Black Holes and the Multiverse

Home Search Collections Journals About Contact us My IOPscience

Black holes and the multiverse

This content has been downloaded from IOPscience. Please scroll down to see the full text. JCAP02(2016)064 (http://iopscience.iop.org/1475-7516/2016/02/064)

View the table of contents for this issue, or go to the journal homepage for more

Download details:

IP Address: 165.123.34.86 This content was downloaded on 01/03/2016 at 21:11

Please note that terms and conditions apply. JCAP02(2016)064 hysics P le b ic t ar doi:10.1088/1475-7516/2016/02/064 strop and Jun Zhang A b . Content from this work may be used 3 osmology and and osmology Alexander Vilenkin C a,b 1512.01819 primordial black holes, Cosmic strings, domain walls, monopoles

Vacuum bubbles may nucleate and expand during the inﬂationary epoch in the rnal of rnal Article funded byunder SCOAP the terms of the Creative Commons Attribution 3.0 License . ou An IOP and SISSA journal An IOP and Departament de Fisica Fonamental iUniversitat Institut de de Barcelona, Ciencies Marti del iInstitute Cosmos, Franques, of 1, Cosmology, Barcelona, Tufts University, 08028574 Spain Boston Ave, Medford, MA,E-mail: 02155 U.S.A. [email protected], , [email protected] [email protected] b a Any further distribution ofand this the work must title maintain of attribution the to work, the author(s) journal citation and DOI.

