pressure of order p ∼ ρm v 2 , where v 2 ∼ GMbag /R is the mean squared velocity of matter particles. The collapse will halt at a radius R = Rbag where this pressure balances the wall tension:
2 GMbag σ ∼p∼ . 4 Rbag Rbag
(49)
The relation (49) is satisfied for a radius which is comparable to the maximum radius, Rbag ∼ Rmax ,
(50)
and much larger than the gravitational radius, GMbag ∼ tH Rbag .
(51)
The velocity of matter inside the bag is of the order of the virial velocity, and we do not expect any substructures to develop by gravitational instability inside the bag. Next, let us consider the case of a “permeable” wall, such that matter can freely flow from one side of it to the other. For tH tσ , this will simply behave as a test domain wall in an expanding FRW. Once the wall falls within the horizon, it will shrink under its tension much like it would in flat space, and the mass of the black hole can be estimated as Mbh ≈ 9πCm σt2H .
(52)
The coefficient Cm can be found by means of a numerical study. This is done in Appendix A. 18
Hi R 105
104
1000
100
10
Hi t 1
10
100
1000
104
105
FIG. 5: Evolution of the radius of a test domain wall in a matter dominated universe, as a function of cosmological time, for different initial radii. The green line corresponds to the horizon crossing time tH , when R = H −1 .
A test wall can be described by using the Nambu action in an expanding universe. The mass of the wall is defined as M (t) = 4πσ √
a2 r 2 , 1 − a2 r02
(53)
where the comoving radial coordinate r(t) satisfies the differential equation 2
r00 + (4 − 3a2 r0 )Hr0 +
2 2 (1 − a2 r0 ) = 0. ra2
(54)
Here a prime denotes derivative with respect to cosmic time t. The condition r0 (ti ) = 0 is imposed at some initial time such that R(ti ) ti . Fig. 5 shows the evolution of the domain wall physical radius R = ar as a function of time. We find that while the wall is stretched by the expansion the mass M (t) is increasing. Defining tH through the relation R(tH ) = H −1 = (3/2)tH ,
(55)
the wall starts recollapsing around the time tH and the mass M (t) approaches the constant value given by Eq. (52) once R tH . We find that, in the limit Ri ti , the numerical coefficient approaches the value Cm ≈ 0.15. 19
(56)
This result is in agreement with a more sophisticated calculation done in Ref. [17]. We note, however, that the result (56) only applies for sufficiently small walls. Such walls collapse at < tσ , and the resulting black holes have mass tH ∼ < Mcr ∼ 1/G2 σ. Mbh ∼
(57)
For larger walls, self-gravity is important, and the scaling Mbh ∝ σt2H no longer holds, as we shall now describe.
2.
Large domain walls surrounded by dust
If a domain wall has not crossed the horizon by the time tσ , its radius starts growing exponentially and decouples from the Hubble flow, forming a wormhole. To illustrate this process, we may consider an exact solution where a spherical domain wall of initial radius R∗ > tσ ,
(58)
is surrounded by pressuereless dust. The solution is constructed as follows. At some chosen time t∗ , we assume that the metric inside and outside the wall is initially a flat FRW metric with scale factor a ∝ t2/3 . We shall also assume that the wall is separated from matter, on both sides, by empty layers of infinitesimal width. (As we shall see, these layers will grow with time.) By symmetry, the metric within these two empty layers takes the Schwarzschild form
−1 2GM ± 2GM ± 2 ds = − 1 − dT + 1 − dR2 + R2 dΩ2 . R R 2
(59)
The mass parameter M − for the interior layer can in principle be different from the mass parameter M + for the exterior. Let us now find such mass parameters by matching the metric in the empty layers to the adjacent dust dominated FRW. By continuity, the particles of matter at the boundary of a layer must follow a geodesic of both Schwarzschild and FRW. Such geodesics originate at the white-hole/cosmological singularity, and they end at time-like infinity (see Fig. 6). The Schwarzschild metric is independent of T , and geodesics satisfiy the conservation of the corresponding canonical momentum: T˙ =
2GM ± 1− R 20
−1
C± .
(60)
Here C± is a constant, and a dot indicates derivative with respect to proper time. Initially, the geodesic is in the white hole part of the Schwarzschild solution (Region I in Fig. 6), and C± is positive or negative depending on whether the particle moves to the right or to the left. Using (1 − 2GM ± /R)T˙ 2 − (1 − 2GM ± /R)−1 R˙ 2 = 1, we have 2GM ± + C±2 − 1. R˙ 2 = R For geodesics with the escape velocity, we have
(61)
8
C± = 1.
(62)
The equation of motion (61) then takes the form R˙ R
!2
=
2GM ± . R3
(63)
This has the solution R ∝ t2/3 , where t is proper time along the geodesic of the dust particle, matching the behaviour of the flat FRW expansion factor. The geodesics at the boundaries of the two infinitesimal empty layers initially coincide at R = R∗ . Assuming that the initial matter density is the same inside and outside the wall, comparison of (63) with the Friedmann equation leads to the conclusion that M + and M − have the same value: Mbh ≡ M ± =
4π 3tH ρm (t∗ )R∗3 = . 3 4G
(64)
Here, Mbh is simply the total mass of matter inside the domain wall, which is also the mass which is excised from the exterior FRW, and tH is the time at which the boundary of such excised region crosses the horizon. The dynamics of the bubble wall in between the two empty layers is determined from Israel’s matching conditions, which lead to the equations [9] −1 2GMbh R ˙ T =± 1− , R tσ 8
(65)
We conventionally choose the constants C± to be positive so that the geodesics originating at the white hole singularity move to the right, from region I towards region II in the extended Schwarzshild diagrams (see Fig. 6). Note that we should think of the two empty layers on both sides of the wall as segments of two separate Schwarzschild solutions. In Fig. 6 these are depicted side by side. The solution for the interior vacuum layer is on the left, and the solution for the exterior vacuum layer is on the right.
21
ma di↵ ons = 2G in and erent ider M , T wa a t ! t i ll i s t m ⇠ ( he ot e-sc pher 1 R nd her ale ica G = l us con For ) 1. is th s are dom 2GM tc R r f a e o e os m ⌧ qu r acc lev in w , T ! ick mall a el e nt all ys t ,t ly olo r t at i h s on o the embe 1 ⇢ gy the Here hrink tretch e rep t i m dy dde = lea mat , we w s un ed b ulsiv n d e 2 -sc am d y t d ale i c s i n a GM a b s to er ar ill b er it cos e fiel m e d o s s d ⇢= o t p f he ue pr un ten olo ca sph aby 2 to such atiall eri univ form d it imari sion gica n be the a w y fl GM cal ers atio whi ly i and l ex ign r= a a B r n t r ep u l l . eg e, a n o l e i an d e f o te fo pan ore l s i v O n e m at t 0 a d re w ion o nd i f a w ts siz rested rms sion. d. I er eg i a e ust e c f of n t or E n i rav s the dom 1.1 cos onsi vac he a mho grow n the black vent this i tat cos ina ca s u mo der uum mb le. h o ion mo ted l og ien Th faste ppo ole o ally, se, f M t . h al Co l s e o a tc tF ed rt y. w i f r t fiel ogica univ n si e h ↵e c t⌧ RW ust han e lim radi en hin d d er to of l sca erse. gS u n wh i c t h e i t , w u s R t h e w R t he ft t he l e t ⌧ i T v a h S h c he e rs Sch hw wa mbi ere = 2 all f dom ⇠ H wo t d e , a wa o a s e G R l w m n 1 t rzs l s w he ain M r zs e a orig t e ain , In c ⇠ x c d i t i w r h n t h p e o 2 w h all Le ild ild ma all, left ally ansi . In t /t in ma m i to l et et r wit in t on r this itr ⌧ the h n wa ad ec ic h a he i ate us c as t . ori l ds 2 oor firs z on l i s us ✓ bla nter . As1 e, th = di n tc td , io ck e it ate os m i sc wh hol r of we Ssh wal 1 2 s, t u ss ere l t G e a h o r hi s M ◆ ep log t he ⌧a Rrem e w p ll s t ak y nd R ma ⇢ = nan allhge ee, t els e h dT 2 st ⇢a tch 0 t 2G t c or ✓ he yst eics ain is re i + f or di nCg o= 0 M t defi e o l d 1 2 m in d n ed an ↵er on f Stc= S R GM ◆ a d s h om 2 e d s t by n id w = = 1 R t he ⇠ the t ti er ti aHrzsc0 ai hR d⌧ 2 r el c o F ( G ot m e a s n dR 2 1 il= d a o 2 h p n + t Su GM q u fo r ion w ) 1 er sca he = 2G + bt r R r i s al R2 ck m . is les rica 2G M d⌧ ac t ⌧ R d a l li d t h ar l M b h th⌦ 2 H y lly i ng ⇢2 = e s . e t , e d e n + bh l h T r ( 3) e o s a m s R2 ±d ad a e , r i n t r , t c c r e l e ma ! du a fro d e T w h s el v i n ⌦2 d⇢ m 1 sp bab to tter e w ks u tch e r e r an . st 2G ( e a e h w 4 t n t R wh = M ✓ ), w tio t a er y u he aro ill d d pu co l e o e b = i b l s R ⌥d an B c a n n eh r u y ls ex p r e a 2G sm T+ t i m t h e em d efo l r ive form nd e pr its (c1o ive ave 1 2 r es n i n a M GM ◆ d y b ed re eg rse at it im te )sm fie si o e ol 1 t d R i e n bh sc n a d .1 us w on , a io wh ar si ol ld ✓ g nf og ,T 1 al m ed 2G R or ratio t e nd n o ile ily on ogi ca C o e c c M 1 T ic in y ! f a i i n a ca n os on f o M d i s dR ons as n con 2G t n n f t u l m si at e of a af , i de va th wo s s ere d ex b e M ◆ d o s 1 R t l o er cu e r i z st f o p i g t o su sp ch un a r = 3 1 gy t u a(2 m e r a cti nt h R th in ⇢ th ch ati wi t h e m m ) h o gr e d i m s n s i n or . on I a e = ( e n g s a all e o ⌧ o b h . n d l 2 a o S d w e b n R e + r i L f w Si n R y S 2 c e . . e t e ↵ b . . ⌧ s / em hw ⇢) nt T f he la E I ±T 2G p u al 3 ch ec giv fl and en a ce t ar l s i l . O at ai FR he ast op ck ve n t w M =4 bso t h e en 0 o ⇢ t ( z v n e a h d h 2 i p m r f sc GM can rbe W u r e ne a GM me n ( e rz w os ole tua is ⇢ t h t s h g ✓ d c 5 h tri @1 = sc ild be ra is tte oo u n t w h a i t e o l l y ca e ) 1/ 3 er b c ( ). A ( y s n h v d co r f t 3 fou e rd i v hi l i m r , w e, , m i t a he d om 2) a s ild ) R ◆ ds 2 3 t ⌧ s ns e i a n e c h e n o r f i s s c on d d h h h t t i a o c e s a 3 a m t. 2 io os ric /2 fro i ft e nd GM in to te = wa am t, w ius en r t yn d s i n r n w ✓ m m i s n a chr int t w a , = s R ⇢ h e ⌧ l a ol t e (3) t he ( 4) th al or bie ere S he ono egr + fi a a 0 o d d r l 1 e r is as: g R w = ig nt ,l e or i r e l u s , at i o u R i d ca u n in e R ta de 2G st et 2G all ⌧ l gi n e o t h e n co Su a 2G x l k i f fi f v f l t p u M s e M c l a of ne M b t h ca er s ◆ s l i n n st l t w y i an t os R t he s th tantra d fir i t h n si . e l e se. es a ⇠ ls w , t m by e I o do t h s h cti1 of nt h t 2 t n ⇢ t i d n o e h f t a con as T o m ⇠ T / ng c h lo di ds 2 2 o or t h e 1 rm i n do b l e i n r at t h i t R ai H s ✓ sta wbee g ( + s c 5 ( m d a n e t ⌧ y ) 3 t u i = r c nte he n a 2G) n at e l k e r i . A c as t h e ai n ss wa 1, A xpsp re bs 1 f M a o e d⌧ . h ro e . t t h r s a d ll o , o he ioT re tiaa or wa 2G r m le of w th ⇢, nshe ⌧ 2 ss ln c bed iz ll e = e m ( M + r t io ion o o 4 em h e sh w 2G by ◆ at n, i s ), R n d⇢ r ± a ch n a w al l al l M w fo tegdin i 1 1
⇢0
R
Sp
=
he