Jaume Garriga, J Black holes and the multiverse Received December 28, 2015 Accepted February 3, 2016 Published February 25, 2016 Abstract. early universe. After inflationcome to ends, rest the with bubbles respectbubble quickly to itself dissipate the depends Hubble their flow on kineticcritical and the energy; eventually value, resulting form they the black black holes. bubble holeforming mass. collapses The a fate to If of baby a the the A universe, mass singularity. is similar which Otherwise, smaller black is the thanduring hole bubble connected a inflation. interior formation certain to As inflates, mechanism the andomain operates illustrative exterior wall example, for FRW scenario, we assuming region spherical studiedOur that the by domain domain results black a walls walls indicate hole interact wormhole. that, nucleating massscenario with depending spectrum matter can in on only have the the gravitationally. significant modelseeds astrophysical for parameters, supermassive effects black black holes and holes.paper produced can is The in even very mechanism this generic serve of andBaby black as has hole universes dark important formation implications matter inside described forall in or super-critical the this vacua as global black allowed structure of holes bynon-trivial the the spacetime inflate universe. structure, underlying eternally with particlewormholes. a and physics. multitude nucleate If of The eternally a bubblescould inflating resulting black be of regions hole multiverse regarded connected as has population by evidence a with for very the inflation and predicted forKeywords: mass the spectrum existence is of a discovered, multiverse. it ArXiv ePrint: JCAP02(2016)064 21 22 – 1 – eq eq > t < t σ σ t t 4.2.1 3.2.1 Small3.2.2 walls Large domain walls surrounded by radiation 19 18 3.1.1 Small3.1.2 walls Large domain walls surrounded by dust 15 13 2.4.1 Small2.4.2 bubbles surrounded by Large radiation bubbles surrounded by radiation 11 4.2.2 12 2.3.1 Small2.3.2 bubbles surrounded by Large dust8 bubbles surrounded by dust9 4.1 Size4.2 distribution of domain Black walls hole mass distribution 19 20 3.2 Walls surrounded by radiation 18 3.1 Domain walls surrounded by dust 13 B.1 Matching SchwarzschildB.2 to a dust Domain cosmology wallB.3 in Schwarzschild Domain wall in dust 31 33 35 2.4 Bubbles surrounded by radiation9 2.1 Initial2.2 conditions4 Dissipation2.3 of kinetic energy5 Bubbles surrounded by6 dust 1 Introduction A remarkable aspect of inflationarylarge-scale structure cosmology to is small that quantum itremain fluctuations attributes small in the through the origin early most universe ofcent of [1]. times. galaxies the and The Here, cosmic fluctuations we history willinflation and explore could become the also possibility nonlinear lead that only to non-perturbativewill in quantum the show effects relatively formation that during re- of spontaneous structure nucleation on of astrophysical vacuum scales. bubbles Specifically, and we spherical domain walls during 4 Mass distribution of black holes 19 3 Gravitational collapse of domain walls 5 Observational bounds 6 Summary and12 discussion A Evolution of test domain wallsB in FRW Large domain wall in dust cosmology 29 23 31 25 Contents 11 Introduction 2 Gravitational collapse of3 bubbles JCAP02(2016)064 , = b , its 1 πGσ 3 H const. − b develop H ≈ = 2 1 i σ & − ρ σ . H per unit area 1 H R − b , then internal b 1 & ρ H − b − R i H ρ R = R during the slow roll). At F b ρ We will be interested in the case 1 . i < ρ b ρ – 2 – We will show that the black hole that is eventually formed contains 2 is Newton’s constant. If the bubble expands to a radius G , where 2 0. Once a bubble is formed, it immediately starts to expand. The difference in / 1 > is the wall tension. We will show that domain walls having size 3) / b σ b ρ The physical mechanism responsible for these phenomena is easy to understand. The The nature of the collapse and the fate of the bubble interior depends on the bubble Bubbles formed at earlier times during inflation expand to a larger size, so at theThe end situation with domain walls is similar to that with vacuum bubbles. It has been We now briefly comment on the earlier work on this subject. Inflating universes con- Higher-energy bubbles can alsoThis be low formed, but energy their internal nucleation inflation rate is takesFor typically place simplicity, strongly in inside suppressed this [3]. the discussion bubble, we even disregard though the inflation gravitational in effect the of exterior the bubble wall. We shall see 1 2 3 πGρ region has already ended. later that if the wall tension is sufficiently large, a wormhole can develop even when inflationary expansion of the universe is driven by a false vacuum of energy density the inflationary epochmasses. can result in the formation of black holes with aBubble wide nucleation spectrum in of includes this vacuum states vacuum of can a occur lower energy [2] density, if the underlying particle physics model size. The positive-energy vacuum inside the bubble can support inflation at the rate of inflation we expectwith to a have wide a spectrumcritical wide of mass masses. spectrum have of We inflating will bubble universes show sizes. inside. that black They holes willshown with form in masses black above refs. holes afalse [4] certain vacuum. that The spherical wallsholes domain are when then walls stretched they can bywalls come the spontaneously is within expansion nucleate known of the to in the be horizon. the universe repulsive and [5,6]. inflating Furthermore, form This the black causes gravitational the walls field to of inflate domain at the rate when vacuum tension on the two sidesof of the the wall, bubble so wall the(or results bubble until in expands a the with force energy acceleration. densitylater This of times, continues the until the inflating the vacuumto end bubble drops of the continues below inflation surrounding to matter, expandvacuum while in but it its it is interior. isa also Eventually, black being slowed this hole. down pulled leads inwards by to by momentum gravitational the transfer collapse negative and pressure the of formation of inflation does not occur, and the bubble interior shrinks and collapses to a singularity. where a ballooning inflatingwormhole. region in On the its other interior, hand, which if is the connected maximum expansion to radius the is exterior region by a tained inside of blackfocused holes on have black holes beena in discussed different asymptotically by mechanism flat of a or cosmologicalmation number de in wormhole of Sitter cosmological formation. spacetime, authors spacetimes The has while [7–9].a possibility also ref. cosmological been of [7] Refs. discussed scenario wormhole considered 8,9] [ in for- where ref.uum it [14], bubbles can but was be without qualitatively suggesting realized.the discussed in Cosmological resulting ref. black black [15], hole hole but formation masses. no by attempt vac- Black was holes made to formed determine by collapsing domain walls have been (8 interior begins to inflate. a wormhole structure, whilecosmological smaller horizon. walls collapse Once toinflating again, a domain there singularity walls soon is connected after a to they the critical enter exterior mass the space above by which a black wormhole. holes contain JCAP02(2016)064 (2.1) (2.2) (2.3) (2.4) , , 2 2 . / / 1 1 1 ) ) − b ) m , πGρ σ πGσ πGρ The metric inside the bubble is the de Sitter 8 , t 8 / b 4 / t = (2 – 3 – 1 = (3 = (3 − i σ 1 t 1 − b H − H ≡ H σ ≡ ∼ t b t t is the Hubble radius at the time when inflation ends. The relation (2.4) 1 − i H is the matter density, the scale associated with the vacuum energy inside the bubble, ∼ i m t ρ In the present paper, we shall investigate cosmological black hole formation by domain The paper is organized as follows. Sections2 and3 are devoted, respectively, to the The interaction of bubbles with matter is highly model dependent. For the purposes The dynamics of spherically symmetric vacuum bubbles has been studied in the thin A more detailed and pedagogical treatment in the asymptotically flat case was later given by Blau, 4 Guendelman and Guth [9]. walls and vacuum bubblesspacetime that nucleated structure during of the suchblack inflationary hole black epoch. mass holes We distribution shall andthe in study particle estimate the physics the their model present parameters. universe masses.holes and We produced shall derive We in see observational this shall that constraintsobserved way for also on can in some find serve parameter galactic values as the centers. the dark black matter or as seedsgravitational for collapse supermassive of black vacuum holes distribution bubbles of and black domain holesobservational produced walls. bounds by on Section4 the suchdeals collapsenumerical distribution. of with study domain Our the of walls, conclusions mass test and areof domain section5 summarized walls thewith in is spacetime the section6.A deferreddiscussed structure to in appendix describingA, appendix B. and the some gravitational technical collapse aspects of2 large domain walls Gravitational are collapse ofIn this bubbles section we describeembedded the in gravitational the collapse matter ofrelevant bubbles distribution for after of the inflation, a dynamics. when FRW they These universe. are are Three the different cosmological time scale scales will be discussed in refs. [16–18],these but papers, the so their possibility estimateno of of wormhole wormhole black is formation hole formed). masses has applies been only overlooked to the in subcritical case (when where of illustration, here wereflecting shall boundary assume that conditions matter atbe is the discussed created bubble in only the wall. outside followingof For section, the the the we bubble, wall. shall case and consider As of has the we domain case shall where walls, see, matter which this iswall will on leads limit both by to Berezin, sides a Kuzmin rather and Tkachev different [8]. dynamics. and the acceleration time-scale due to the repulsive gravitational field of theIn domain what wall follows we assumetaneously, that and the for the definiteness inflaton we transfers shall its also energy assume to a matter separation almost instan- of scales, where guarantees that the repulsiveare gravitational force subdominant due effects to at the the vacuum end energy of and inflation, wall and tension can only become important much later. JCAP02(2016)064 0 ρ (2.8) (2.5) (2.6) (2.7) (2.9) (2.10) (2.11) . 3 i . 3 R 3 for pressureless 2 6 R / 2 Gσ 2 = 2 Gσ π 2 β 8 π , which has been replaced 8 i − t − 2 / 2 1 . / ] ) 1 2 i ] 2 2 R , and that the vacuum energy Ω = 0, so the right hand side of (2.5) i 2 b R d at time . The difference in energy density ρ 2 2 b b H i dS r ρ is the radius of the bubble wall, and H ≈ 3 R i − M . dt, + ) R R − 2 σ i 0 2 ) t t i r , so the negative sign must be discarded. i ( . t + 1 2 dr 2 for radiation, or ( R m m and the bubble wall tension are continuous M a + 1 2 i / 2 i m )( ρ ρ ˙ 2 t ˙ R − R [ ( ˙ H R πρ 2 i 2 1 , [ = 1 1 4 – 4 – a 3 4 2 0 R β p ρ ≈ + ≡ i πσR 2 = i ˙ πσR R , so that dt M i + 4 dτ t − + 4 3 i 3 R = b R 2 ) i πρ ds ρ . The metric outside the bubble is the Schwarzschild-de Sitter 4 3 1 − − b ≈ b H ρ 3 i ( is the proper time on the bubble wall worldsheet, at fixed angular R π ) τ i 4 3 t ( . Assuming the separation of scales (2.4), the right hand side of eq. ( 2.6) = i m t is the scale factor, where πρ M β 4 3 t , where ∝ (which holds during inflation, when the energy density in the parent vacuum is higher than in the ) is the vacuum energy outside the bubble, i t (SdS), with a mass parameter which can be expressed as i ( 5 ρ a dR/dτ < ρ Let us now calculate the Lorentz factor of the bubble walls with respect to the Hubble b Here we are ignoringIn slow principle, roll both corrections signs for to the the square vacuum root energy are allowed during for inflation. the second term at the right hand side. However, ρ ≡ 5 6 ˙ Here, we have assumedalmost that instantaneously the at transfer the of time energy from the inflaton to matter takes place new vacuum) only the plus sign leads to a positive mass for by a bubble of a much lower vacuum energy density Here, matter, as may be theminimum case of if a the quadratic energy potential. is The in proper the form time of on a the scalar worldsheet field is oscillating given around by the flow at the time is dominated by the kinetic term and we have 2.1 Initial conditions A bubble nucleating in devanishes Sitter during space has inflation. zero mass, in Assuming the that transition from inflation to the matter dominated regime, we have outside the bubble is completely negligible after inflation, is the “excluded” mass of matter in a cavitygoes of into radius the kinetic energyto of the the second bubble and wall third and terms its in self-gravity eq. corrections, corresponding (2.6). metric R position. Eq. (2.5) can be interpreted as an energy conservation equation. (dS) metric with radius Here, Note that the left hand side of eq. (2.6), We wish to considerbubble the is given motion by of the bubble relative to matter. The metric outside the JCAP02(2016)064 1. 1. & < 0 i (2.12) (2.15) (2.16) (2.17) (2.18) (2.13) (2.14) R ar i H , 2 , , it is straightforward to show that i γ i σ t α, t , t m The corresponding proper acceleration 2 0 ρ 0 ≈ = , 7 . ∼ , and we have r i . t t ar t 2 2 2 2 i t dt a 0 vα γ ∼ − + r σ σ m t ˆ 2 i r − = ρ a 3 i 2 σ 0 1 U t t 1 ˆ 0 HR − √ – 5 – vv ∼ − U ∼ 3 1 ∼ − 2 m = γ t γ ρ q dγ α ∆ = ∼ dτ ≡ dR r i dt dγ ˆ 0ˆ γ T that hits the domain wall, and a reflected fluid which moves in the ˆ µ U . The typical size of bubbles produced during inflation is at least of β/t = is the radial outward velocity of the wall. The second equality just uses the 0 /a 0 ar a is the four-velocity of matter in an orthonormal frame in which the wall is at rest, 1. = ˆ µ = , although it can also be much larger. Hence, at the end of inflation we have v i U t H i γ Here we use a simplified description where the momentum transfer is modeled by superimposing an 7 opposite direction. Thismore description realistic should setup, betransfer. shock valid waves in will the form limit [10, of11], a but very we weak expect fluid a self-interaction. similar behaviour In for a the momentum incoming fluid with four velocity with In the rest frame of ambient matter, we have where kinematical relation between acceleration andand proper acceleration acceleration are for parallel the to case each when other. velocity Combining (2.15) and (2.16) we have Solving (2.17) with thethe initial wall loses condition most2.13) ( of at its energy on a time scale order 2.2 Dissipation ofThe kinetic kinetic energy energysurrounding of matter. the Let us bubbleon now the estimate walls wall due the will to time-scale momentum for transfer be this per unit to dissipated area happen. in by the The radial momentum force direction acting can transfer be estimated to as the where a prime indicates derivative with respect to On the other hand, theCombining (2.9) second term and in) (2.12 the we numerator have of (2.12) is bounded by unity, and using (2.4),tic, we find that the motion of the wall relative to matter is highly relativis- where of the wall is given by hatted indices indicate tensorof components matter in is that highly relativistic frame, in and the we radial have direction. used that the motion JCAP02(2016)064 2 ) i /t σ (2.19) (2.22) (2.21) (2.20) t ( , nested bh ∼ 2 i M γ , which is still i ∼ t . 3 when the bubble is which we shall take shell R i γ t 0 σ 2 the relative shrinking ρ /t i ≈ t 2 i Gσ t t 2 is the energy scale of the π ∼ w 8 i η 0 + ∆ t − i R t 2 . , / , the first term will be dominant. i 3 i 1 = b ] R η M t 2 R , where i 2 b p and i H i . /M M η πσH − ) 2 i i , i η t w i ( R σ 3 is the energy scale in the vacuum inside the w t + 4 η t a + 1 η is proportional to the initial volume occupied b b , with most of this energy contained in a thin 2 replaced by the vanishing vacuum energy after i η + – 6 – , at the time ≈ ˙ i R i bh πρ [ 2 i c 2 b M ρ R t t 2 3 4 r i M 1 in a time of order ∆ ≈ H expanding with a Lorentz factor ∼ ≈ 2 πσR ∼ ≈ ) , where i 4 σ b shell γ bh ˙ bh η E R + 4 /t i M M 3 R R b )( for the Schwarzschild metric within the empty layer can be esti- πρ /G 3 4 i bh t ( = M ∼ bh is the inflationary energy scale. The second term is Planck suppressed, and i M η shell M ), and we have = 0, i t 0 is the excluded mass given in (2.8). On dimensional grounds, the first term in ( ρ m i ρ M After the momentum has been transferred to matter, the bubble starts receding relative Since the bubble motion will push matter into the unperturbed FRW region, this results The mass parameter where parenthesis in (2.20) is of order bubble wall and mated from the initial condition so in the absence of a large hierarchy between to the Hubble flow. As we shall see, for large bubbles with relative to the Hubble flow.a subject Possible astrophysical for effects further of research. this propagating shell are left as speed becomes relativistic inillustrate a this time-scale point, which let is us much first smaller consider than the2.3 case the of Hubble pressureless scale. Bubbles matter. surrounded To Once by the dust bubble wallinfinitesimal has layer devoid stopped of withto matter, respect with be to a zero. negligible the Sincethat vacuum Hubble the energy time flow, matter on it outside we is of this surrounded can layer by use cannot an eq. affect (2.5) the motion with of the wall, from at rest with respect to the Hubble flow. Using the separation of scales (2.4), we have A remarkable feature ofby the (2.20) bubble, is withparameters. that a universal Under proportionality the factordensity condition which depends (2.4), only this on factor microphysical is much smaller than the initial matter bubble, whereas the second term is of order layer of mass in a highly overdenseThe shell total surounding energy the of bubble, the exploding shell at is a highly relativistic speed. and that the wallmuch less slows than down the to Hubble time at the end of inflation. inflation, The motion of the bubble is such that it eventually forms a black hole of mass in an empty cavity of co-moving radius JCAP02(2016)064 . i M ) σ (2.33) (2.24) (2.32) (2.23) (2.29) (2.31) (2.25) (2.27) (2.28) (2.30) (2.26) /t i t ( is of course ∼ σ t bh ∼ M cr GM . i 2) / 2 defined by g + − , . In the absence of a hierarchy , H 2 3 σ 2 , / 2 z (1 g 1 3 / /η . / 3 2 − p } 8 4) + . − E, , b / 1) 2 2 σ 3 2 H M 2 , t 2 3 σ 2) g − H R / σ . . / ) = 2 t 3 ∼ H 2 6 z m − σ ) bh { z b G + ( g τ, z 2 t + 4 2 σ + 16 2 H bh z σ t (1 H V − 2 2 b 3( 4 + − − H 6 m , but the situation where GM H 1 Min ≡ H √ + H z b – 7 – 2 4 = t 3 3 GM 2 ¯ ∼ g = g M (2 = = ∼ − cr ¯ 2 + ˜ . However, we may also consider the limit when the τ 8 + (1 M 3 ˜ i , while τ cr dz d = z H 2 b are defined by p = M ) = 1 in the whole parameter range. GM h 2 τ E /η z ) cr GM p ( 2 1 ∼ b V M M /t ≡ m and ˜ i z t 3 m ∼ ( z z ≤ b ≈ t = 3 bh z M 2, and 1 we may expect σ ≤ η g ≤ and is larger or smaller than a critical mass defined by b η bh the proper time on the bubble wall trajectory, and M τ The qualitative behaviour of the bubble motion depends on whether the Schwarzshild It was shown by Blau, Guendelman and Guth [9] that the equation of motion) (2.19 On dimensional grounds Although the expression for the critical mass is somewhat cumbersome, we can estimate it as where the rescaled variables The shape of the potential is plotted in1. figure The maximum of the potential is at where between In this case wevacuum energy have inside the bubble is completely negligible, and then we have with where Note that 0 also possible. mass can be written in the form The potential and the conserved energy in eq. (2.23) are given by , 1 = (4) (5) (6) (1) (2) (3) ⇢ H 0 r ⇠ = t r . =0 t coordinate. The ⌧ ⇢ , the domain wall is , /t ⇢ 2 t const. r t . 2 = ⇠ ⌧ ⌦ 1 A ⇢ d R . In this case, the wall repels 2 R t GM R GM ⌧ 2 + GM t 2 dR. dR, =2 2 s 1 1 R . S 1 2 dR R , 1 ◆ ◆ = ⌦ 3 d / ⇢ 2 1 ◆ ) tanh R R R GM GM 2 2 + R GM GM 2 2 ⇢ (2 1 1 !1 d R 3 GM ✓ ✓ / 2 2 1 ,T R ✓ s ) GM bh R R ⇢ 2 GM GM + + 2 2 2 + 2 + / GM s s ⌧ 2 3 ( dT ⌧ ◆ + 3 2 d ◆ =2  , can be found from (3) as: 1 = R dT dT R (1) (2) (3) (4) (5) (6) = ⇢ ⇢ GM = R ± ⌥ H GM 2 0 r 2 R 2 ⇠ ✓ = t and ds 1 3 = = !1 r ⌧ 1 0 @ . ✓ ⇢ ⌧ =0 t ,T d d ect of the domain wall, let us first discuss the matching of Schwarzschild coordinate. The bh ⌧ ↵ bh