ri t 2G ca di C = 0 M ld an ↵er on t S R om t d en si d = t = ⇠ t h e t t i er i 0 Baby"universe" Parent"universe" ai H m a R ( co F G ot e n he -sc sph 1= = q u n f o or ) w 2 a e 1 r ick rm R al . is les ric 2G GM t h H l y al l ⌧ li th a al M b e r e d bh h , T l e e m er e sh r y s t , n Schwarzschild"singularity" o a a r cc el m a ds at , w ink tre th ! du e a e t b le va in sp a to ter e w s u ch e r 1 st ra nt w he by th a il n ed ep R tio t a d co o ll = an B rica un e f rou l be er by uls n d ef l r ive orm nd p its co ive 2G sm t i m t h e em o r a r M 1. e- d y b ed du re egio se, ati it w ima ten smo fiel ol bh sc n a d 1 st we n an on h ril sio lo d og ,T al m ed C i c o g y l n d c o c e a e f i on ic in os on o M f a i i n a ca n ! y d i s n t si d f n m t l u s b at e of a 1 Spherical domain wall in dust cosmology ol ide va th wo s si ere d f ex e er 1 ch t c og r e r z s o p i o su c sp th i dust"FRW" y. th uum am mh e g ted rm an gno a t ⇢ n he h a t i a e e g = bi ole row in s a sion red . Sc l e l S r w e . ↵ y . t ep a hw nt T s f he bla . E I ch ec 2G ul ll. Rfl= ar t w FR he ast op ck ve n t siv O at 2GMSM R = 0 R = 2GMbh , T ! 1 R = 2GMbh , T ! 1 ⇢ = 2GM ⇢ tr o z a e n d h f sc 1 r Spherical ho dust po in 1 = 2GM m e W u rwall tu is cosmology e ⇢0ne t = th domain zs hi domain wall in dust cosmology co i H bh gr i at0t t1= t⇢Spherical e u n s t w t h a s i t e l e o al l c a ld = ch or sConsider a e y d n s v f a spherical domain wall embedded in a spatially flat matter d c r t i m di , h l e o v i d h i o i r ld t ta e d m w , m er iiden=fy" s2 ns et na om tio co time-scales ai se ch w he it, adiu he for di↵erent ric are relevant to the dynamics of such a wall. One is the co t te = t . n a n o in ⇢ = s, s T ! t 1 ⇢n= r R = 0rR==0 2GM, m =w2GM, R ✓= 2GMS R Twe! a1 r = r0T⇢! = 1 R = 2GM, T ! 1 ⇢ = 2GM ⇢ m2GM R al sother S const. s R a = 0 R = w2GM, at is the =acceleration bi he R the ⌧and the th ol= 2GM time-scale due to the repulsive gravitationa a o a S fi e l r e d o1tg= 0d t = ti 0 is ⇢10 t = 0 t = ti u l, re rig nt e ⇢ e R = w 0 l r R i d l t a i 2 t ⇠ (G ) . u c st et wallle embedded n ak efi = universe. ll ⌧ flat matter G al dominated ni a spherical Consider domain spatially ofConsider domainTwo wall embedded in a spatially flat matter ft all exp in a2GM M ◆ a spherical es us f For ne r0 ve repulsive c t One Rdi↵erent ⌧ t s,ccosmological the field t ⌧ R ththe w y i an of tsuch a awall. R l d r thdi↵erent l a time-scales oare relevant to the dynamics is scale t ⇠can Hto1be ,theignored. s ⇢ time-scales are relevant dynamicsIn of this such case, a wall.for One is the , fi s s i t h n si . ⇠ l e m e e r by e t = w . s o I h d t d conformally stretched by cosmological expansion. Eventually, when the wall 2 t t n t ol fo the other time-scale repulsive field domain wall eand the om a duehe to nthe other thethe acceleration time-scale due to the repulsive gravitatiof T 2 is the acceleration ⇠ isTof th ds 2 and /t ithi gravitational di rm og do bl in rat thquickly w 1+ ✓ e s a n 1 H i shrinks under its tension and forms a black hole of radius RS = 2GM t ⇠ (G ) . c ac te e. s c ⌧ tt ⇠ o m (G ) in . y end"of"infla=on" re = us 1 he ai k ri la ascase, w s ignored. 1 , A d⌧ t For R ⌧ t , the repulsive field can be In this for t ⌧ domain wall is n o Here, we will be primarily interested in the opposite where . IR tio For R ⌧ t , the repulsive field can be ignored.limit, In this case,Rfor tt ⌧ h . al d⌧ 2 th ho w ol r o s w e, t 2G l ns e e riwall a h conformally stretched Eventually, when the falls within the horizon, it f = conformally stretched by cosmological expansion. Eventually, when the wal e the matter around it while its size grows faster than the ambient expansion r + 2 Mby◆ cosmological expansion. m l e z re th s on l i 2 at R its tension and forms a black m of eradius ha wRaquickly s t /t ± quicklyGshrinks under hole 2GM ⇠ ⌧wormhole. t.its tension The shrinks under and forms a black hole of radius R = 2GM S =formation , M c S n leads to the of a dust which was originally in the int w l 1 dT hi i an a ll s l r R we will be primarily interested in the opposite = s ng ll Here, limit, where R t . twe In this the wallinterested repels be primarily in the opposite whereaRblack t . d⇢ 2 at cbaby and in the ambient FRW universe we arelimit, left with e, epHere, FIG.dR6:2 A large domain wall causing the birth ofwill a case, baby universe. go euniverse, e o ⌥ y l it while its size grows faster thanf Sthe ambient expansion rate. As we shall see, this 2G the matter around s t e the matter around it while its size grows faster than the ambient expansion s + dT h te s i region + of of vacuum. is s ch spherical M R2 d nleads R2 of a wormhole. The dust which was the of the wall goes into The dust which was originally in the + to interior the formation of a wormhole. ✓ to the formation R leads wa originally in toin d⌦ Before we consider the e↵ectcysted of the let us first discuss the m d 2 r ⌦ a baby universe, and in the ambient FRW universe we are left with a black hole remnant in domain a FRW wall, a 2 a baby universe, and in the ambient universe we are left with a blac z 1 . sc a dust cosmology. 2G R ✓ and . and h 2 spherical spherical ild 2 region of of vacuum. G region of of vacuum. M M ◆we consider the e↵ect of the domain wall, R Before 2GM bhfirst discuss Before let us the matching of Schwarzschild 1 2 R we consider the e↵ect of the domain wall, let us first discuss the ˙ + − 1. (66) R = ⌧ 2G a dust cosmology. 1 + and 1.1 Matching Schwarzschild to a dust cosmology and a dust cosmology. R t M ◆ σ ⇢) 2 dR R /3 , 1 Consider the Schwarzschild metric The double ± in (65)torefers the embedding the domain wall as to seen fromcosmology the (2 1.1 dMatchingsign Schwarzschild a dusttocosmology 1.1 of Matching Schwarzschild a dust by G (1 R ✓ ◆ ✓ ◆ M ) . a 2GM 2GM 1 2 2 2 1/ ) Consider the Schwarzschild metric Consider the Schwarzschild metric s interior and exterior Schwarzshild solutions respectively. The embedding is different, since ds = 1 dT + 1 dR + R2 d fro h 3 , ✓ ◆ ✓ ◆ 1 m i ft ✓R ◆ ✓ R ◆ 1 2GM 2GM 2GM 2GM 2 2 2 (3 in 2 2 = 1 the (2extrinsic dT 2 + 1 curvaturedR R2 d⌦ . = we 1(1) dT +side 1 dR2 + R2 ) there has to be adsjump in of +the worldsheet as one In coordinates, this ds takes the go formfrom R ) R Lemaitre as the R R : or ta ig In coordinates, takes the form In Lemaitre coordinates, this takes the C form = 2GM nh the wall to the this other. Comparing (65) and (66) with (60) and (61) with 1 and in of Lemaitre s ds2 = d⌧ 2±+ d⇢2 + R2 d⌦2 . (3 1 of ) 2 R 2GM 2GM 2 th ± ds R = >d⌧t2 + d⇢2 +moves R2 d⌦2 . faster than the dust particle (2) ds2 = atd⌧the + edged⇢2 + R2 d⌦2 . M for e = Mbh , we see that, σ , the 2G R 1 R wall (4 R where ⌧ and ⇢ are defined by the relations ⇢ be ) M A co⌧ and ⇢ are defined by the relations d where s are defined the by the relations di by of theordexpanding interior ball of matter. As seenwhere from⌧ and the⇢ outside, wall moves in✓ the ◆ ⇢, na a s in 2GM 2GM 1◆ s ✓ ◆ 1 ✓ te s a 1 d⌧ = ±dT 1 dR, ( t 2GM 2GM h 2GM 2GM 5) ar ift R into1 R d⌧ exterior = ±dT FRW Hubble 1 directione.opposite to the flow.dR,Consequently, the wall runs d⌧ (3) =never ±dT dR, e s T R R R R ge in ✓ ◆ 1 he s s o d th R 2GM ✓ ◆ 1 ✓ ◆ 1 es e R by (66) 2GMthereafter. is well described d⇢ = ⌥dT + R1 2GM dR. ic T matter and its motiond⇢ = ⌥dT + 1 dR. d⇢ (4) = ⌥dT 2GM + 1 R dR. s, c 2GM R (6 2GM R an oor ) > t Eq. (66) has no turning points and the radius continues to expand forever. d di For R σ n Subtracting (3) from(3) (4), we(4), have ⌧ Subtracting (3) from (4), we have Subtracting from we have i s at e τ /t th . σ 2/3 The expansion reaches exponential behaviour R ∝ e after the time whenthe term 2/3 e 2/3 3 first 3 3 R = (⌧ + ⇢) (2GM )1/3 , R (5) = R (⌧ ⇢) + ⇢)(2GM )1/3),1/3 , = + (⌧ (2GM 2 2 2 in the right hand side of (66) can be neglected and the second one becomes dominant. Here
III"
IV"
II"
I"
where an integration constant has been absorbed by a shift the origin theconstant ⇢ coordinate. Thebeen where an of integration constant has absorbed a shift in the origino wherein an integration has been absorbed by abyshift in the origin expression T as a time functionon of the ⌧ andworldsheet. ⇢ can be found from (3) as: τ is the for proper This expansion is of course much faster than that expression for T as a function of ⌧ and ⇢ can be found from (3) as: expression for T as a function of ⌧ and ⇢ can be found from (3) as: 0
✓
◆
s
s
1
0
0
s
1 ss 1 RR tanh 1 A 2GMtanh 2GM
s ✓ ◆3/2 3/2 ◆R R place R A (see Figs. 6✓and of the Hubble ±T flow, and 1takes in R the baby universe 1 7) 3/2. LooselyR = 4GM @ + tanh 1 ⇢, 1 @(6) R ±T = 4GM +R 3
2GM
2GM
2GM
±T = 4GM @
3
2GM +
3 2GM speaking, a wormhole in the extended Schwarzschild solution forms and closes on a 2GM time-
2GM
with R given in (5). A second integration constant has been absorbed a shiftinin(5). the AT second coordinate. with by R given integration constant has been absorbed by R coordinate given A (2) second integration constant been spatial absorbed by a Since the metric (2) is synchronous, the lines of constant with spatial are geodesics, and ⌧ is the the Since in the(5). metric is synchronous, lines of has constant coordinate
22 the metric (2) is synchronous, the lines of constant spatial coordinate a Since
FIG. 7: A space-like slice of a baby universe with an expanding ball of matter in it, connected by a wormhole to an asymptotic dust dominated FRW universe. The wormhole is created by the repulsive gravitational field of an inflating domain wall, depicted as a red line.
scale ∆t ∼ GMbh . This coincides with the time tH when the empty cavity containing the black hole crosses the horizon. An observer at the edge of the cavity can initially send signals into the baby universe, but not after the time when H −1 = 2GMbh . In the above discussion, we did not make any assumptions about the time t∗ . The exact solution illustrating the formation of a wormhole is obtained by a matching procedure, and can be constructed for any initial value of t∗ . As we have seen, the condition R∗ > tσ ensures that the wall will not run into matter. The wall evolves in vacuum, and the Hubble flow of matter remains undisturbed after t∗ . Moreover, it is easy to show that for t∗ tσ and R∗ > tσ , the wall will be approximately co-moving with the Hubble flow at the time t∗ , with a relative velocity of order v ∼ (t∗ /tσ ) 1.9 9
This can be seen as follows. Denoting by U µ the four-velocity of matter at the edge of the cavity and by W µ the four-velocity of the domain wall, the relative Lorentz factor is given by s 4 2 2 2 9t 9t 3t 9t ∗ ∗ γw = −gµν U µ W ν = 1 − ∗2 1 + 2∗ − ∓ . (67) 4R∗ 4tσ 2R∗ 4R∗ tσ
where we have used (60), (61), (65) and (66), with 2GMbh = (4/9t2∗ )R∗3 . Using t∗ tσ and R∗ > tσ , we
23
Nonetheless, in order to make contact with the setup of our interest, we must consider a broader set of initial conditions such that the wall can be smaller than tσ at the time ti when inflation ends, Ri < tσ . In this case, the wall may initially expand a bit slower than the matter inside. Consequently some matter may come into contact with the wall. However, since superhorizon walls are approximately co-moving for t tσ , the effect of this interaction will be small, and we expect the qualitative behaviour to be very similar to that of the exact solution discussed above provided that the size of the wall becomes larger than < tσ . tσ at some time t∗ ∼ B. 1.
Walls surrounded by radiation Small walls
Let us start by discussing the situation where the domain wall is “impermeable”. Small walls with tH tσ continue to be dragged by the Hubble flow even after they have crossed the horizon. Unlike dust, radiation outside the wall may continue exerting pressure on it, and initially its effect cannot be neglected. The radius of the wall will stop growing once the wall tension balances the difference in pressure ∆p between the inside and the outside: ∆p ∼
σ . R
(68)
After that, the pressure outside decays as (tH /t)2 and becomes negligible compared with the pressure inside, which is given by p ∼ (1/Gt2H )(tH /R)4 . From (68) we then find that the equilibrium radius is of order Rbag ∼ (tσ /tH )1/3 tH tH .
(69)
The mass of the bag of radiation is of order Mbag
tH ∼ G
tH tσ
1/3
.
(70)
Therefore the gravitational potential at the surface of the bag is very small Φ ∼ (tH /tσ )2/3 1. have γw = 1 + O(t2∗ /t2σ ), or v ∼ (t∗ /tσ ) 1.