JCAP02(2016)064 GM ⇢ = , the domain wall is , /t ⇢ 2 t 2 GM const. r t . =4 GM . 2 i cr = ⇠ ds ⌧ t ⌦ 1 A T the ⇢ d and =2 M R . In this case, the wall repels 2 =2 ± (2.38) (2.35) (2.37) (2.34) (2.36) R t GM ) R R 1 GM max ⌧ b max 2 + GM R = 2 (lower t R bh as a function of 2 dR, dR. =2 2 s g 1 1 , the repulsive field can be ignored. In this case, for σ/ρ H R . M S 1 ( are defined by the relations =0 2 dR T R , t 1 ◆ ◆ = ⌦ i ∼ 3 ⇢ R R t d / . ⇢ t 2 1 ◆ 1 ) . tanh R R R ⌧ GM GM = 4 S i 2 2 ) z + t and R R which corresponds to = 0. For GM GM R 2 ) 2 ⇢ ˙ z ⌧ (2 z 1 1 R !1 GM G d R 3 ( GM ✓ ✓ ( / =0 2 2 1 For Here, we will be primarily interested in the opposite limit, where Before we consider the e V Consider a spherical domain wall embedded in a spatially flat matter dominated universe. Two ,T erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale R t ✓ s ⇠ ) GM =2 bh R R ↵ ⇢ 2 GM GM + 0 3 + . 2 2 2 where Subtracting (3) from (4), we have where an integrationexpression constant for has been absorbed by a shift in the origin of the a baby universe, andspherical region in of the of ambient vacuum. FRW universe weand are a left dust with cosmology. a black hole remnant1.1 cysted in a Matching Schwarzschild toConsider the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form 1 Spherical domain wall in dustR cosmology ⇢ di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by field cosmologicalquickly expansion. of shrinks the under Eventually, when its domain the tension wall and wall forms falls a withinthe black the matter hole horizon, around of it it radius leads while to its the formation size of grows a faster wormhole. than The the dust ambient which expansion was rate. originally in As the we interior shall of the see, wall this goes into + 2 + = 1 (middle curve) and . Once the bubble radius reaches / GM i s s ⌧ 2 3 g t 2 max ( dT ⌧ ◆ . + 2 3 d ◆ =2 . 3 ,  / 2 can be found from (3) as: . 1 is the value of 1 / i πσR R dT dT R 1 = ⇢ GM i = R ) R ⌥ ± GM 2 i z i 2 2 R 2 ) + 4 2 i ✓ R i b i and ds ρ σH 3 1 H = = R !1 i H 3 b ⌧ , and therefore the bubble wall decouples from 1 3 max ( 0 @ σ ρ 3 H ✓ ⇢ ⌧ / – 8 – R ,T ( ≈ 2 d d b ect of the domain wall, let us first discuss the matching of Schwarzschild , where ≈ t bh ≈ ∼ ↵ bh the time it takes for the empty cavity of co-moving GM m = 0 (upper curve), i πρ i ∝ z = R i H max 3 4 g R H R ) t 2 GM ∆ t max t 1 =4 R R GM ≤ , and the bubble trajectory has a turning point. In this ( ≈ R ds a T 1 . From eq. (2.19), we find that for . This is in contrast with the behaviour of the scale factor =2 i 3 =2 we have bh ± / > E R < R 1 1 2 M ) / τ i we have , the bubble collapses into a Schwarzschild singularity. This is 3 as a function of z m ) 2 , the repulsive field can be ignored. In this case, for H z i ∝ are defined by the relations =0 / ( T 3 max t H i ) ⇢ R i V R t i . R max 5 1 R 25 30 10 15 20 - H ⌧ = ( R S - - - - - = i ) z t and R R ) . The bubbles grow from small size up to a maximum radius R ( i ⌧ ) to cross the horizon, we have z GM G i b ( ( R t =0 ( For Here, we will be primarily interested in the opposite limit, where Before we consider the e V Consider a spherical domain wall embedded in a spatially flat matter dominated universe. Two erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale 1, the first term in (2.20) dominates, and the turning point occurs after a /ρ t ⇠ , we have b =2 i ↵ /a 0 cr i /ρ Subtracting (3) from (4), we have where an integrationexpression constant for has been absorbed by a shift in the origin of the where the matter around itleads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust region in ambient which of expansion the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant1.1 cysted in a Matching Schwarzschild toConsider the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by field cosmologicalquickly expansion. of shrinks the under Eventually, when its domain the tension wall and wall forms falls a within black the hole horizon, of it radius 1 Spherical domain wall in dustR cosmology ⇢ i R σH b = < M σH . The potential in eq. (2.27), for /ρ c i r bh σH M the Hubble flow in a time-scale which is at most of order radius its turning point at in the matter dominated universe, represented in figure2. Denoting by bubble radius behaves as which happens after a time-scale much shorter than the expansion time, ∆ In the last two cases, Figure 1 small fractional increase in the bubble radius curve). 2.3.1 Small bubblesFor surrounded by dust case it is easyleft to show of that the with potential the condition barrier, (2.4), the bubble trajectory is confined to the For Finally, for the initial radius then recollapse. Thewe turning have point is determined from (2.19) with For 1