24
Next, we may consider the case of “permeable” walls. Like in the case of dust, the dynamics of small walls is well described by the Nambu action in a FRW universe. The mass of the resulting black holes can be estimated as Mbh ≈ 16πCr σt2H ,
(71)
where in a radiation dominated universe tH is determined from the condition R(tH ) = 2tH .
(72)
The numerical coefficient is found by using the numerical approach described in Appendix A, and we find Cr ≈ 0.62.
(73)
Note that, although permeable walls do form black holes, both in the dust dominated and radiation dominated eras, their masses are much smaller than the masses of the corresponding bags of matter trapped by impermeable walls.
2.
Large domain walls surrounded by radiation
For tH ∼ tσ the estimates for Mbh and Mbag which we found in the previous subsection converge to the value Mbh ∼
tH . G
(74)
For tH tσ a wormhole will start forming near the time tσ , and some of the radiation may follow the domain wall into the baby universe. Nonetheless, the size of the resulting black hole embedded in the FRW universe cannot be larger than the cosmological horizon, and therefore Mbh ≤ tH /2G. Continuity suggests that the estimate (74) may be valid also for large bubbles in a radiation dominated universe. A determination of Mbh for this case requires a numerical simulation and is left for further research.
IV.
MASS DISTRIBUTION OF BLACK HOLES
In the earlier sections we have outlined a number of scenarios whereby black holes can be formed by domain walls or by vacuum bubbles nucleated during inflation. The resulting mass distribution of black holes depends on the microphysics parameters characterizing the 25
walls and bubbles and on their interaction with matter. We will not attempt to explore all the possibilities here and will focus instead on one specific case: domain walls which interact very weakly with matter (so that matter particles can freely pass through the walls). A specific particle physics example could be axionic domain walls, whose couplings to the Standard Model particles are suppressed by the large Peccei-Quinn energy scale. The walls could also originate from a ”shadow” sector of the theory, which couples to the Standard Model only gravitationally.
A.
Size distribution of domain walls
Spherical domain walls nucleate during inflation having radius R ≈ Hi−1 ; then they are
stretched by the expansion of the universe. At t − tn Hi−1 , where tn is the nucleation
time, the radius of the wall is well approximated by10
R(t) ≈ Hi−1 exp[Hi (t − tn )].
(75)
The wall nucleation rate per Hubble spacetime volume Hi−4 is λ = Ae−S .
(76)
In the thin-wall regime, when the wall thickness is small compared to the Hubble radius Hi−1 , the tunneling action S is given by [4] S = 2π 2 σ/Hi3 .
(77)
This estimate is valid as long as S 1, or > H 3. σ∼ i
(78)
With this assumption, the prefactor has been estimated in [19] as A ∼ σ/Hi3
2
.
(79)
The number of walls that form in a coordinate interval d3 x and in a time interval dtn is dN = λHi4 e3Hi tn d3 xdtn . 10
(80)
As in previous Sections, we assume that Hi−1 tσ , so wall gravity can be neglected during inflation.
26
Using Eq. (75) to express tn in terms of R, we find the distribution of domain wall radii [4], dn ≡
dN dR = λ 4, dV R
(81)
where dV = e3Hi t d3 x
(82)
is the physical volume element. At the end of inflation, the distribution (81) spans the range of scales from R ∼ Hi−1 to Rmax ∼ Hi−1 exp(Ninf ), where Ninf is the number of inflationary e-foldings. For models of inflation that solve the horizon and flatness problems, the comoving size of Rmax is far greater than the present horizon. After inflation, the walls are initially stretched by the expansion of the universe, R(t) =
a(t) Ri , a(ti )
(83)
where a(t) is the scale factor, ti corresponds to the end of inflation and Ri = R(ti ). The size distribution of the walls during this period is still given by Eq. (81).
B.
Black hole mass distribution
As we discussed in Section III, the ultimate fate of a given domain wall depends on the relative magnitude of two time parameters: the Hubble crossing time tH when R(tH ) = H −1 , where H = a/a ˙ is the Hubble parameter, and the time tσ ∼ Hσ−1 = (2πGσ)−1 , when the repulsive gravity of the wall becomes dynamically important. Walls that cross the Hubble radius at tH tσ have little effect on the surrounding matter and can be treated as test walls in the FLRW background. The mass of such walls at Hubble crossing (disregarding their kinetic energy) is MH ≈ 4πσH −2 ∼
tH M(tH ) M(tH ), tσ
(84)
where M(t) =
1 2GH
(85)
is the mass within a sphere of Hubble radius at time t. Once the wall comes within the Hubble radius, it collapses to a black hole of mass M = CMH with C ∼ 1 in about a Hubble time. 27
In the opposite limit, tH tσ , the wall starts expanding faster than the background at t ∼ tσ and develops a wormhole, which is seen as a black hole from the FRW region. Assuming that the universe is dominated by nonrelativistic matter, we found that the mass of this black hole is M ∼ M(tH )
(86)
and its Schwarzschild radius is rg ∼ tH . The boundary between the two regimes, tH ∼ tσ ,
corresponds to black holes of mass M ∼ 1/G2 σ ∼ Mcr , where Mcr is the critical mass
introduced in Section III. We note that depending on the wall tension σ, the critical mass Mcr can take a wide range of values, including values of astrophysical interest. If we define the energy scale of the wall η as σ ∼ η 3 , then, as η varies from ∼ 1 GeV to the GUT scale
∼ 1016 GeV , the critical mass varies from 1017 M to 104 g. > tσ > teq , where teq is the time of equal matter and The estimate (86) applies for tH ∼ ∼
radiation densities. For walls that start developing wormholes during the radiation era, tσ < teq , we could only obtain an upper bound, < M(tH ) M∼
(tσ < teq ).
(87)
We shall start with the case tσ > teq , which can be fully described analytically. Note, < 1 GeV , which is rather however, that this case requires the energy scale of the wall to be η ∼
small by particle physics standards. The critical mass in this case is Mcr > 1017 M . 1.
tσ > teq
Let us first consider black holes resulting from domain walls that collapse during the radiation era, tH < teq , and have radii between R and R + dR at Hubble crossing. For such domain walls, the Hubble crossing time is tH = R/2, and their density at t > tH is 3/2 dR R . dn(t) ∼ λ t R4
(88)
The mass distribution of black holes can now be found by expressing R in terms of M , R ∼ (M/σ)1/2 , and substituting in Eq. (88):
σ 3/4 dM dn = λ 2 . t M 7/4 28
(89)
A useful characteristic of this distribution is the mass density of black holes per logarithmic mass interval in units of the dark matter density ρDM , f (M ) ≡
M 2 dn . ρDM dM
(90)
Since black hole and matter densities are diluted in the same way, f (M ) remains constant in time. We shall evaluate it at the time of equal matter and radiation densities, teq , when Eq. (88) derived for the radiation era should still apply by order of magnitude. With ρDM (teq ) ∼ B −1 Gt2eq , where B ∼ 10, we have 1/2
1/4 f (M ) ∼ λ(σ)3/4 Gt1/2 ∼ Bλ eq M
Meq M 1/4 3/4
.
(91)
Mcr
Here, Meq ∼ teq /2G ∼ 1017 M is the dark matter mass within a Hubble radius at teq . For black holes forming in the matter era, but before tσ (tσ > tH > teq ), Eqs. (88) and (91) are replaced by
2 dR R , dn = λ t R4 1/2 M f (M ) ∼ Bλ . Mcr
(92) (93)
Finally, for black holes formed at t > tσ , the mass is
M ∼ M(tH ) ∼ R/G.
(94)
Substituting this in Eq. (92), we find f (M ) ∼ Bλ.
(95)
The resulting mass distribution function is plotted in Fig. 8.
2.
tσ < teq
We now turn to the more interesting case of tσ < teq . (We shall see that observational constraints on our model parameters come only from this regime.) In this case, domain walls start developing wormholes during the radiation era, and we have only an upper bound (87) for the mass of the resulting black holes. Domain walls that have radii R < tσ at horizon crossing collapse to black holes with M < Mcr . The size distribution of such walls and the mass distribution of the black holes 29
tΣ > teq logfHML
M 12
M 14 Mcr
Meq
logMM
FIG. 8: Mass distribution function for tσ > teq .
are still given by Eqs. (89) and (91), respectively. For tσ < R < teq , let us assume for the moment that the bound (87) is saturated. Then, using R ∼ GM in Eq. (88), we obtain the mass distribution f (M ) ∼ Bλ
Meq M
1/2
,
(96)
where B ∼ 10, as before. This distribution applies for Mcr < M < Meq . With the same assumption, for walls with R > teq we find f (M ) ∼ Bλ.
(97)
The resulting mass distribution function is plotted in Fig. 9, with the parts depending on the assumed saturation of the mass bound shown by dashed lines. The assumption that the bound (87) is saturated for M > Mcr appears to be a reasonable guess. We know that it is indeed saturated in a matter-dominated universe, and it yields a mass distribution that joins smoothly with the distribution we found for M < Mcr . A reliable calculation of f (M ) in this case should await numerical simulations of supercritical domain walls in a radiation-dominated universe. For the time being, the distribution we found here provides an upper bound for the black hole mass function.
30
tΣ < teq logfHML
M 14
M -12
Mcr
Meq logMM
FIG. 9: Mass distribution function for tσ < teq . V.
OBSERVATIONAL BOUNDS
Observational bounds on the mass spectrum of primordial black holes have been extensively studied; for an up to date review, see, e.g., [20]. For small black holes, the most stringent bound comes from the γ-ray background resulting from black hole evaporation: < 10−8 . f (M ∼ 1015 g) ∼
(98)
For massive black holes with M > 103 M , the strongest bound is due to distortions of the CMB spectrum produced by the radiation emitted by gas accreted onto the black holes [21]: < 10−6 . f (M > 103 M ) ∼
(99)
Of course, the total mass density of black holes cannot exceed the density of the dark matter. Since the mass distribution in Fig. 9 is peaked at Mbh = Mcr , this implies f (Mcr ) < 1.
(100)
These bounds can now be used to impose constraints on the domain wall model that we analyzed in Section IV. As before, we shall proceed under the assumption that the mass bound (87) is saturated. The model is fully characterized by the parameters ξ = σ/Hi3 and Hi /Mpl , where Hi is the expansion rate during inflation. The nucleation rate of domain walls λ depends only on 31
4.0
Mcr = 103 M⊙
3.5
Mcr = 1024 g
Mcr = 1015 g
ξ
3.0
2.5
2.0
1.5
1.0 -20
-18
-16
-14
-12
Log 10
-10
-8
-6
Hi MPl
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).
ξ, λ ∼ ξ 2 e−2π
2ξ
(101)
(see Eqs. (76), (77), (79)). The parameter space {ξ, Hi /Mpl } is shown in Fig. 10, with red and purple regions indicating parameter values excluded by the constraints (98) and (99), > 1, where the semiclassical tunneling calculation respectively. We show only the range ξ ∼
is justified. Also, for ξ 1 the nucleation rate λ is too small to be interesting, so we only show the values ξ ∼ few.
The dotted lines in the figure indicate the values Mcr ∼ 1015 g ≡ Mevap and Mcr ∼
103 M ≡ 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 ξ ∼ 1 in the entire range shown in the figure, and therefore Mcr ∼ ξ −1 (Mpl /Hi )3 Mpl depends 32
essentially only on Hi . The horizontal upper boundaries of the red and purple regions correspond to the regime of M > Mcr , where the mass function (96) depends only on λ. The corresponding constraints < 10−15 for the red region (evaporation bound) and λ < 10−26 on λ can be easily found: λ ∼ ∼ for the purple region (accretion bound). As we have emphasized, the mass function (96) that we obtained for super-critical domain walls represents an upper bound on the black hole mass function, and thus the red and purple regions in Fig. 10 may overestimate the actual size of the excluded part of the parameter space. We have verified that the regime tσ > teq does not yield any additional constraints on model parameters. A very interesting possibility is that the primordial black holes could serve as seeds for supermassive black holes observed at the galactic centers. The mass of such seed black holes should be M > 103 M [22] and their number density n0 (M ) at present should be comparable to the density of large galaxies, nG ∼ 0.1M pc−3 . The region of the parameter space satisfying this condition is marked in green in Fig. 10. (As before, the horizontal upper boundary of this region may overestimate the black hole mass function.) The solid straight line in Fig. 10 marks the parameter values where f (Mcr ) = 1, so the parameter space below this line is excluded by Eq. (100). This line lies completely within the red and purple excluded regions, suggesting that the bound (100) does not impose any additional constraints on the model parameters. We note, however, that the observationally excluded regions are plotted in Fig. 10 assuming that the expansion rate Hi remains nearly constant during inflation. Small domain walls that form black holes with masses Mevap ∼ 1015 M are produced towards the very end of inflation, when the expansion rate may be substantially reduced. The tunneling action (77) depends on Hi exponentially, so any decrease in Hi may result in a strong suppression of small wall nucleation rate. The evaporation bound on the model parameters can then be evaded, while larger black holes that could form dark matter or seed the supermassive black holes may still be produced. For example, with ξ ≈ 2, a change in Hi by 20% would suppress the nucleation rate by 10 orders of magnitude. The black holes that form dark matter in our scenario can have masses in the range 1015 g < Mbh < 1024 g. For Mbh < 1015 g the black holes would have evaporated by now, and for Mbh > 1024 g the line f (Mcr ) = 1 in Fig. 10 enters the purple region excluded by the gas
33
accretion constraint.11 We note that Ref. [23] suggested an interesting possibility that dark < Mbh < 105 M . Most matter could consist of intermediate-mass black holes with 102 M ∼ ∼ of this range, however, appears to be excluded by the gas accretion constraint (99).
We note also the recent paper by Pani and Loeb [26] suggesing that small black holes could be captured by neutron stars and could cause the stars to implode. They argue that < Mbh < 1023 g from being the this mechanism excludes black holes with masses 1017 g ∼ ∼
dominant dark matter constituents. This conclusion, however, was questioned in Ref. [27].
VI.
SUMMARY AND DISCUSSION
We have explored the cosmological consequences of a population of spherical domain walls and vacuum bubbles which may have nucleated during the last N ∼ 60 e-foldings of inflation. At the time ti when inflation ends, the sizes of these objects would be distributed < Ri < eN ti , and the number of objects within our observable universe would in the range ti ∼ ∼
be of the order N ∼ λe3N , where λ is the dimensionless nucleation rate per unit Hubble volume. We have shown that such walls and bubbles may result in the formation of black holes with a wide spectrum of masses. Black holes having mass above a certain critical value would have a nontrivial spacetime structure, with a baby universe connected by a wormhole to the exterior FRW region. The evolution of vacuum bubbles after inflation depends on two parameters, tb ∼
(Gρb )−1/2 and tσ ∼ (Gσ)−1 , where ρb > 0 is the vacuum energy inside the bubble and
σ is the bubble wall tension. For simplicity, throughout this paper we have assumed the separation of scales ti tb , tσ . At the end of inflation, the energy of a bubble is equivalent to the “excluded” mass of matter which would fit the volume occupied by the bubble, Mi = (4π/3)ρm (ti )Ri3 , where ρm is the matter density. This energy is mostly in the form of kinetic energy of the bubble walls, which are expanding into matter with a high Lorentz factor. Assuming that particles of matter cannot penetrate the bubble and are reflected from the bubble wall, this kinetic energy is quickly dissipated by momentum transfer to the surrounding matter on a time-scale much shorter than the Hubble time, so the wall comes to rest with respect to the Hubble flow. In this process, a highly relativistic and dense 11
The allowed mass range of black hole dark matter is likely to be wider in models where black holes are formed by vacuum bubbles.
34
exploding shell of matter is released, while the energy of the bubble is significantly reduced. If the bubble is surrounded by pressureless matter, it subsequently decouples from the Hubble flow and collapses to form a black hole of mass 2 ti ti Mi Mi , Mbh ∼ 2 + tb tσ
(102)
which is sitting in the middle of an empty cavity. The fate of the bubble depends on whether its mass is larger or smaller than a certain critical mass, which can be estimated as Mcr ∼ Min{tb , tσ }/G.
(103)
For Mbh < Mcr , the bubble collapses into a Schwarzschild singularity, as in the collapse of usual matter. However, for Mbh > Mcr the bubble avoids the singularity and starts growing exponentially fast inside of a baby universe, which is connected to the parent FRW by a wormhole. This process is represented in the causal diagram of Fig. 3, of which a spatial slice is depicted in Fig. 4. The exponential growth of the bubble size can be due to the internal vacuum energy of the bubble, or due to the repulsive gravitational field of the bubble wall. The dominant effect depends on whether tb is smaller than tσ or vice-versa. If the bubble is surrounded by radiation, rather than pressureless dust, there is one more step to consider. After the bubble wall has transfered its momentum to the surounding matter and comes to rest with respect to the Hubble flow, the effects of pressure may still change the mass of the resulting black hole. We have argued that for Mbh Mcr the black hole mass, given by Eq. (44), is not significantly affected by the pressure of ambient matter. However, for Mbh Mcr this may be important, since some radiation may follow the bubble into the baby universe, thus affecting the value of the resulting black hole mass. This process seems to require a numerical study which is beyond the scope of the present work. On general grounds, however, we pointed out that the mass of the black hole is < tH /G, where tH is the time when the co-moving region corresponding to bounded by Mbh ∼ the initial size Ri crosses the horizon.
The case of domain walls is somewhat simpler than the case of bubbles, since only the scale tσ is relevant. If the wall is small, with a horizon crossing time tH < tσ , and assuming that the only interaction with matter is gravitational, the wall will collapse to a black hole of mass Mbh ∼ σt2H . Alternatively, if the wall is impermeable to matter, a pressure supported bag of matter will form, with a substantially larger mass which is given in Eqs. (51) or (70). 35
These estimates apply respectively to the case where matter is non-relativistic or relativistic at the time when the wall crosses the horizon. Larger walls with tH > tσ will begin to inflate at t ∼ tσ , developing a wormhole structure. If at t ∼ tσ the universe is dominated by nonrelativistic matter, the mass of the resulting black hole is Mbh ∼ tH /G, where tH is now the time when the comoving region affected by wall nucleation comes within the horizon. For tσ in the radiation era, the estimate of the mass is more difficult to obtain, since some radiation may flow into the baby universe following the domain wall. This issue requires a numerical study and is left for further research. As in the case of bubbles, for this case we < tH /G. were able to find only an upper bound on the mass, Mbh ∼
A systematic analysis of all possible scenarios would require numerical simulations, and we
have not attempted to do it in this paper. Nonetheless, for illustration, we have considered the case of a distribution of domain walls which interact with matter only gravitationally. Also, awaiting confirmation from a more detailed numerical study, we have assumed that < tH < teq , lead to black holes large walls entering the horizon in the radiation era, with tσ ∼ ∼ which saturate the upper bound on the mass, Mbh ∼ tH /G. Under these assumptions, we have shown that black holes produced by nucleated domain walls can have a significant impact on cosmology. Hawking evaporation of small black holes can produce a γ-ray background, and radiation emitted by gas accreted onto large black holes can induce distortions in the CMB spectrum. A substantial portion of the parameter space of the model is already excluded by present observational constraints, but there is a range of parameter values that would yield large black holes in numbers sufficient to seed the supermassive black holes observed at the centers of galaxies. For certain parameter values the black holes can also play the role of dark matter. Our preliminary analysis provides a strong incentive to consider the scenario of black hole formation by vacuum bubbles, which has a larger parameter space. Note that the black hole density resulting from domain wall nucleation is appreciable only if ξ = σ/Hi3 ∼ 1, where Hi is the expansion rate during inflation. This calls for a rather special choice of model parameters, with12 σ ∼ Hi3 . The narrowness of this range is related to the fact that the nucleation rate of domain walls is exponentially sensitive to the wall tension. 12
On the other hand, this relation may turn out to be natural in certain particle physics scenarios, once environmental selection effects are taken into account, see e.g. [24]. We thank Tsutomo Yanagida for pointing this out to us.
36
FIG. 11:
The eternally inflating multiverse generally has a very nontrivial spacetime structure,
with a multitude of eternally inflating regions connected by wormholes. This is reminiscent of a well known illustration of the inflationary multiverse by Andrei Linde, which we reproduce here by courtesy of the author (see also [28] and references therein). In the present context the links between the balloon shaped regions should be interpreted as wormholes which are created by supercritical bubbles or domain walls.
On the other hand, the bubble nucleation rate is highly model-dependent, and the parameter space yielding an appreciable density of black holes is likely to be increased. We note also that string theory suggests the existence of a vast landscape of vacua [25], so one can expect a large number of bubble types, some of them with relatively high nucleation rates. Super-critical black holes produced by vacuum bubbles have inflating vacuum regions inside. These regions will inflate eternally and will inevitably nucleate bubbles of the highenergy parent vacuum with the expansion rate Hi , as well as bubbles of all other vacua allowed by the underlying particle physics [2, 3, 12, 28]. In some of the bubbles inflation will end, and bubbles nucleated prior to the end of inflation will form black holes with inflating universes inside. A similar structure would arise within the supercritical black holes produced by domain walls. Indeed, inflating domain walls can excite any fields coupled to 37
them, even if the coupling is only gravitational. In particular, they can excite the inflaton field, causing it to jump back to the inflationary plateau [29, 30], or they may cause transitions to any of the accessible neighboring vacua [31]. We thus conclude that the eternally inflating multiverse generally has a very nontrivial spacetime structure, with a multitude of eternally inflating regions connected by wormholes.13 This is reminiscent of a well known illustration of the inflationary multiverse, by Andrei Linde which we reproduce in Fig. 11. In the present context the links between the balloon shaped regions should be interpreted as wormholes. We note that the mass distributions of black holes resulting from domain walls and from vacuum bubbles are expected to be different and can in principle be distinguished observationally. Black hole formation in the very early universe has been extensively studied in the literature, and a number of possible scenarios have been proposed (for a review see [18, 32]). Black holes could be formed at cosmological phase transitions or at the end of inflation. All these scenarios typically give a sharply peaked mass spectrum of black holes, with the characteristic mass comparable to the horizon mass at the relevant epoch. In contrast, our scenario predicts a black hole distribution with a very wide spectrum of masses, scanning many orders of magnitude.14 If a black hole population with the predicted mass spectrum is discovered, it could be regarded as evidence for inflation and for the existence of a multiverse. We would like to conclude with two comments concerning the global structure of supercritical black holes, which may be relevant to the so-called black hole information paradox (see Ref. [35] for a recent review). First of all, given that the black hole has a finite mass, the amount of information carried by Hawking radiation as the black hole evaporates into the asymptotic FRW universe will be finite. For that reason, it cannot possibly encode the quantum state of the infinite multiverse which develops beyond the black hole horizon. Second, the supercritical collapse leads to topology change, and to the formation of a baby universe which survives arbitrarily far into the future. In this context, it appears that unitary evolution should not be expected in the parent universe, once we trace over the degrees of freedom in the baby universes. Finally, we should stress that the formation of wormholes, 13 14
A similar spacetime structure was discussed in a different context by Kodama et al [7]. It has been recently suggested [33] that black holes with a relatively wide mass spectrum could be formed after hybrid inflation [34]. However, the spectrum does not typically span more than a few orders of magnitude, and its shape is rather different from that in our model.
38
leading to topology change and to formation of baby multiverses within black holes, does not require any exotic physics. It occurs by completely natural causes from regular initial conditions.15
VII.
ACKNOWLEDGEMENTS
This work is partially supported by MEC FPA2013-46570-C2-2-P, AGAUR 2014-SGR1474, CPAN CSD2007-00042 Consolider-Ingenio 2010 (JG) and by the National Science Foundation (AV and JZ). J.Z. was also supported by the Burlingame Fellowship at Tufts University. A.V. is grateful to Tsutomu Yanagida for a useful discussion, suggesting the possible relevance of our scenario for dark matter and to Abi Loeb for several useful discussions.
Appendix A: Evolution of test domain walls in FRW
In this appendix, we solve the evolution of test domain walls in a FRW universe numerically and find the numerical coefficient Cr and Cm defined in Sec. III. The metric for a FRW universe is ds2 = a2 (η)(−dη 2 + dr2 + r2 dΩ2 ), where η is the conformal time. The scale factor is given by p η , a(η) = ηi 15
(A1)
(A2)
This does not contradict the result of Farhi and Guth [36], that a baby universe cannot be created by classical evolution in the laboratory. That result follows from the existence of an anti-trapped surface in the inflating region of the baby universe, the null energy condition, and the existence of a non-compact Cauchy surface. With these assumptions, Penrose’s theorem [38] implies that some geodesics are past incomplete. This is a problem if our laboratory is embedded in an asymptotically Minkowski space, which is geodesically complete. However, it is not a problem in a cosmological setting which results from an inflationary phase. Inflation itself is geodesically incomplete to the past [37], if we require the inflationary Hubble rate to be bounded below. In the present context, such geodesic incompleteness is not related to a pathology of baby universes, but it is just a feature of the inflationary phase that precedes them. It should also be noted that the infinite Cauchy surface in the FRW universe of interest is not necessarily a global Cauchy surface. For example, the spacetime to the past of the end-of-inflation surface t = ti could be described by a closed chart of de Sitter space, in which case all global Cauchy surfaces would be compact and Penrose’s theorem would not be applicable.
39
where p = 2 for the dust dominated background and p = 1 for the radiation dominated background. We choose ηi = p, so that Hi = 1. The domain wall worldsheet is parametrized by η, and the worldsheet metric can be written as ds23 = a2 (η)
−1 + r02 dη 2 + r2 dΩ2 .
(A3)
In this Appendix, prime indicates a derivative with respect to the conformal time η. The domain wall action is proportional to the worldsheet area, Z √ S = 4πσ dτ 1 − r02 a3 r2 ,
(A4)
and the equation of motion is a0 2 r00 + 3 r0 (1 − r02 ) + (1 − r02 ) = 0. a r
(A5)
We solve this equation with the initial conditions r(ηi ) = Ri ,
r0 (ηi ) = 0.