sthe is

⌧

(6)

coordinate.

(5)

⇢

oriae The coordinate.

⇢ GM

(4)

R s

A @

(3) ◆ ✓

s s GM 2 GM 2 GM 2 3

1 0

tanh (6) , ⇢ tanh + GM =4 T

R R 1 R

2 / 3

1 GM

(1) (2) (3) (4) (5) (6) R

(2) ⇢ s

a efudfo 3 as: (3) from found be can ⇢ and ⌧ of function a as T

H for expression

0 , is the

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where oriae The coordinate.

⇢ ⇠

⌧ /

◆ )

= 1 t

 r /

coordinate. 3

2 GM

dR. (1)

= R (5) , ) GM (2 ) ⇢ + ⌧ (

◆

2 T . ✓

3 / 1

3 =0

3 / 2

3 3

)

R coordinate. The

⌧ as: (3) from found be can dR, 0

JCAP02(2016)064⇢

◆

⇢ ⇢

utatn 3 rm() ehave we (4), from (3) Subtracting

, the domain wall is

1 2

⌧ ,

/t 2 (



✓ ⇢

and

2 GM

const. r 1

◆ ✓ t

. ⌧

⌥

s GM 2

= GM

⇠

⌧

(4) dR. 1 + dT = ⇢

d ⌦

2 GM

± 1 A

R 2

⇢ ✓ d R GM 2

. R

R 1

. In this case, the wall repels 2 1

⌦

2 , the

⌦ t

GM ◆ ✓

d R

d GM

± ⌧ s

R R

ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given +

2 2 +

bh R

GM 2

(3) dR, 1 dT = ⌧

d t

2 R

dR, dR.

=2 2

+ dT ⌥ s

GM 2 GM 2 +

2 1 1

R and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since . 1

S of function a as

1 , while the

2 2 2 dR

⇢ R dR T

d GM

1 with ◆ ◆ =

⌦

= 3

◆

R d /

dT ⇢ ±

and ⌧ where r endb h relations the by deﬁned are

⇢ 2 1 1 ∼

⇢ ◆

)

d tanh R R

R GM

R GM GM t

2 2 2 +

. ⌦ d R + ⇢ d + ⌧ d = ds (2) GM

2 ⌧ 2

⌧

2 2 2 2 2 2

2 GM ⇢

✓

d (2

1 1

!1 1 d xrsinfor expression R

, 3

✓ ✓

⌧ /

= 2 . This leads to the 1 when inﬂation ends, a hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where 2 1

nLmir oriae,ti ae h form the ,T takes this coordinates, Lemaitre In

} 2 R

H ✓

s i )

ds GM

⇠

2 σ bh

/t R R t

◆ ⇢

dT GM GM

◆ ✓ ◆ ✓ 0, initially expands with a

+ t

t 2 2 ⇠ R

R , t

= 2 +

nti ae h alrepels wall the case, this . In

= ds (1) . ⌦ d R + dR 1 + dT 1 b

h oanwl is wall domain , the / > R GM

⇢ s s

⌧ 2

t 3 t 2 2 2 2 2

GM 2 GM 2

t ( 0 dT

⌧

{ ◆ r

1 utatn 3 rm() ehave we (4), from (3) Subtracting

⌧ + 3 2 GM d GM

ρ ◆

2 = 

= can be found from (3) as:

R ✓

osdrteShazcidmetric Schwarzschild the Consider 1 =2

R R

dT dT

⇢ relations the by deﬁned are = ⇢

GM S ⌧ = R

Min ± ⌥ GM

⇢ 2

=0 0 2 R 2

t ✓

. acigShazcidt utcosmology dust a to Schwarzschild Matching r1.1

= and ds

and ∼

1 3

= = !1

2 , 1 ⌧ ⌧ =ds 1 0 @ t

✓ . The thin relativistic expanding shell of ⇢ ⌧

os.r const. ,T r d , and transitions to the higher energy d

n utcosmology. dust a and ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild = R

↵ where bh

⇢ H GM "FRW" ↵ e the consider we Before c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect = H

2 GM =0

=4 GM

peia eino fvacuum. of of region spherical

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching dust the discuss ﬁrst us let wall, domain the of ect ds 2

↵ t ⇠

T =2

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW =2 ambient the in and universe, baby a

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In

= will eventually occur, leading to a multiverse R 1

⇢ ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads – 9 – i

as a function of

, the repulsive ﬁeld can be ignored. In this case, for H

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the are deﬁned by the relations =0 H T

const. r

!1 ⇢

R t

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, nti ae h alrepels wall the case, this In . t " R .

1 metric Schwarzschild the Consider ⌧

⇠ =

⌧

= S S

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly . t /t t GM =2 R , so the trajectory is unbounded and the bubble wall

) , so the black hole forms on a time-scale much shorter

t and

2 ⇢ bh . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 given in (5). A second integration constant has been absorbed by a shift in the R

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally ⌧

GM G

⌧ ⌧

⌧ R

GM (

max

eoew osdrtee the consider we Before h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive the , t R For h oanwl is wall domain the , t R

t > E Here, we will be primarily interested in the opposite limit, where Before we consider the e Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two

erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale cosmology. dust a and

t ⇠ ⇠

=2 =2

singularity ↵

" ) 0

GM . ) G ( t

R where an integrationexpression constant for has been absorbed by a shift in the origin of the with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and Consider the Schwarzschild metric In Lemaitre coordinates, this takes the form where Subtracting (3) from (4), we have quickly shrinks under its tension and forms athe black matter hole around of radius itleads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust in ambient region which expansion of the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant1.1 cysted in a Matching Schwarzschild to a dust cosmology 1 Spherical domain wall in dustR cosmology ⇢ di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by ﬁeld cosmological of expansion. the Eventually, when domain the wall wall falls within the horizon, it

2 1 . m &

peia eino fvacuum. of of region spherical t

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and z

!1 , the domain wall is i

" ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

⇠ (

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the

R rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent ↵ di , H t

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads V bh 1 =

t ,T 2

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here,

⌧ / bh ⌧ h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in ⇢ embedded wall domain spherical a Consider 1

. )

Bubble

R /t i GM 2 1 z ) z ( V

⌧

) =2

Schwarzschild radius of hole black a forms and tension its under shrinks quickly

R t R

For

1 i

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

( =0 r

R ⇠

H H

i we have

(

, in a dust dominated spatially ﬂat FRW. At the time bh

i 0

GM =2 H t = t =0 t ⇢

t ⇠

t 1

!1 ⌧

cr t =

bh bh rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent S cr 0

2 = ⇢ ,T GM =2 R ,T GM =2 R =0 R GM =2 R = ⇢ r = r =0 r const. = ⇢ GM n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and

!1 t ↵

S M

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider GM

,T di . In this case, the wall repels

=0 H GM

t t t

bh 0

=2 > M

⇢ . Causal diagram showing the formation of a black hole by a small vacuum bubble, with