(A6)
The solutions are plotted in Fig. 5 in terms of the cosmological time t. We regard a black hole as formed when the radial coordinate drops to r < 10−4 ti and determine its mass from Eq. (53), which in terms of the conformal time η takes the form a2 r 2 M (η) = 4πσ √ . 1 − r02
(A7)
This mass grows while the wall is being stretched by Hubble expansion, but remains nearly constant during the late stages of the collapse. According to Sec. III, the numerical coefficients Cm and Cr are defined as Cm, r =
p2 Mbh , 2 (1 + p) 4πσt2H
(A8)
where the Hubble-crossing time tH satisfies Ri a(tH ) =
1+p tH , p
(A9)
The dependence of Cm,r on Hi Ri = Ri is shown in Fig. 12. We see that the coefficients approach the values Cm ' 0.15 and Cr ' 0.62 for Hi Ri > 10. 40
Cm 0.5
Cr 1.4
0.4
1.2 1.0
0.3 0.8 0.2
0.6 0.4
0.1 0.2 0.0
0
10
20
30
40
50
60
Hi Ri 0.0
0
10
20
30
40
50
60
Hi Ri
FIG. 12: The dependence of Cm and Cr on Hi Ri = Ri . For Hi Ri > 10, we find Cm ' 0.15 and Cr ' 0.62. Appendix B: Large domain wall in dust cosmology
In this Appendix we describe in more detail the construction of the solution represented in Fig. 6, which corresponds to a large domain wall embedded in a dust cosmology. The gravitational field of the wall creates a wormhole structure leading to a baby universe. We shall start by cosidering the matching of Schwarzschild to a dust cosmology, and then the matching of two Schwarzschild metrics accross a domain wall. Finally, the different pieces are put together.
1.
Matching Schwarzschild to a dust cosmology
Consider the Schwarzschild metric −1 2GM 2GM 2 2 dT + 1 − dR2 + R2 dΩ2 . ds = − 1 − R R
(B1)
In Lemaitre coordinates, this takes the form ds2 = −dtˆ2 +
2GM 2 dρ + R2 dΩ2 . R
where tˆ and ρ are defined by the relations r −1 2GM 2GM dtˆ = ±dT − 1− dR, R R r −1 R 2GM dρ = ∓dT + 1− dR. 2GM R 41
(B2)
(B3) (B4)
Adding (B3) and (B4), we have
3 R = (tˆ + ρ) 2
2/3
(2GM )1/3 ,
(B5)
where an integration constant has been absorbed by a shift in the origin of the ρ coordinate. The expression for T as a function of tˆ and ρ can be found from (B4) as: ! r 3/2 r R R R 1 + − tanh−1 − ρ± , ± T = 4GM 3 2GM 2GM 2GM
(B6)
with R given in (B5). A second integration constant has been absorbed by a shift in the T coordinate. Since the metric (B2) is synchronous, the lines of constant spatial coordinate are geodesics, and tˆ is the proper time along them. The double sign in front of T corresponds to two possible choices of geodesic coordinates ρ± . We shall consider geodesics which start from the singularity at R = 0 at the bottom of the Kruskal diagram, see Fig. 13. In the region R < 2GM the T coordinate may grow or decrease depending on which spatial direction we go, while tˆ and R both grow towards the future. A sphere of test particles at ρ = ρ0 has the physical radius given by R ∝ (tˆ + ρ0 )2/3 .
(B7)
Thus, co-moving spheres with different values of the ρ coordinate have a similar behaviour, with a shifted value of proper time. The metric (B2) can be matched to a flat FRW metric ds2 = −dt2 + a2 (t)(dr2 + r2 dΩ2 ).
(B8)
The co-moving sphere with ρ = ρ0 will be glued to a co-moving sphere with r = r0 . Continuity of the temporal and angular components of the metric accross that surface requires R = a(t) r0 ,
(B9)
a(t) ∝ t2/3 ,
(B10)
with t = tˆ + ρ0 , and
corresponding to a dust cosmology. Such cosmology has a matter density given by ρm = 1/(6πGt2 ). Using (B5) with ρ = ρ0 , it is then straightforward to show that M=
4π ρm R3 . 3 42
(B11)
t c l w see i n a at e , t in . m th iona t ith ⌧ t the all goes in s w i t , ⌧ s e sh l l t ed t sm fl ne ita ild R fall t2 / cas we wa cys o ch ly . O rav co l s l s he nt i zs a ll ⌧ al g ⇠ r i t h A t Tw 1 , t t t s a e wa r e w GM In te. of mna u p a w l si v . h s o c r f a e H al l d . ra io re s S th 2 a h pu , n er t ⇠ n w s e e n = t on t e r l e of i i n su c r e a v i S o i i h c n g l is h w s R R ns e i 1 ded of the in is un cale ma al ll it s ch 1) a th lly, diu ere xpa th lack d o t ( s w d d a ! b e mi c e t o w n, te al In t u a r a w h t e y i n a b m n , T em n a d u i n r i zo e na gic the 1 ai a l d. ven e of it, bien all th i h M al l d y l e e t r E ol l i m m gi n w i m o m l o of pe s 2G w the -sca h no . a i om Sp re do e h do mo (2)ld us ig ion ck site he or left n to c d = e l i h s s l e r e R t he a l a e, R ma t t i m h di be ns bla po n t was re gy te co fi = t p a n pa ri ic , t h i n t. e w l s e a 1 do van on at he nal lo 2 . rs t ca ex s a e o th ich we h r di C 0 R i fi c e t l t m i o l ! t t al go o a ⌧ e a h t is ti ld al rm th er w e an ↵er on = us h l d , T eric re re elera ⌧ ls w3) /t se, e sh all m 2 d⌦ fie gic d fo in fast st ers fla ne ita s t d t ent side 2GSp 2 et l a M p h s a cc ( l o R e o u d w cy v c a v o l R w i t n f M ⇠ he t i r l, a c 2Gm s le lly O ra s i v o a s t e ws d n + al ⌧ all ⇠ this ) As the ant ul osm ion ere gro The u co F (G ot me-1 a sp , T gy st 1 R2 GM tia all. e g = r aai sca the w t p o f 4 h o e c W w n (. n d e n is q u nfo r 2 sp a w si v re y ens int ze e. R ) 1 er sca he !0 R sid ol ai or e GM In te r o m F ick rm R a l du M◆ he b t ly si ol . is les ric=a 1on tim her wa sm , f n t h = 2 t . r a er i o e r e om = a ch epu th arR l C R nt ot 1 . l , t hed its ari its mh ient e o d t h H l y al l y ⌧ o n s n G c l r ⇢ n u r lt i tc er im ile o b e e d e= io nt o g l e e m er e sh r s t , he ca h e S ac re o↵mer th2e ) st l i ✓ 2 R 1 ed i of 2s . the ⌧ stnre und e pr wh a w am s , w s R R ans he i k h l ) hi n ce ledvi ai GM ft u a ads at , w ink tret the i 5 o G e f d d d a d l er a n n ( , R l y k s u b i t o t h (c . th lly diu ere xp t lac sp bab to ter e w s u che re a m ect at at nat ⇠wa orT al in isll d on n w 2 + 1 ! bed micRs2 ed⌦to h e y t h ar i l l n d d p u e In tua f ra wh nt e y in a b io tto lFl r! m m r e w t un ati d i cuu e↵ to a2 + du n T m l , e o c m nwall , n a . i n an B rica uni e fo oun be er i by lsiv 1 n tSpherical e th efmo y sh wdomain l ◆ , Th e T e in dust cosmology o r a M d M all dyd⇢ le th im e on bkl 1 e, r a oros , a of v th d efo l r ve rm d pr ts cos e ild ed◆ E1 ve dRle mit bi ina ith . r e d h r f r c c s r i fi e a e m t e i i e M e t c m 2G -sc yqnui ddHe tt he rs o of ide . h o l am i g w du re w gio se, atio t w mar ens mol eld 2G w hGe sca 1. no . us na zs td rico al am edma o t ive on lo ns gy ig Mn k te . e or ft 1 st e n o an n hil ily ion og ca R = m in o t2 eRisc ordi e ✓ e ar ics e in t un egi cog olo e n2sGio Rlac ◆ os1i dRth as e le Co r a t 2t + im d co co f o d of e i in a ica n l R m d w b o o s h M f m t y ue Tt o=ad a1y l r we m 1 c sm n s f i n a t s t e n d R l = h ns 0b T = 1 da st pp han h w ar n ✓ pa a b oM at 1 hedo van d⌧on ⇢ id ol ide vac th wo siz res fo exp e ig Sc crhiilc t lef su babsp ica re cos c e fir ca ex 1 suniverse. er ch og r e r a ospherical in dominated t a r Consider 2eG Rr t Two le= matter ti s !kescat l rflat hi e w he h th a ch hearti domain zes =a spatially l fo ust wallingembedded m M e t d a 2 t y. th uum am mh e gr ed ms nsi nor th e r u in e 2 r l a w i e One e sare a a Ball d f es 2iGc fo ✓ iscale ed s such e e n stt ⇠t H r1s, h dynamics s ler is the , T taa rwall. n a o time-scales g bi ole ow idi↵erent to cthe In re p w relevant dof s ficosmological Sc e↵ . hw ph l et i n o en . T s t h b l n . . I t al dya fl Mthisphe s ar dcce tion ve log nRd ed 1s fa dus ive Le atime-scale hw Sch ec Ev nis thepuacceleration ScS ac other l. at faande othe l, t g o s due to the repulsive gravitational field of the domain wall n G n t l i l i a a h 3 , 2 m e s e w R t a r k , s s F a s e / 2 M u ar l l l iv On m w ⇢, s m o 1 wa o ai g1y RW e d ter pp h 1ent thi of th e zs ar 1 dthRe ) = ateer a -sca the e re epu cosdT sion stere GgrM Th W tr os )ole . ua s ust ⇠t (G lo A gr e.1is att der 1 th ch he n e 2 z c n t ± . h i M ◆ i h e t o R R n i l r s e t u n a a e s e ild i a co z le F 3 wall i y t for y R e of t l,y, these repulsive G a si r vi1 tfield ch ni wh For n he 0 rd sid In be ignored. ⌧ do te ⌧ m R M M i ft i n o de y t +s, sithehodomain r case, or im this ta hoen dcan ve ic n thR li⌧ b the dt b= t 2/ (2 omis ft i as: oss m m rad wh , f re ild m = cooon Eventually, di ds 2 tioC co om ri⌥l dTiwithin rs h or cosmological t rm the et d it hi 3) c e⌧ wall t t othe 1 n. ed when e itstretched its afalls en horizon, ai iu en by , ⌧ conformally expansion. the 2G 2Ga sh e ge na C i e e , n s h i r R w n d 1 s a e r ( t n r n m i an = al t etc de rim ile 2 wo mb we as m wh s R th t ⌧ o te at ait ⇢) th re hole a m st ✓ c e ofd) efiradius ol forms wa quickly y Rar e bi eunder = d s, e shrinks its tension and a black R = 2GM ⇠ t /t ⌧ t. h a h fi h e r o + o S o a p S n u ↵ d f y ar ri en re t Gre ✓ d b ate n ⌧ st u e ⇢ w f a e 3 ( ⌧ o eld gi em ⇢ i ll, th b fr d = wa R du 1 gi will e we un dopposite ta 1 cLain the e t ar R . d a d i t d s b ( h t o R l d Here, be primarily interested limit, where R t . In this case, the wall repels in ⌧ 2 s b i l y n l ef a e x et ofIn l st e G l i 2G ta an d⇠⇢ or all ink ill d one n t um= ↵e2c rbe foun o a R or rd de t lly pa t M fal t ca vers than th sfaster 2bs+ oo ke co M ◆ r expansion w un tirate. n the theus matter it while its grows ambient wiaround v i AscuRwe eshall sosee,e thiss ls size t F fin n , m l e t a c a e h a s i e M ⇠ 2 . fi r R s sm t n t io I o mae h nd va he w th d t . ⌧ ed th o y s we in ab n b into d G ◆ ndT al 2 ⇠ rwhich leadsrsto t the hformation a of of the e Tw was t wall e r winterior l e, r ar the a he n r ofn tah twormhole. nf originally ol n cgoes d⇢ /t ithi e d Theomdust a hil/2 + 2 M bee ati dT by e di o k , o M fo e H a r , f b a e o n r p o 2 c i e c o 3 c◆in ac inwh e f remnant s s sc n t is i lefte with ⇢ ds 2 la and rm b cysted gy a baby universe, te ae black th universe we are 4) s hole ⌧ th mFRW + ✓ 2G R us ck ter ine. thecaambient 2Gha nt a wa 1, e qu H at thm ( iver n o nsid gyh.as and rzs ri i o A s e t. e h i n s = re m toro n gio co lot ⌧ ll a R met ✓ tant nsRta orm 1 spherical region vacuum. ro s th of hof ,t wa o o l + a 2 f e l w r w✓ d M e 2G e d⌧ d⌧ 2 o n of Schwarzschild tio f f t e he s ) u re matching ll ns1 co m th addiscuss re the G Before we consider e↵ect of theizodomain wall, let us first e 0h ns l we msta n of M ◆ (3 y the n, i s d⌧ at co of m he sha wa + 2 = Sc @1 c3hil2 th R lieng bab rica re ccoosn ctio (4 ch na wa ll ll it n es G and a dust cosmology. s s o g 1 r ), d⇢ t i o stn un n M in = e nt ll se ep = z n e ± t f c 2 i 2 a r k l a g h e we M o f R i dT s ra cy go e, t els dR s ta of grs e = d⇢ 2 sp B a rdauti s a ch 4Ghwa ds bt ha st es h i 2 ns te d th is Sc t =Sc ed in to s a dust cosmology Su ve i n u s, io eg a +1.1 Matching Schwarzschild ⌥ h + a 2G d t h t t t T dT T d wa R2 in o R2 o , la s n M ±the anin or M ✓ rz es d⌦ + d⌦ re a co on R an f dT 2 2 r e se chr at ch Consider the Schwarzschild smetric e e sion .1 ± h . n . r n i 1 i d t R A e es ld i y 1 d 2 s h s . y r R 2G G ✓ ✓ ◆ ✓ ◆ r = ) s o b M as on = (5 ) i 2GM 2GM 1w exp2 M ◆ co (1) 3 ed ⌥ 1 R be ds2 = 1 dT 2 + 1 dR + R2 d⌦C2 . in (2 e d⌧ n r c ( e 2 1 2 n fi R R i t nd n ⌧ G e e = ai + M ◆ d i v et r ⇢ ab d g m ⇢) 2/ e R R ca s o R he m , d⇢ 3 In Lemaitre coordinates, this takes the form ar 1 Le n rb (1 ⇢ b e ed n i t h ce t I (2 d ) d w ◆ R fo by G v n . 2GM 2 3/ un M an Si ha 2 a ds2 = d⌧ 2 + d⇢ + R2 d⌦2 . (2) ⌧ ) 1/ 3 e s df e R w r o sh i f r + , e (2 m t ), ) (3 in (4 wh where ⌧ and ⇢ are defined by the relations 2G R ) a th m nt s: e M ro s or ha ✓ ◆ )f ta ig (3 2GM 2GM 1 in nh (3 ta s b s ) d⌧ = ±dT 1 dR, (3) nt ee g 1 of in R R sp n a th n ct (4 µ s at b e a io ✓ ◆ ) r 2G R 1 i a so ⇢ at l c rb R 2GM 1 bt r M A co u g o o ed a d⇢ = ⌥dT + 1 dR. (4) or S te rd b 2GM R di in r T ⇢, in y a na at (5 n fo t µν µ;ν s e a e ) . ar hift Subtracting (??) from (??), we have e sion r T e e s he ge i n o d th wh p r e 2/3 3 x es e T 1/3 e i cs R = (⌧ + ⇢) (2GM ) , (5) , a co (6 2 ) n d or d ⌧ i na n αβ by a shift in the origin of the ⇢ coordinate. The where an integration constant αβ has been absorbed is te th . t expression for T as a function of ⌧ and ⇢ can be found from (3) as: e wi −1 1/2 0 1 s s S ✓ ◆ n r ρ 3/2 1 R R R A ±T = 4GM @ + tanh 1 ⇢, (6)
o
ic C R al an ↵er on = s do t d t ent ide 2G M ⇠ he t i r m a m , co F (G ot e- s T ai p h o n n q u fo r ) 1 er sca he ! R 1 r r ick m w . is les ica ⌧ al a l R t th H y lly t h e ar e l d = li s e e l ea m r e h r st , o a r n cc ele ma 2G a ds at , w ink ret th e M v i du l er a n sp bab to ter e w s u che e re n , w a he y t h a i l nd d p st tio t t al T ! l r u o r n an B ica uni e fo oun be er i by lsiv t h l em co t d efo l r ve rm d pr ts co e im e 1 sm i a e r fi t s i e- d y b ed du re w gio se, atio t w ma ens mo eld 1. sc n a d e ⇢ = ol 1 al m d st e n o an n hi rily ion log c og e Co ica an i cs i n 0 d of le c c f i d o o a M sm ns of in a its nte nd l ue o ns y be a e at w f i v id ol de ac th o siz res fo xp ig t o su sp er ch og r r e t a r th ch ati th y. th uum am mh e gr ed ms nsi nor in e a a e e g In bi ole ow in a on ed. . re w lly S e . t e . b Sc ↵e s ch p Le h n l ul all. fla t Th fa e o ac Ev In wa hw m 1 ct si v O t F R e st k e t p ai r o n h e h zs ar e n e ma f t 1 W du r po o tu is tr S c gr i s t t l t s e s z a e h h ph e wh u n t w h a i t e o l l y c as ild Rsc av t er co fr , do S i v h n hi e or er i t a he d l e i , = m w t m i a e r e m di d l p c h f tio co om d h et 0 d s2 R ic ai h rse h w e it, iu en or ⌧ na d r C R a n a sm i n t a i n i an e = a ↵✓e con =o = wh s R t t te m w a l r l fi ol ate wa i d e s o bi e S he ⌧ s, 0 n d r e si d 2 a d c d r C a n ⇢ t e R G o t l e l d ogi d r i e R d w l e r a n t = ↵ hi i t o , M m 1 ar c u ⇠ he timan r a er ,uns = al st l e t l d e l e gi n t e R 2 ⌧ e s of al n i v l e 2 a G d ( i 2 T a de co ak F G otGht e- t spnt t!dce G sc e in us o ft w lly xpa t M fal t t o s M n h r h M ⇠ h e fin Su o qu fo s r R ) R1 er ◆cal e ertim 1 e al e r s e . w fir ma ith in nsio . I ⇠ ls w , th ed bt ick rmth co . F is(G es ot ica e sams , T d a l st i n a t h n n t 2 i t e om t ⇠ T ra by th H ly allqe f ⌧nfo or dTthe) arheer l -dsc R o=plh ! ct ai H wo al o2egr 1 l indis w bl e in rat this /t hin dom in te h e m erdes 2shr yusioctrkmt ,rm R 2+ac1. ✓ reis om n l e i c a ca us a ck ter e. c a ⌧ t h e ai g ade a , wt = in rely tha ⌧ ce levt asi a GyM wa 1, R d a l l h k n t s i o A se . (3 n ler ane re , d = l s rte tte hee Hs e tchs ely wh u h l 1 h t b ll s t r , r ) l w T w h o s a l u s o o a t h w o a r r e e a e s ex er d⌧ph by tthi aad mildl⌧ 2 ned, drin ptur , t t2ioG tocc arllel !m 2G fro r i z al t e m in le of we the a o a r e l a e pr e e b h n t e w c r l e on i s k en so tb m r va in M os at du em he sha wa es an an B =ica usnpiv bfaso utno tee+rp r2eitw ys cu sticvhe eRr Mtim◆thleer em ,i (4 si o i sit ons efid ep e dat bnet 1wa , T mchi st na wa ll ll r d efo l r heer brym dthi raimGM t ) d y 1 d l m n d t i e , ⇢ ± e l a e r t u n ! a s s ⇢ n te re d gi rie, uti e w Roa bn eor bld ls sc noan doe l= n l e o l l e w t p g d y a 1. u s c l e l a n fo gr o t g c r o m f d e B e o e e T i u i w o c s a , h n d h o i l tii st d e enf o lan ivne o rimle nlddy⇢ 2 opnr itgsic ccaon ve R 2e r T at goyf S os yst oes th ls ha 1 = 1 ie m i c o rd w Cov an tael smb fie + due cms eo n day 2bGed ch m ed int is M 1⌥d cosma douns rfe o2fGegiion rsfea, w aittis tn+twer im -fsc ns Md ⇢ as ion n d e e l nes a o Si i t h d sau apm e wa ol in o = a t .1 T 2 to id ols stide wevM ac n✓tohe anor onsize hReist2 rifloyr sxiopna loiggn R a co nc R d a c ⇢ c l t e + o t rz og a ld er C ch fu ns e gy cR ins nasnio icoare and⌦ 2 he h dau iicasll =in2 0 ors th counum f oamdminho of agr ee⌦idt i2 m sc y nc t a thon in M th giv . l t b d o r y w f a n hi o e . m . s n G a b l s e e s d e . n e w t 1 . e en i o t In ld Scide g Sa2tGc R ✓ oloe↵ ide 2 vacien the.e Twors f sitzhe rebsla f.oE expIn e ig pul atlol. f flsua M s ± m R n p h e t t r t hwr a c Le s r G a c u v e T h m a n c c et i n h g O a o e M i k m t o = n s hwin e t h F u t a t v m o t t y e d p h M t g h n h f⌧ s m n i ric (5 s = h h e a m . Rof e ◆m RW d o er ropo iho s tu iois re 3 rz e ba gr e er is aatwt ally 4G sc Sc argz 1S 2 th e↵ 1. ubieust le. th wssit n tle oa ball n.ca d. a e iItn (2 ). A hi h an f e he f lay, Ese I avit eputh aerll d fl n e G Mw 0 nd enre Lce 2 (⌧ + s ) T w e c n t l c e . o at h M ◆ docdt d wa iv F hi he t aslim o ra ck w ve, f n i s se ⇢ abomo a h@e 1 ✓ m er R ch dhe te it ppdi hhe n or th atio lsivco O ⇢) d 2/ m rz hilRw mRof sy co ca sordit r se W w u a r , o us o n tu t is et sc d a r na e sm neina ma ai, t s 23 nc nd 1 n rbinrae 3 e⌧ w gr ol iste tt r l n t s m t s z t a l a w e h h i ⌧ c R h i t h h o w = c b e hr ( l s R s u t ild wae e n o w bi an er e S of e ly as 1) fie avog td er ✓ (2 e d tceos o 2Gahned ◆ achdR on int ar iv ri h en e li = r w , wR e, ld it ic heu d fo by, rt ll,do G d m diuld. e er ginic t th R m 2 adall h ⌧ f ou egr M re⇢ 3/ m un M 1s 2 hdin of atai l s cnoiv om e l lef se a h ex e it, G iu f e or e a 2 ⌧ a t a s at s s, at t 1 rae d ) r t on c s er i a 2 i t M n w p l t tco t w ly a a am t w s ll /3= G ns ic u t th ion fro sh taeks n+dde s , wi e i s ns b . h ⇠ R s th t ⌧ he alale mose. nat wa osa , R M ✓◆ ift e,s n e w fi e h t fi e i m o ( i I S o h ar thri nen2 n re t 2 = it w e R domfiet ⇠ log T ed rs ll ⇢ne Su md l i n co t, a e e gi r t) th R /t hin a do 1d ard R (3 intthhies ns ai ld H icawo un bt oul s e d l 2 l ⌧ blle inna at ex is 2 e e ) s n of 1 l b i T t ra otg sct G dy 22G acft te ll e. p ca ⌧GMthl famai t as the tfoarkm of an wa t , sc iver ct ✓ d u u M e t y e c + k M y r s a 2 n t : l t t s s h e s fi w Suin A i l c o . ◆ l l h e al e s e n s , o e h o s s e n h i o h R ⇠ w i fi s . , g r s s = n r o t t t o e t n bt ( 1 ig th m do t . hr(es adrnela leh ofth wioe n thIen t 2 rizwit allhe st as i wh t ra 3) e 2 3 n o a o m ⇠ Tw b a d b s t / d h h r m t t i G e d ) n d s w d h f s r e l n y T c i ⌧ t e h f i 2 o ai H o 2 M og 1on ex er , in om mbl e in haat alis aist tinrom t en ⌧ ds +2 of rm t ◆ ✓ n w a n l ⌧ t l pr e a e c h s s t t R + l 2 y u c c h a t pa a = e . g ( h r a a e G wa 1, k a h s e es w n s n r i ( l i 1 nsg (3 4) 4) t h l igo eeA, s psee t. e h n si o h i n 1= M e ⇢ d⇢ t i a b so ± 1 2GrelR ll t r c , ) , l o w o R or hofe ylse eos thw s th r d l dR2 ex n efr te fro we d⌧ d2⇢ c s MatioA i z al l te f G 2 ies = c bde⌧ T S e g e o
ol
at
III"
IV"
II"
I _"
+"
b"
Sp
he
r
a"
=
0
1
FIG. 13: Lemaitre coordinates in the extended Schwarzschild diagram. The left panel shows the di
R
curves ρ = 0 for the two possible choices of the double sign in Eq.(B6). In the left panel, we adopt the lower sign, indicated by −, which corresponds to radial geodesics traveling towards region II from the white hole singularty at R = 0, for different values of ρ. Note that such geodesic coordinates only cover half or the extended Schwarzschild solution.
Therefore, the mass parameter M in the Schwarzschild solution corresponds to the total
mass of dust that would be contained within the sphere of radius R, where the two solutions
are matched. The relation (B11) can also be derived by imposing that the dust particle at the edge of the FRW metric should be a geodesic of both FRW and Schwarzschild. This is explained in Subsection III A 2.
Since the normal n to the the matching hypersurface ρ = const. or r = const. has only
radial component, and the metric is diagonal, the extrinsic curvature K
= n
takes a
simple form given by
K
where ∂ = a ∂ = (R/2GM )
1 = ∂ g 2
(B12)
∂ is the normal derivative, and α and β run over the 3
2GM
2GM
2GM
temporal and angular coordinates. It is straightforward to check that Kαβ has only angular with R given in (??). A second integration constant has been absorbed by a shift in the T coordinate.
components, and thatSince these are(??) theis synchronous, same on the both sides ofspatial the coordinate matching hypersurface: the metric lines of constant are geodesics, and ⌧ is the KΩΩ0 = gΩΩ0 /R.