=2 (1)

peia oanwl nds cosmology dust in wall domain Spherical 1 . R bh S bh GM R H t

M R

M =2 cosmology dust in wall domain 1 Spherical formation of afigures3 wormholeand4). which eventually The “pinches bubble off”, inflates at leading the to rate a baby universe (see wormhole closes and any signalssingularity. which are sent radially inwards end up at2.4 the Schwarzschild Bubbles surroundedDue by to radiation the effects ofmore pressure, involved the than dynamics in of the bubbles case surrounded of by pressureless radiation dust. is somewhat Here, we will only present a qualitative inflatinary vacuum withstructure expansion [3, rate ].12 can Initially, send a signals geodesic that observer will end at up the in edge the of baby universe. the However, matter after dominated a region time grows monotonically towards infiniteof size in the the bubble asymptotic is future. exponential, The unbounded and growth starts at the time matter produced by theof bubble dust is near not the represented boundary in represented the by figure. the dashed This line. shell disturbsNote the that homogeneity 2.3.2 Large bubblesFor surrounded by dust than geodesic of a particle escaping to infinity has unbounded large Lorentz factor relativeto to matter, the turning Hubble aroundand flow. due eventually The collapsing to motion into the a isdashed internal Schwarzschild slowed line singularity. negative down At at pressure by fixed(see the and momentum angular appendix coordinates, transfer B). edge the the Such tension of time-like geodesic ofsmaller the slope approaches the dust would time-like bubble correspond region infinity wall, to represents with the a trajectory unit of radial slope. a geodesic particle The at of reason a the finite is radial Schwarzschild that coordinate metric a small bubble (to the left of the diagram) with positive energy density Figure 2 R mass (2)

!1 ,T bh 2 . (3) GM bh ⌦ 2 d =2 GM R (4) R 2 + 1 =2 =0 1 dR R H i S t (5) = ◆ 1 Spherical domaint wall in dust cosmology GM GM 2 R 2 . ⌦ =2 =0 2 d t , the repulsive field can be ignored. In this case, for 1 R 1 . 0 Consider a spherical domaint wall embedded in a spatially flat matter dominated universe. Two R ⇢ 1 ) 2 ✓+ 2 + dR, (6) erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale ↵ ⌧ ect of the domain wall, let us first discuss the matching of⇢ Schwarzschild di (G R ↵ dT d 1 coordinate. The and the other is the acceleration time-scale due to the repulsive gravitational field of the domain wall GM ◆ dR. ⇠ For 2 GM ⇢ t GM◆ R 2 R 2 R + is the 2 ◆ ⌧ conformally stretched by cosmological expansion. Eventually, when the wall falls within the horizon,⌧ it 1 GM coordinate. Here, we will be primarily interested in the opposite limit, where d 2 R quickly shrinks under its tension and forms a black hole of radius 1 T ✓ , GM ⇢ ✓ = 2 1 2 R 3 , the matter around it while its size grows faster than the ambient= expansion rate. As we shall see, this / 2 1 ds ✓ leads to the formation of a wormhole. The dust which was originally in the interior of the wall goess into R ) 1 a baby universe, and in the ambient FRW universeds we are left with a black hole remnant cysted in a R GM Before we consider the e 2 GM A spherical region of of vacuum. dT 3 GM / (2 2 s 2 ± + s1 and a dust cosmology. = dT ) ⇢ ⌧ 1.1 Matching Schwarzschild to a dust cosmology d ⌥ + tanh = ⌧ 3 ( Consider the Schwarzschild metric ⇢ R d 2 = are defined by the relations GM R 2 ⇢ 2 s / + In Lemaitre coordinates, this takes the form can be found3 from (3) as: and ⇢ ⌧ and R ◆ ⌧ where GM 1 2 ✓ 3 0 GM@ =4 as a function of T T Subtracting (3) from (4), we have ±

where an integration constant has been absorbed by a shift in the origin of the expression for

given in (5). A second integration constant has been absorbed by a shift in the R

with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and

sthe is

⌧

(6) coordinate.

(5)

⇢

oriae The coordinate.

⇢ 2

s (4)

(3)

tanh

R s

(2)

, +

3 ◆

) 1

GM GM

dR. ✓

, (2

R @

◆

1 = (1)

(1) 3 (2) (3) (4) (5) (6)

⇢ 3

0 / )

a efudfo 3 as: (3) from found be can

2 R 1

H ⇢

0 is the ⇢

r dR,

⇠ ◆ ⌧

⌧

GM 

(

2 = and

t 1

✓

⌧

1 r R

JCAP02(2016)0643

=4 coordinate.

± T

GM T

. R 2

=0 2 t

✓

s . .

ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given coordinate. The

⌧

⌦

R + R

d d ⇢ γ

, the domain wall is and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

2 , R

R ⌥

/t dT

safnto of function a as GM

⇢ 2 2 t

with + + const. r t T

⇢ 2 2

= d

⇠

⌧

dR ⌦

dT 1 A R

⇢ ◆ ⇢ d

R d . In this case, the wall repels 2

GM R

R t 2 GM R = , where + GM

⌧

2 +

GM 2 ⌧ ⌧ m

GM t xrsinfor expression

d d 2 dR, dR. 2 =2 2

✓ ρ

1 1

R the of origin the in shift a by absorbed been has constant integration an where

1 . S 1

2 = 2

t 1 ◆ ◆

= γ + ⌦

2 ⌧ 3 ,

ds d

⇢

2 1 2

◆ ◆

dT )

∼ tanh R R

H R ⇠ /t GM GM

2 2

t s 2

⇠ + utatn 3 rm() ehave we (4), from (3) Subtracting t R

nti ae h alrepels wall the case, this . In , in a dust dominated GM GM

h oanwl is wall domain , the =

t 2

⇢

3, and therefore the

cr ⇢ (2 t

2 1 1

d R ✓

⌧

GM / 3

r endb h relations the by deﬁned are ✓ ✓ 1

r /

R M 1

⇢ 2 1

= =2 ,T

✓ r

S s )

⌧ GM 1. As the wall advances

bh R R =

and ⇢ R 2

GM GM

≈ +

⌧

t =0 2 2

2 2 + 2 ds + / GM s s ⌧ 2 β bh = 3

( dT

⌧ i ◆ ⇢ where +

3 2 d

◆ =2 M

os.r const. 0 

r can be found from (3) as: /t

= R dT dT

1 , R = ⇢

= σ ⇢ GM = R

± ⌥ t GM

r 2

2 c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect R 2

✓ ↵ H

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In and =0 ds 1 3 2 ∼ = = !1

"FRW" t ⇠ ⌧ 1

0 @ i

= ✓ ⇢ ⌧

,T γ ⇢ d d ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild dust bh ↵ const. r bh GM

= metric Schwarzschild the Consider

!1 2

⇢ GM =4 GM

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 ds " ,T T

, a shock wave forms. The shocked ﬂuid is bh =2

GM ±

2 R . 1

eoew osdrtee the consider we Before m t cosmology. dust a and GM , the domain wall is ρ

= t as a function of , the repulsive ﬁeld can be ignored. In this case, for =2 H

are deﬁned by the relations

=0 ⌧ ⇢ T – 10 – peia eino fvacuum. of of region spherical R t

i /t

2 ⇢

t ⌧

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we R universe FRW t ambient the in and universe, baby a singularity 1

" !1

⌧

= h eusv edcnb goe.I hscs,for case, this In ignored. be can S ﬁeld repulsive , the ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

)

t and t ⇠

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, given in (5). A second integration constant has been absorbed by a shift in the bh ⌧ R h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than t faster grows size its while it around matter the ⌧ ,T

!1 ⌧

bh GM

GM G . In this case, the wall repels

,T . R ( GM

R =0

the size of the bubble starts growing exponentially in a baby

For Here, we will be primarily interested in the opposite limit, where Before we consider the e bh Consider a spherical domain wall embedded in a spatially ﬂat matter dominated universe. Two GM ucl hik ne t eso n om lc oeo radius of hole black a forms and erent time-scales are tension relevant to its the dynamics of such a under wall. One is the cosmological scale shrinks quickly

=2 ) t

⇠

=2 =2 (1)

↵

S For

0 ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched } conformally

G 1

GM R ( Subtracting (3) from (4), we have where an integrationexpression constant for has been absorbed by a shift in the origin of the with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and Consider the Schwarzschild metric In Lemaitre coordinates, this takes the form where R ⇢ di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by ﬁeld cosmologicalquickly of expansion. shrinks the under Eventually, when domain its the tension wall wall and falls forms a withinthe black the matter hole horizon, around of it radius itleads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust in ambient region which expansion of the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant1.1 cysted in a Matching Schwarzschild to a dust cosmology 1 Spherical domain wall in dust cosmology

R ⇠

σ R

=2 H

, t t t

Schwarzschild R =0

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent R

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and t (2)