(B13)
Here Ω and Ω0 run over the angular coordinates. The continuity of the metric and of the extrinsic curvature means that there is no distributional source at the junction. The matching of Schwarzshild and FRW metrics can be done in two different ways, which are represented in Fig. 14. An expanding ball of matter may be embedded in a Schwarzschild 43
M l dinu us pche le evv1 w thrke et hmhee lea a bm ds rte, atteri⇠ a inn M w std t sdp abat o w r nkes(wG estuconthcheeedr -srceap l leeriarcaat anntt R waa ,,TT! t h i i e s ! e = h l t a t r t l o y d o l e u t l s u h l e e F r i e b ) l t b n w cuoscto p o i e l d 2 1 o d o a o r s e n b r t e o s c u sp a abB ic t n oan fo illrunR de . r bi yutclh iv re n toi m thhe GemM sm scm erpr tys osseiv ae rel tim ained mbeb, 11 hned ye alhqe iverfoorrm b d ⌧ rica fuorn reugfoic rs unmaa e pit tiitms tceon me ccfieel evamee-s dywyn dTedd !⇢ ⇢ olosom e o t n l c a fi k e = d e d an1. Be adl u ivwe iornmly, ad iloy riwmh ,atrtei ssimo lo el lerca t-stcal nm l = d l og logolo 1 d1 f rs ersH oat snhdrit nwostreaile lhnyse nolo gicd antio oaelted am ices meidn 12 y y gy b h e i e i i n G c c i r t 2 a f r a i ⇢ f Co a d More etgtch i c n u n r n e e e c , g o i o o i i o a a e l b n o G h e d s h o k p i o = d t a t t l e M , i n d n a fwn n si yse e u d c e n e im u y f o dsepad Sp 1. ns us a wlee snmm ufc s a si ⇢M 2G 1 ide t tcc acds oolfoatsitdneed iveacowufiatlheluenwidtorem idsniztebyreasntedlsifvoarelmexpanbeign e-scateo tnohasm R p he n t h u M ⇢ i r o o c = s i x a r o fi c l Co = fio s eapladsio ignred le detrh ascwho allytaia sc = ⇢ M r th oahsimbnabnstido gtyfh. va tahnreotuhum.wboaemrpb shiziotl s rtgersotwosdm ri n r o p f n u o 0 a n o e l n e c m . c = e a n e g s e e s e a e e l r l di C R putr ll wucfla y stiaon 2 at S ph oloSy uer fcou 1e↵enda mieinm .gT nssdi thosgi blasiao.nE redI ca In s i d t o ↵ h . 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rs c F w an d Su h i s a M 2 o f t e a h e o i h e ↵ , / o t a w . l m r ed , t n e o a c s a p d e G b e h e = R o t e u u H , w l f l b a e l l e n ot ) n i w m t i n o t t r a r t e r h a o s l d a t l bt r dauoin disa w eblay inoraingat ti.s e ⇠⌧an n tBweoM nis iw n fordohu1en t ⇠e p Tit ayinc siv 1, lseids str2icG 1lo mctah⇠ by qhuis for 1Re rfoz2scn1.d r rdiesTnz2sc tim h ⇢ ra I v l e i s d x s d m t e n cull it cren e iino.a Icna pa t 2 the.ifto ai er t 2 e iwthm , do Hr wso ow ickt dm Rgy t⌧rmGhilt t + h✓eiarl coaM m a ct w a L s l r t fi k r a n m i a a e fi h t d s t i 2 h i s r l s a t l l en A / m r m n d h e i s a a n l h t a in rastll, s ti ed mat d = eH lyke =lly ⇠t M e◆m hteim rdea t sm t e, ll titohin wain1a ens o eld s a hoelhfteorwo lrysa1w.i1net,htiss.iotndu hIonirenwtthwgaioo/m g n l ree 1e rseo , oTm 2G e i h r R i l , ( ⌧ e w h e s n o t s e a a h h w ⌧ t i i i o b a s l n ( i i n t d t e G i t r e z h f n o r th ic c-cs 2Gpleh aol! fin l e e c rlaet htrcho dsdtFo2 , 3) wh isw umsd lac re Ctetthh aee. s heecwia rtatcte oen ces lciasofin ntd. o haolrille lyall n gic can a M d t d e c m o i s c e i r d o e n ⌧ v i i i e f fro o. . , ionhoseo, w in a ziot ntis an al nas uas atfiurs k mnornioe w bhlAM r ex S er e la M ear ugn1 da a⌧d atord, ownqu nekfo 2+ertc=R h)eT ✓ m sl t chi tt d ho ansird oa aaslclkaws telle,rrteiho smA prub a st si ri ftvhal bby s=t ted inesa ic s rum h2G r1.2e+ is✓ Relreas ◆inctal sytwRca , T w sn, er d e be l t l t o ⌧ a t i ! 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W b u ol r owo inol au ons e 4 f f ) f ct ◆ . ) r t S e r h o t a y t ( e h o d n s G ⌧ o h l 0 he s ncht einn he m R 1i2n) A nes M de + nd enree Lin 2 (⌧ e nt f t clhliche ezeseSc p2eG zonthelel↵ies 1 unienst we. tThan ssite theeof ballaly, .caEs he = ro in M 2 w e r g r s s wh@ cdt s 1TG ropn= taebg re iv t F hi he t faslim o ra ck w mg ( ⇢ (3) a itsh2eG1 se ⇢ abcmomoaga o + ⇢ em(c4y)he ldowmeahra , cthhhai Rlw lsM ◆ , itdom aot u srat ela2t G R ic+ er R ch dhe te it ppdi hhe 1 ✓ M s e d R c 2 o ( l s c s r l c s l w c s o , t ) s (43) re an w n f r R d z i o M / 6 a l d se W w u a r , o u s o d y ⌧ i o iasl±,d⌧ or ion io M ARan 2 o an r cithnr S oo) : dnRriog◆n ai, f t e s a2 3n tedal etirsnc se s rterp d⇢ 1 3 ss ) r n b b ⌧ e i n t f R z c h d i we u as st mbthwh sit R le 2 2 d h h c r l , n i + d h a ro t d d n t o c c w = e c e b e G h 2 ion exint er s d s h e R s e T e i o t a c i o g w w oo =lid ns ⇢ 2nGa ro+ (1 inof e d geos o w (2 cy i✓n oe ld , t aclshdR in ⌧M m 2G hned ◆ a = e e d arniv oriwh ienan ere e lSi n e M t p b o f , l ⇢ a r t s t a t G n h b t l, om ou y rtfhd rz M dst a s m is dil . re⇢ 3 di ±es2yo ant fo re gr an ha(4 e er ginic t th R m , isRt+ Re. anh Ru5)2 eg he te o M d 1 d u / i e d e l S n r T ss at l ef se a h e x e s i 2 . ⌥d⇢ nd a snc sc ) 1 d 2 nt tr et ai 1 Rs,d2 rat ⇢ c ⌧ar 2 he d 2 ve), st t atdTGafMsco s has a ionio int h t w llywa pa am h e / o ⌦ n i t a s w o 1 ⇢ T = 2 G u i d 2 io s c t 3 w o T = s e a hinf✓st a t n f R o i n + b wi e i s n s c d h e s ⌦ eh ✓ r as fn eg l h (5)o if ksea, r d , Ma ◆ he rdi +defi wa 2 een osa . 2n R + a e + fi t th a n tor io R r m n s c 2 n o R t r r z e . ⇢ c S a r on at n t av G 2 m t l a l i s R A a t ⌥ s d l i o n t h u 2 e G s t i M , h n b g d are heigi c 1 e a ( p f u T st i o n n d R 1 e u b o i s . d 3) es hi es⌦ 2 st dile th ✓ atia orb d⌦ tr dT 2 s eo d M blle in 2GrT lost TR e b nc as a n f t l t t ⇢ s 2 2 o 2 y a a d 2 e acft h G ✓ cuu of. n , ed gyc R es G T l c ct dsas2: e arkm G + ✓ tio ant co M hdeefith . + kw M t o R s s e M i b S i M c o o ◆ s n c o ( y s ◆ n e n hoi u n r o h h f R ±T sm = rig th din 6)a = tfihrs R teadre ns as btg ( R s,1 a c2oo 1 1 R o un as sta s e 3 a
m -sc yqn dd du re w gio se, atio t w mar ens mol eld s 2 R z ft ri al am edma o t ive onolo nsi gy. ig n k te . e o st e n o an n hil ily ion og ca sc = main to t2 eRetd ✓ e ar ics e in t un egi cog olo e n2sGio Rlac ◆ os1i dRth as e le r di m c o c o f o d of e i i n a i c a n + l R m d i w 2 b o s h t p fom n ⌧ t r b y ue Tt o=ad a1y l r we m n t w 1 co sm n s f i n a t s t e n d R l = h b 0 T = 1 a a d s a p a l r at 1 thed va d on ⇢ id an ✓exp1 s a eGoM th ich e ol ide vac th wo siz res fo exp e ig Sc crhii c to lef su babsp ica re cos fi c er i e ch e r og r e l r a spherical in a spatially dominated t a r Consider !kesca rflat h ew th a ch heati domain zs fo st wallingembedded m universe. Ml e2l = rmatter at 2 = th2 eRr wTwo y. th uum am mh e gr ed ms nsi nor th us f t h in i e One e sare ahr e a Balel du esld 2iGca for ✓ iscale ed s such e e t n 1 astt ⇠st H h dynamics s ele is the , Ts taaerwall. n a on time-scales g r1s, bi ole ow idi↵erent o to cthe w re p w relevant dof s ficosmological ya e d r e g R Sc e↵ . . h p l f t . . n b e t h a d S i s p c n a u c io ve lo n ed s d Mth p s nt T f he la E In iv 3 , ll, lnl.d fla atime-scale hw uacceleration ec SS cgravitational ch gi t o s and the other is the due to the repulsive field of the domain wall n G a t i l i c a a h v t e s e o w R t a r k , s s F a s e / 2 s u ar l l l iv On m M th w 1y m o 1 wa o RW e d ter pp h 1ent thi of e zs ar ) = ateer a -sca the e re epu cosdT sion stere GgrM Th W og A e tr e 1 at er 1 os )ole . ua s ust ⇠t (G l g th ch h n e 2 z i c . n t r ± t . h i M i h o R i r y en i n i ze l e F R e s un t w an te o lly as av 1 s t seidr sc t e ild co G a f t , , thee repulsive 0 ordnsid In R do e t b⌧ hoen dcan be ignored. m thisi case, y for t ⌧ t +, sthehodomain m R M ita field hi iv hiFort R li⌧ or (2 omis t in as: ss t 2/3 wall m er ch he mi rad wh , fo o o t ti her d b th ed =its rilydT ts om m = ld di ds 2 f t n c o C c i e d i ) e o s m i t e a i ⌧ o t tr r cosmological , conformally stretched by the wall afalls within the itsh (3 c 1 2G na i horizon, u e , n o 1fin. e when R e C eEventually, in na sm inexpansion. to i = t etchd der rim ⌥ile 2 wor mb we was am wh s R n th t ⌧ te at aitr ⇢) the r hole a m st l and ✓ c e ol forms e radius wa quickly e b ) = s, e shrinks under its tension a black of R = 2GM ⇠ t /t ⌧ t. e h d a h fi h e r o + o S ar ori ien re S = w R ✓ d ⌧ st un e p⇢ w f a e a 3 (⌧ of eld gi edm i↵ t Gre ll, th by fr du tan du gi will e we un dopposite 1 cLain the e1 t be R primarily al .thectwall erepels d (alimit, d .it In this is d a s b h o R l d Here, interested where R t case, ⌧ 2 y n l e e o n t n s l l k l rb e l G i n ⇢ 2G m t al xp 2 ta l f v l t r I u a n e f b f s ⇠ n i d t u e M a a s+o oo o r t i ca rs than d the us matterw around n o ly a itt while its th faster o o e 2 ↵ = R ke f co efi i l n i M ◆ r o u b w n the size grows ambient expansion rate. As we shall see, this v l t F r m sh e e s s e e le e. ta a c st M i t i n n si . I ⇠ s w , t h ne fi R sm ou maetha nd vac Rhe d t o y ab n b into d ⌧ G ◆ ndT al o 2 he d ⇠ rwhich leadsrsto t the hformation ar the a of of the e Tw was t wall r winterior l e, w rin a the n r ofn tah twormhole. nf originally ol n cgoes /t ithi e d Theomdust a hil/2 + 2 M bee ati dT by e d o k , o fo e H a r , f ) se hole bl in at is n o og a baby universe, 2 i sc o wec areic lefter with 3 c◆in inwh e f remnant s sp ⇢ ds 2 rm be cysted te ae black th acand aic 4 ⌧ t mFRW universe t ine the ambient + ✓ us y 2Gha R nt wa 1, e k eri . A cas t he ain qu H at thm ( iver n o nsid gyh.as and rzs s = tr t ta re or s e , . o h h m e l n a 1 spherical region of of vacuum. o i o t R l o s rm ol or wa la r n eg he lot f ⌧ w✓ c mM ✓ s1ta con 2G e of we the d⌧ d⌧ 2 tio fo i zo l l s3t) f y u the he ddiscuss r matching o Schwarzschild e mstoan of h n f m d t re theth e↵ect G Before we consider of the domain wall, let us first e 0 l o c s ns l M i ( w 2 n w i 1 n at c o m e ha a + 2 ◆ ,i s = S @ c3h th R lienag bab ica re ccoosn tio ch na wa ll ll G t and a dust cosmology. s on es 1 re M in es nt dust ± ll se cosmology ct a her efo sotn func ing M rz 2ati = lin 1 Spherical domain wall in 2 = k p g a R e dT g u cy o , t el s s p B dati a dR s tr ta of gdrs the h 4Ghwa = d⇢ 2 s d e e b s c 2 h r s t s s a te Sc is =Sc Su i n u s, +1.1 Matching Schwarzschild ⌥ hi + d int to 2G dteg a at hw t a dust cosmology dT R2 in o R2 ,t s nd no anin for T M ±tThe ar s el a o o e d n + R =R0 MR ✓= 2GM, T d r a c ⇢ z2GM r t a ⌦T2 ! T R = 0 R = 2GM, T ! 1 R = 2GM, T ! 1 = 2GM ⇢ = const. r R2GM, =Consider 2GM, ! 1 ⇢ = ⇢ = const. r = 0 r = r ⇢ = ⇢ t = 0 R= R= 1 R = 2GM, T ! 1 ⇢ = 2GM ⇢ = const. r = 0 r = r ⇢ = ⇢ t = 0 ⌦! e s 0 0 0 0 2 01 r n e s ch ch the Schwarzschild metric e 1 . na . r e ssi o 1 ild th A n id Twoembedded Consider a spherical domain wallinembedded in a flat spatially flat matter universe. 