↵

S {

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider

"" di

!1 GM

,T 0

⇢

bh =2 Min R 2 . (3)

GM bh ∼ ⌦ Inﬂa=ng bubble" " 2 d

=2 GM cr R (4) R cosmology dust in wall domain 1 Spherical t 2 + 1 =2 =0 1 dR R H i S t (5) = ◆ 1 Spherical domaint wall in dust cosmology GM GM 2 R 2 . , the wormhole “closes” and any signals⌦ which are sent radially inwards end up at the =2 =0 2 d bh . A space-like slice of an inflating bubble connected by a wormhole to a dust dominated flat t , the repulsive field can be ignored. In this case, for . Formation of a black hole by a large vacuum bubble, with 1 R 1 . 0 Consider a spherical domaint wall embedded in a spatially flat matter dominated universe. Two R ⇢ 1 ) 2 ✓+ 2 + dR, (6) erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale GM ↵ ⌧ ect of the domain wall, let us first discuss the matching of⇢ Schwarzschild Initially, as mentioned in subsection 2.1, the Lorentz factor of the bubble wall relative G R ↵ d di ( dT 2 1 coordinate. The and the other is the acceleration time-scale due to the repulsive gravitational field of the domain wall GM ◆ dR. GM ⇢ ⇠ For ◆ 2 2 t GM R∼ R 2 R is the into ambient radiation withinitially at energy rest density in the frame of the wall, and its proper density is of order universe, which is connecteda by geodesic a observer wormholebaby at to universe. the the edge parent Thist of dust is the dominated represented FRW dust universe. bySchwarzschild region singularity. the Initially, can blue send arrow signals in through the the wormhole figure. into However, the after a proper time is the Lorentz factor ofradiation, the shocked the fluid shock relative to front the moves ambient [11]. outward at In the a frame subsonic of shocked speed Figure 4 to the Hubble flow at the end of inflation is of order FRW universe. analysis of this case, sinceare a beyond detailed the description scope will of require the further present numerical work. studies which spatially flat FRW.Instead, In at this the case, time the bubble does not collapse into the Schwarzschild singularity. pressure of the shocked gas has time to equilibrate while the wall is pumping energy into it. Figure 3 + 2 ◆ ⌧ conformally stretched by cosmological expansion. Eventually, when the wall falls within the horizon,⌧ it 1 GM coordinate. Here, we will be primarily interested in the opposite limit, where d 2 R quickly shrinks under its tension and forms a black hole of radius 1 T ✓ , GM ⇢ ✓ = 2 1 2 R 3 , the matter around it while its size grows faster than the ambient= expansion rate. As we shall see, this / 2 1 ds ✓ leads to the formation of a wormhole. The dust which was originally in the interior of the wall goess into R ) 1 a baby universe, and in the ambient FRW universeds we are left with a black hole remnant cysted in a R GM Before we consider the e 2 GM A spherical region of of vacuum. dT 3 GM / (2 2 s 2 ± + s1 and a dust cosmology. = dT ) ⇢ ⌧ 1.1 Matching Schwarzschild to a dust cosmology d ⌥ + tanh = ⌧ 3 ( Consider the Schwarzschild metric ⇢ R d 2 = are defined by the relations GM R 2 ⇢ 2 s / + In Lemaitre coordinates, this takes the form can be found3 from (3) as: and ⇢ ⌧ and R ◆ ⌧ where GM 1 2 ✓ 3 0 GM@ =4 as a function of T T Subtracting (3) from (4), we have ±

where an integration constant has been absorbed by a shift in the origin of the expression for

given in (5). A second integration constant has been absorbed by a shift in the R

with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and JCAP02(2016)064 , τ t ∼ ∆ ∆ t γ m given (2.40) (2.39) (2.41) ρ . This 2 2 bh ) = γ / i 1 t M t ∼ − τ t ∝ ∆ s ) = ( t ( t γρ a between the wall and the w , the radius grows as γ max R R γσ. . + i , t 3 i R / t ∆ 1 , the bubble initially grows until it reaches a , after which it collapses to form a black hole τ m cr ρ ∝ 2 – 11 – max L M γ R R ∼ , we expect that the wall will only be in contact with σ i i , this energy will be distributed within a layer of width t bh , with a tendency for the gap between the bubble and i γ . Here, we have neglected the rest energy of the ambient i t t t M & t relative to the shocked fluid. . This is the characteristic time which is required for the pressure 2 i − t γ , the bubble wall will be recoiling from ambient radiation at mildly i i t t . By that time, the shock will have dissipated, and the wall will have, at 2 2 i ) σ ∼ , the expansion of the universe, and of the bubble, start playing a role. The /t i t G/t . After time of order i t i t . As mentioned after eq. (2.37), for ( ∼ & bh is the proper time ellapsed since the shock starts forming, and ∆ ∼ t G/t m is proper time on the wall. Note that the ambient radiation of the FRW universe τ M ) ρ ∼ τ m i ρ Eq. (2.40) shows that the bubble wall looses most of its energy on a time scale ∆ For i σ i γ ( near the position of the wall, and its density will be comparable to the ambient energy γ i The left hand side of thisright relation hand represents the side initial is energy the of the energy wall. of The the first layer term of in shocked the fluid. This is of order means that eventhe in Hubble the flow absence onthe the ambient of Hubble timescale external flow pressure, to grow the in bubble time. motion In what decouples follows, from and awaiting confirmation from Consider a small region of size near the wall.(including the At expansion such of the scales,energy universe) can the conservation be per wall neglected. unit looks In surface the approximately of frame the flat, of ambient bubble and radiation, wall gravitational can effects be expressed as of mass where expands faster than that, as a function of its own proper time, since where ∆ approximately by eq. (2.20).turning point, For at some maximum radius t density most, a mildly relativistic speed with respect to the ambient Hubble flow. is the corresponding interval injust the the ambient energy frame. of Thefluid, the second which wall term at is in time of the order right hand side is σ/ shocked fluid will beis at most of order one. The initial energy of the wall per unit surface is wall’s motion becomes then dominateddensity in by its the interior. effect Again, ofis the its larger fate tension, of or and gravitational smaller of collapse depends than the on vacuum a whether energy critical the2.4.1 bubble size. Small bubblesShortly surrounded after by radiation relativistic speed, due tothe the aftermath pressure of wave which therarified proceeds shock. radiation inward For dripping at ahead theradiation, of speed the the of dynamics pressure sound, proceeds wave. in our as Neglecting discussion if the in the possible bubble subsection effect were.2.3 of in such In an that empty case, cavity, along the the bubble lines has of a conserved energy of the shocked fluidthe to rough accelerate estimate theof (2.18), wall the which away is from wall it,travel based cannot at and on grow the is a too speed actually of simplified large, sound. in fluid though, agreement Hence, model. because with the relative the The Lorentz backward factor recoil pressure velocity wave can only JCAP02(2016)064 (2.44) (2.45) (2.43) (2.42) . Hence, σ , t b t i crosses the horizon R . During that time, i cr . R i t GM when the co-moving scale H t , bh . This condition led to a highly . H i t 3 i ρ , we have requires numerical analysis and is R cr GM cr M i ∼ b H M t ρ t . Domain walls will be essentially at rest . i , i H πσH ρ σ bh t 2 i i t R t = bh R M . + 4 + b M b ∼ ρ i bh – 12 – 2 b R πρ H t i t t 4 3 GM ∼ ∼ bh bh M GM . On general grounds, we expect that the radius of the black hole bh crosses the horizon M i , the situation is more complicated. A wormhole starts developing at R cr M , when the growth of the bubble becomes exponential. On the other hand, the cr bh Eventually the wall will collapse, driven by the negative pressure in its interior and by GM M ∼ is at most of the size of the cosmological horizon at the time some radiation can flowestimate together (2.42) with the for bubble into the baby universe,corresponding thus to changing the the black hole radiusthe is time much of smaller horizon crossing. than the co-moving2.4.2 region affected by Large the bubblesFor bubble surrounded at by radiation t wormhole stays open during a time-scale of order ∆ where in the last relation we have used that for The study of modificationsleft to for further eq. research. (2.42) for 3 Gravitational collapse ofIn domain the walls previous sections we considered bubbles with at the time and we can write (2.42) as boosted bubble wall atdomain the walls, which time correspond when to inflation the ends. limit Here we shall concentrate instead on the wall tension.and We bubble can wall estimate contributions, the as mass in of eq. (2.20): the resulting black hole as the sum of bulk numerical studies, we assume thatthereafter. the influence of radiation on the wall’s motion is negligible In a radiation dominated universe, the co-moving region of initial size with respect to the Hubbleof flow bubbles at the we end assumedtransfered of that inflation. to the matter Another inflaton difference thatwhere is field lives that matter lives also for is only outside. the created outside case both Here,choice, the inside since we and bubble, domain will outside walls so explore the areand the its domain supposed we alternative energy wall. may to possibility is This expect be seemsthe the the to presence physics result be of of of a matter both spontaneous natural inside symmetry domains the breaking, to domain be wall essentially has the a same. dramatic As effect we on shall its see, dynamics. JCAP02(2016)064 . σ t (3.5) (3.4) (3.1) (3.2) (3.3) H t and then σ t H t , after which the walls σ t is very small . ∼ t max 2 H t V/M GR ∼ . . ∼ 1 H . t where this pressure balances the wall is the mean squared velocity of matter 2 2 bag bag , 4 bag 3 max bag / 3 . Depending on their initial size, their fate R R 1 / /R R max GM ) 1 GM a ) σ R = bag σ ∝ ∼ ∼ t /t ∼ ) R p 2 H H R t – 13 – t GM ( ( , which can be estimated through the relation ∼ bag max ∼ ∼ ∼ R R ( max 2 bag σ Φ v R V R max − R ∼ − , the ball of matter inside the wall starts collapsing under its , where 2 max 2 max start creating wormholes at the time v R m σR σ t ρ ∼ are initially at rest with respect to the Hubble flow, and are conformally p is the mass of the mater contained inside the wall, and we have neglected H i t t /G , the graviational field of the walls is completely negligible. Walls of super- σ H t t R ∼ t bag M Let us first consider a situation where matter is reflected off the wall with very high For ). However, part of the kinetic energy will be invested in increasing the surface of the R ( The relation (3.4) is satisfied for a radius which is comparable to the maximum radius, probability. In thisit case, keeps even pushing if outwards.description, the its Initially, wall positive thisV tension kinetic matter exerts has energy an the exactlydomain inward escape balances wall, pull, velocity. up the matter to negative In a inside Newtonian a maximum of size potential Newtonian will be different. As we shall see, small walls enter the horizon at a time The fate of thewith wall matter, after which horizon is crossingtwo highly depends contrasting model crucially limits. dependent on [13]. the To nature illustrate of this its point interaction we may consider Here a small initial contribution of the domain wall to the energy budget. From (3.1) we have horizon size stretched by the expansion of the universe, they begin to decouplethe from domain the wall, Hubble theylong flow. may lived Depending either remnants on in collapseLarger the the to walls interaction form a with of of black matter pressure holeundergo supported with exponential singularity, bags or expansion of in they matter, baby may bounded universes. develop by3.1 the wall. Domain wallsThe surrounded case by of walls dust surroundedthe by wall dust can is be particularly completelysmall simple, ignored. walls, since for the As effect which we of self-gravity did matter is for outside negligible, bubbles,3.1.1 and here large we may walls, distinguish for Small between which walls Small it walls is are important. frozen in with the expansion until they cross the horizon, at a time At that radius the gravitational potential of the system Φ After reaching the radius own weight, helped also bya the force pressure exerted of by order theparticles. wall tension. The As collapse a willtension: result, halt matter develops at a radius JCAP02(2016)064 (3.8) (3.9) (3.6) (3.7) (3.10) ) = 0 i t ( 0 r t ) is increasing. i t H ( M 5 10 . . The condition t ) = 0 2 4 0 r 10 2 , , a H 2 . 0 t r . − .5 Figure shows the evolution of the 2 2) bag i 2 2 H a / t (1 r R 2 2 σt − a 2 m 1 ra 1000 = (3 ) i √ H 1 + t πC t , this will simply behave as a test domain wall ( − 0 9 σ πσ t R – 14 – ∼ H ≈ Hr ) satisfies the differential equation ) t ( 2 bag bh 0 ) = r 100 ) = 4 H r t t M H 2 ( as a function of time. t a GM ( 3 M R ar − = 10 R + (4 00 r can be found by means of a numerical study. This is done in appendixA. 1 R . i m 1 5 4 H 10 − C 10 10 100 1000 H through the relation = H t R . Evolution of the radius of a test domain wall in a matter dominated universe, as a function We find that while the wall is stretched by the expansion the mass Next, let us consider the case of a “permeable” wall, such that matter can freely flow A test wall can be described by using the Nambu action in an expanding universe. The , when H is imposed at somedomain wall initial physical time radius such that where the comoving radial coordinate Here a prime denotes derivative with respect to cosmic time Defining in an expanding FRW. Oncemuch the like wall it falls would within in the flat horizon, space, it and will the shrink under mass its of tension the black hole can be estimated as from one side of it to the other. For The velocity of matterexpect any inside substructures the to bag develop is by gravitational of instability the inside order the bag. of the virial velocity, and we do not The coefficient Figure 5 of cosmological time, for differentt initial radii. The green line corresponds to the horizon crossing time mass of the wall is defined as and much larger than the gravitational radius, JCAP02(2016)064 (3.16) (3.13) (3.15) (3.14) (3.11) (3.12) , the numerical . i 2 t Ω d 2 i no longer holds, as we R R 2 H + ) approaches the constant 2 σt t ( ∝ dR M 1 bh − , its radius starts growing expo- σ . M t . ± ± 1 C σ. R 1 − 2 = 1, we have − GM 2 2 ± 2 ˙ . /G R C 1 , 1 ± − 15 σ . and the mass + − ∼ 1 ) 0 R ± H > t cr GM t ≈ /R 2 ∗ + – 15 – R M ± m R 2 GM − C . 2 1 . We find that, in the limit dT GM bh = H 2 t 2 M = ± ˙ − R ˙ T (1 R GM R − 2 2 ˙ − T for the interior layer can in principle be different from the mass ) . We shall also assume that the wall is separated from matter, on 1 3 − / /R 2 t M ± − ∝ = , and geodesics satisfiy the conservation of the corresponding canonical 2 a GM T for the exterior. Let us now find such mass parameters by matching the 2 ds − + M , and the resulting black holes have mass is a constant, and a dot indicates derivative with respect to proper time. Initially, σ t ± , we assume that the metric inside and outside the wall is initially a flat FRW metric ∗ . C t To illustrate this process, we may consider an exact solution where a spherical domain By continuity, the particles of matter at the boundary of a layer must follow a geodesic is positive or negative depending on whether the particle moves to the right or to the H t ± both sides, by empty layers oftime.) infinitesimal width. By symmetry, (As the we shall metric see, within these these layers will two grow empty with layers takes the Schwarzschild form wall of initial radius value given by eq.coefficient (3.7) approaches the once value nentially and decouples from the Hubble flow, forming a wormhole. is surrounded by pressuerelesstime dust. The solution iswith constructed scale as factor follows. At some chosen Here the geodesic is inC the white hole part of the Schwarzschild solution (region I in figure6), and left. Using (1 of both Schwarzschildsingularity, and and FRW. they Such endindependent geodesics at of originate time-like atmomentum: infinity the (see white-hole/cosmological figure6). The Schwarzschild metric is shall now describe. 3.1.2 Large domainIf walls a surrounded domain by wall dust has not crossed the horizon by the time The mass parameter parameter metric in the empty layers to the adjacent dust dominated FRW. the wall starts recollapsing around the time This result is inhowever, agreement that with the a more resultat sophisticated (3.11) calculation only done applies in for ref. sufficiently [17]. small We walls. note, Such walls collapse For larger walls, self-gravity is important, and the scaling

sthe is

⌧

(6)

coordinate.

(5)

⇢

oriae The coordinate.

⇢

2 (4)

(3)

tanh

R (2)

+ 3

) ◆ 1

, 3

, GM

(1)

1 dR.

= 1 GM

◆

2 (5) (6) (1) (2) (3) (4) (1) (2) (3) (4) (5) (6) ✓

⇢ R

⇢

@ 1

H 3

0 is the

/ is the R )

2 a efudfo 3 as: (3) from found be can r dR, r

⇢ 0

⇠ 1 ◆ ⌧

⇠ ⇢ ⌧

= =

1 t 2

⌧

2 ( 

r ✓ r and R GM coordinate.

1 coordinate.

3 ⌧

T =4

. the is GM =

2 . GM

⌧

=0 T ± =0 2 t R

✓ .

R coordinate. The

2 ⌧

⌦

2 coordinate. The

⌦

d ⌧

d ⇢

R coordinate.