1. Two di a spatially flat Consider a dominated spherical 2G R Consider heuniverse. swall domain wall embedded a spatially matter dominated e domain . s sy y mat rin 2G a spherical ✓ ✓ ◆ ✓ ◆ r ) n w o b 1 i p 5 M 2GM M ◆ di↵erent time-scales are relevant to the dynamics x 2 relevant 1 , n1(,dynamics of 2GM such a wall. One is the cosmological scale to⇠ H ) co of(1) eare di↵erent time-scales such a wall.ed One is dt di↵erent time-scales are relevant to the dynamics of is2 the scale ⇠ C2H 1 R ds2such = a1wall. OnedT + 1cosmological dR + Rt2 d⌦ . to the (2 i re c 2G 1 and the other is the acceleration time-scale due to the n fin gravita R repulsive R i t gravitational field of the domain wall i e r and the other acceleration time-scale a due to the repulsive + and theR other isd the acceleration time-scale due to the repulsive gravitational fieldisofthethe domain wall M ◆ de i v et 1. g m ⇢) 2/ e R t ⇠ (G ) r t ⇠ (G ) 1 . R he m , 3 t ⇠ (G ) 1 .1 In Lemaitresolution coordinates, thisbe takes the form to Le FIG. 14: A can matched a dust FRW at ta hyper- ⇢ a t ρ is= (1 can ForSchwarzschild R ⌧ t , the repulsive field be ignored. In this case, for t ⌧ R ⌧ solution t , the domain wall n beρ0ignored. ithfield I (2 e d ) For R ⌧ t , the repulsive can In this case, for t ⌧ d c w G by n For R ⌧R.t , the repulsive field can be ignored.expansion. In this Eventually, case, for2 twhen ⌧ Rthe⌧wall t ,falls the2within domain is it 2GM conformally stretched by cosmological the wall horizon, M an Si 2 2 2 a conformally stretched by cosmological expansion. Eventually, when the ds = d⌧ + d⇢ + R d⌦ . (2) 1/ ⌧ ) surface. This can be done in twoanddifferent ways. The shows an expanding 3 quickly under its expansion. tension forms a black hole of radius R = 2GM ⇠ t2 /tthe ⌧horizon, t. conformally stretched byshrinks cosmological Eventually, when theleft wall falls within it ball of dust f r o sh RSpanel re hole of radius R = 2 , e (2 quickly shrinks under its tension and forms a black m i ft S ) interested we will be primarily in theofopposite limit, where R ⇠ tt2./tIn this its tension and forms a black ⌧ t.case, the wall repels (3 in quickly shrinks underHere, wh where ⌧ and ⇢ are definedhole by the radius relationsRS = 2GM ) a th Here, we will be primarily interested in the opposite limit, where R theprimarily around it whileininterior, itsthe sizeopposite grows faster than the ambient wewall shall see, this matched tomatter a Schwarzschild while the right panel shows a rate. blackAs hole interior embedded in s: e Here, we will be interested limit, where R t expansion . In this case, the s matter fr or the around while its repels size grows faster than the ambient expan ✓ ◆it 1of leads to the formation of a wormhole. The dust which was originally in the interior the wall goes into i gi an 3) (3 grows faster than the ambient expansion 2GM rate.2GM (in arounda it while its size As we shall see, this n h 1 s the matter ) leads to the formation of a wormhole. The dust which was originally d⌧ = ±dT dR, (3) g t baby exterior. universe, and in the ambient FRW universe we are left with a1 black hole remnant cysted in a a dust FRW of R interior R tin a b leads to of a wormhole. The dust which was originally in the of the wall goes into ththe formation c a baby universe, and in the ambient FRW universe we are left with ( spherical region of of vacuum. s 4) e ✓ ◆ 1 ra 2G R 1 ⇢ andBefore in theweambient FRW universe we arewall, left let with a black hole remnant cysted in a Rdiscuss 2GM bt consider the e↵ect of the domain us spherical first the matching of Schwarzschild region of of vacuum. M aA baby universe, co u ed d⇢ = ⌥dT + 1 dR. (4) or S 2GM R b a dust cosmology. di ofand spherical region of vacuum. Before we consider the e↵ect of the domain wall, let us first discuss ⇢, na na y a ( t t e sh 5 e. and the a dust cosmology. Before we consider matching of Schwarzschild )the e↵ect of the domain wall, let us first discuss ar ift Subtracting (??) from (??), we have re Th e 1.1 Matching Schwarzschild to a dust cosmology in a dust cosmology. ge and he e o d th w 2/3 p 3 es e T ex Consider the Schwarzschild metric i cs R = (⌧1.1 + ⇢) Matching (2GM )1/3 ,Schwarzschild to a dust cosmology (5) , a co ( 2 6) ✓ ◆ ✓ ◆ 1 1.1 nd oMatching Schwarzschild to2 a dust cosmology rd Consider metric 2GM 2GM 2 2 the Schwarzschild ⌧ i na ds = 1 dT + 1 dR R2 d⌦in2 . the origin of the ⇢(1) where an integration constant has been absorbed by a+shift coordinate. The is te R R ✓ ◆ ✓ ◆ . Schwarzschild metric Considerthethe 2GM 2GM 1 2 expression for T as a function of ⌧ and ⇢ can be found from (3) as: 2 2 ds = 1 dT + 1 dR + Lemaitre coordinates, this ◆ takes the ✓ form 0 ◆ 1 ✓ in Schwarzschild 1 R R 2. InDomain wall s s ✓ ◆ 2GM 2GM 3/2 2 R2 . R ds2 = 1 dT 2±T +2 =14GM2 @ 12GMR dR + R22 d⌦ (1) tanh 1 ⇢, (6) 2 In 2+ Lemaitre coordinates, this Atakes the form Co ns
1.
1
M
1
a"
where ⌧ and ⇢ are defined by the relations
R=
extrinsic curvature Kµν = nµ;ν are given by the simple expression 0 ✓ ◆3/2 s with constant R given in has (5). been A second integration beenorigin absorbed in the T coordinate. a shift where an integration absorbed by constant a shift has in the of by the ⇢ coordinate. The 1 R R 0 ±T = 4GM 1 lines 2GMcoordinate + Since the metric (2) is synchronous, the of constant spatial and @ ⌧ is the ΩΩgeodesics, ˙ gare µ from ˆ expression for T as a function of ⌧ and ⇢Kcan be found (3) as: 3 2GM 2GM 0 0 = n ∂ g = ρ ˙ + t . (B16) ΩΩ µ ΩΩ 2s R 0 1 sR
R R A µ The normalization of1 the Rradial tangent vector requires that ±T = 4GM @ + tanh u1 Since ⇢, (2) is synchronous, (6) the lines of constant spatial coordin the metric 3 2GM 2GMr 2GM 2GM 2 tˆ˙ =been1 + ˙ .a shift in the T coordinate. (B17) with R given in (5). A second integration constant has absorbed ρ by R Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and ⌧ is the
✓
b"
Spherical domain wall in dust cosmology
R
ds =
s
Subtracting (3)d⌧from we have =(4),±dT
2/3 The normal to the worldsheet sis then ✓3given by ◆ 1 1/3 R (⌧ +2GM ⇢) (2GM ) , (5) R= 2r Subtracting (3) from (4), we have (4) d⇢ = ⌥dT + 1 dR. 2GM R 2GM where an integration constant has n been=absorbed by a − shift the0origin ρ, ˙ tˆin˙, 0, . of the ⇢ coordinate. The 3 (B15)2/3 µ R = (⌧ + ⇢) (2GM )1/3 , expression for T as a function of ⌧ and ⇢ can be found from (3) as: R Subtracting (3) from (4), we have 2
3/2 R2/3 temporal R directions, Rthe Since the normal is in the radial components of the where 3 @ 1 and A angular ±T = 4GM + 1/3 tanh 1an integration ⇢, constant has been (6) absorbed by
(⌧ + , expression2GM 3 ⇢) 2GM (2GM ) 2GM for T as a function of ⌧(5) and ⇢ can be found from (3) as: 2
0
✓
d⌧ +R3
2GM 2 Let where us now motion of a domain wall in a spherically symmetric vacuum. In Lemaitre coordinates, thisconsider the the form ⌧ and ⇢takes are defined by the relations ds2 = d⌧ 2 + d⇢ + R2 d⌦2 . with R given in (??). A second integration constant has been absorbed by a shift in the T coordinate. R s Since the metric (??) is2GM synchronous, the constant spatial coordinate are geodesics, and ⌧ is by the ✓ linesisof tangent ◆ 1 coordinates, a radial vector which to the worldsheet is given 2 2 2 2GM 2 2 2GM ds = d⌧ d⇢ + R d⌦ (2) (3) d⌧ + = ±dT 1 . where ⌧ and dR, ⇢ are defined by the relations R R R s
In Lemaitre s uµ = (tˆ˙, ρ, ˙ 0, 0), where a dot indicates derivative with ◆respect to the wall’s proper time τ : ✓ 1
d⇢ = ⌥dT +
2GM 1 R
✓
◆
R 2GM
s
◆3/2
44
1
1 dρ ˙2GM ρ = ◆ 1. dτ dR,
2GM R
Spherical domain wall in dust cosmology Dust"FRW"""
exterior solution, as indicated in the left panel, or a black hole interior can be embedded in
a dust FRW exterior, as indicated in the right panel. In this case, the black hole connects
through a wormhole to the asymptotically flat region IV .
d⇢ + R d⌦ .2GM R2GM
s
2GM
dR.
R
1
(2)
d⌧
= (4) ±dT
d⇢ (3) = ⌥dT +
s (B14) ✓
2GM R
✓
R 2GM
1
2GM R
1
2GM R
◆ 1
◆ 1
tanh
1
dR
dR
a shift in the ori
s
2G
with R given in (5). A second integration constant has been absorbed
Israel’s matching conditions for a thin domain wall of tension σ imply that the extrinsic curvature should change by the amount [KΩΩ0 ] = −
2 gΩΩ0 , tσ
(B18)
as we go accross the wall in the positive ρ direction. Here tσ = (2πGσ)−1 .
(B19)
By continuity of the metric accross the wall, the proper radius of the wall as a function of proper time, R(τ ), should be the same as calculated from both sides. This requires that the trajectory of the wall as seen from the “inside”, ρ− (tˆ− ), should be glued to a mirror image of this trajectory, ρ+ (tˆ+ ). Here, the double sign ± refers to the choice of Lemaitre
coordinate system in Eq. (B6), see Fig. 13, but ρ+ (tˆ+ ) and ρ− (tˆ− ) have the same functional form, which we shall simply denote by ρ(tˆ). The extrinsic curvature of the domain wall will
be the same from both sides, but with opposite sign. Combining (B16), (B17) and (B18), we have 2GM ρ˙ + R
r
1+
2GM 2 R ρ˙ = , R tσ
(B20)
which has the solution s −1 " 2 # 2GM R R R 2GM ρ˙ = 1 − − + + −1 . R tσ 2GM t2σ R
(B21)
The trajectory of a domain wall gluing two segments of the Schwarzschild solution is depicted in Fig. 15. Note that for R = tσ we have ρ˙ = 0, and it is straightforward to check that ρ˙ > 0, for R > tσ ,
(B22)
ρ˙ < 0, for R < tσ ,
(B23)
a property which will be important for the following discussion. A similar calculation in the Schwarzschild coordinates gives a simpler form for the equations of motion [9], R˙ 2 = −V (R), −1 2GM R T˙ = ± 1 − . R tσ 45
(B24) (B25)
domain"wall"trajectory"
FIG. 15: The trajectory of a domain wall between two segments of Schwarzschild solutions with √ the same mass M > Mcr = tσ /(3 3G).
where 2GM − V (R) = 1 − R
R tσ
2
.
(B26)
Eq. (B24) is analogous to the equation of motion for a non-relativistic particle of zero energy in a static potential V (R). The maximum of this potential is at Rm = GM t2σ
1/3
.
(B27)
The value of V (Rm ) is negative (or positive) for M larger (or smaller) than a critical mass Mcr given by: tσ Mcr = √ . (B28) 3 3G For M < Mcr , a domain wall which starts its expansion at the white hole singularity will bounce at some R < Rm and recollapse at a black hole singularity. Here we are primarily interested in the opposite case M > Mcr , where the expansion of the wall radius is unbounded and continues past the maximum of the potential at Rm , towards infinite size (see Fig. 15).
3.
Domain wall in dust
As mentioned around Eq. (B22), for R > tσ the wall recedes from matter geodesics with the escape velocity, on both sides of the wall. Because of that, we can construct an exact solution where an expanding ball of matter, described by a segment of a flat matter dominated FRW solution is glued to a vacuum layer represented by a segment of the Schwarzschild solution. This in turn is glued to a second vacuum layer also described 46
by a Schwarzschild solution of the same mass on the other side of the wall, as described in Subsection B 2. Finally, the second layer is glued to an exterior flat matter dominated FRW solution. The result is represented in Fig. 6.
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