+ 2

, the domain wall is the in shift a by absorbed been has constant integration second A (5). in given

R 2 ⇢

R /t

, the domain wall is

⇢

+ 2 R t GM (6) dT , s

⌥

+ /t

⇢ const. r 2

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

2 t safnto of function a as

2 const. r ⇢

. dR ⇠

d ⌧

⌦ T

with = 2

= ◆ 1 A

R d

⇠ ⌧

± ⌦ R . In this case, the wall repels 2 ⇢ 1

1 A

⇢ ⇢

d GM R

R t

GM A

R R . In this case, the wall repels 2

2 GM

⌧ +

2 +

, R

GM GM

⇢ R

⌧ GM 2 2 dR, dR.

⌧ GM 1

⌧ d

✓ 2 +

. d 1 1

R GM . 1 t S

1 t ,

GM xrsinfor expression 2 2

⌧ dR, dR. dR =2

1 R s

1 ◆ ◆ 1 1 ⌦ s

R the of origin the in shift a by absorbed been has constant integration an where

(5) .

S 1 R

2 d

= / 2

⇠ dR

ds 2

oriae The coordinate. 2 1

/t R

◆

1 ) ◆ ◆ =

JCAP02(2016)064⇢

⌦ dT ⇢

t tanh R R 3 2

t R

⇠ GM GM d

= ⇢

nti ae h alrepels wall the case, this . In 2 2

2 tanh 1

h oanwl is wall domain , the

⇢ +

◆

R R

)

t GM

0 tanh R R t GM

R 2

GM GM GM

r 2

⌧

⇢ 2

GM utatn 3 rm() ehave we (4), from (3) Subtracting

GM 2 2 (2

1 1 = s

2 d +

R 3

R GM

✓

r ✓ ✓

GM GM

R / 2

!1 1

2 2 1 ◆ ⌧ r endb h relations the by deﬁned are +

S ⇢

(4)

R 1 1

✓ ⇢ R =0 !1 d s )

R GM

t GM

R R

✓ ✓ ⇢ 2

GM GM 2 GM = +

@ 2 / + ✓

2 2 1

2 3 2 (3.18) (3.19) (3.17)

and 2 ,T

2 +

2 R

✓

3 ⌧

R / s

ds ) GM GM, T ,

s s

⌧ 2

3 R R 0

os.r const. ⇢

2 (3) ( dT

GM GM

⌧

3 +

) ◆

/ + + 1 3 2 1

= 2 2 d

◆

+  ⇢

where

GM can be found from (3) as:

a efudfo 3 as: (3) from found be can

GM / GM

s s (2 ⌧ 2

⇢ 3 dT dT

R R

= ⇢

( dT GM

⌧ =4

± ◆

± ⌥

GM 2

+ 3 2 T

and

GM d

)

ie n() eoditgaincntn a enasre yasiti the in 2 shift a by absorbed been has constant integration second A (5). in given

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect ◆

2 3

⇢ 2

⌧

" /  ✓

↵ R ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

2 can be found from (3) as:

, + 

◆ and ds 2

dR.

1 ⌧ 1 3 = = = R dT dT

R (

= = ⇢

!1 form the takes this coordinates, Lemaitre In GM

(5) (6) (1) (2) (3) (4)

= with ⌧

⇢ ⇢ 0 @

± ⌥

GM 2 R (2) 3

2 = ✓ ⇢ R H ⌧

2 0 is the

✓

R d d

r ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild

✓ and ds

⇠

GM ⌧

1 3 ↵ = =

◆

!1 2 GM safnto of function a as dR,

!1 t = 1

= ⌧

1 T

universe 0 @

GM, T

r osdrteShazcidmetric Schwarzschild the Consider GM coordinate. 2

✓ 2 ⇢ ⌧ ,T 1 =4

s d

ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild

bh ds

T R =2

. bh

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 T

. Assuming that the ini-

= ✓ t ↵ "FRW" bh GM

=0 2 ∗

⇢ + ± (1)

R 1

GM =

0 ⌥ coordinate. The

⌧ dT r for expression

R R eoew osdrtee the consider we Before 2 GM

Parent cosmology. dust a and have the same value:

=2 =4 GM ⇢ as a function of

= 1 ,

. dust

, the domain wall is s

= r , the repulsive ﬁeld can be ignored. In this case, for the of origin the in shift a by absorbed been has constant integration an where

ds R =0 ,

GM are deﬁned by the relations

/t =

T − 2

. ⇢

=2 2 ⇢

H t

peia eino fvacuum. of of H region spherical i d G

R ⇢

⌦ t . =0 ± t

!1 . const. r ±

" d 2 4

R dT

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and 1 universe, baby a

1 R 3

t ⇠ M ⇠

⌧ ⌧ R

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the =

⌦

1 A

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads +

bh )

as a function of

. d ,T

t t

and

= given in (5). A second integration constant has been absorbed by a shift in the

. In this case, the wall repels 2 ⇢

R ⇢ ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, ⌧ bh , the repulsive ﬁeld can be ignored. In this case, for

= H

⌦

2 2 h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

d are deﬁned by the relations =0 const. r d

⌧ ⌧

R GM GM G d t

GM . R .

t 3 ∗ R R

= 2

and (

⇢ ⌧

=0 R

R 1 t

⇢ . + 2 + II"

Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two Here, we will be primarily interested in the opposite limit, where Before we consider the e =2

) ±

erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale GM

utatn 3 rm() ehave R we (4), from (3) Subtracting

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly t t 1

⇠ +

=2 2 2

dR, dR.

2 +

GM =2

⌧ ◆ ) ↵

= dR s 1 For

3 0 ⌧

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally 1 1

2 R 2 d ∗ .

( )

S R 1 Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and 1.1 Matching SchwarzschildConsider to the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form where Subtracting (3) from (4), we have where an integrationexpression constant for has been absorbed by a shift in the origin of the with GM 1 Spherical domain wall in dustR cosmology ⇢ di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by cosmological ﬁeldquickly expansion. of shrinks the under Eventually, when its domain the tension wall and wall forms falls a withinthe black the matter hole horizon, around of it it radius leads while to the its formation sizea of grows a baby faster wormhole. universe, than The andspherical the dust region in ambient which of expansion the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall dust this goes with cosmology. into a black hole remnant cysted in a t ⇠ and

2 singularity t

2 dR given in (5). A second integration constant has been absorbed by a shift in the

R R is the time at which the boundary of such " R M .

1 = i (

1 ◆ ◆

⌦

⌧

. GM t 3

=0 t

G , the domain wall is t d

= / I"

= 2 R

ds m

2 (

8 = ✓ 2 III" 1

t H

=0 GM

⇢ ◆ rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent )

t n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is ⇢ other the and

2 ρ

t Before we consider the e Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two Here, we will be primarily interested in the opposite limit, where

tanh R R

erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale R = 1 ↵ ⌧ 1

S t

GM GM

⇠ osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain =2 spherical a Consider

di = ↵

2 /t 2 2

⌧ π + 3

0 +

=0 R R

t ±

◆ 4

GM relations the by deﬁned are

GM GM t Subtracting (3) from (4), we have where an integrationexpression constant for has been absorbed by a shift in the origin of the with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by ﬁeld cosmologicalquickly expansion. of shrinks the under Eventually, when domain its the tension wall and wall forms falls a withinthe black the matter hole horizon, around of it it radius leads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust in region ambient which of expansion the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant1.1 cysted in a Matching Schwarzschild toConsider the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form where 1 Spherical domain wall in dustR cosmology ⇢ 2

2 – 16 –

2 ⇢ dT

0 ⇢ C

(2 ⇢ ⇠ !1 ⌧ 1 1

=2 t ! R d

to be positive so that the geodesics originating at the white R

and 3 IV"

,T GM ✓ ✓

R / ˙ GM ⌧ !1 2

bh . In this case, the wall repels 2 1 R R

is proper time along the geodesic of the dust particle,

✓

t GM R

✓

=2 2 s

) GM C t GM 1 R R where

⇢ S 2 Schwarzschild

GM GM " +

R M

R (1) 2 2 peia oanwl nds cosmology =2 dust in wall domain 1 Spherical " 2 + = 2 +

R /

GM, T

s s

⌧ 2

3 ≡ 2 . ds ( ⌧ t dT ⌧

◆ + 3 2

◆

 "

bh can be found from (3) as:

!1 /t

⇠ (2) , where

dT dT

nti ae h alrepels wall the case, this In . R

,T t R

= ⇢

2 GM = t R 3 universe

M bh form the takes this ± coordinates, ⌥ Lemaitre In GM 2 "

, / inﬂa=on

2 1 ✓

R GM

⇠

GM bh is wall domain the ,

H t

and ds 1 3 t

⌧ = =

=2 "of" 2

!1 . t

=2 S GM ⌧

iden=fy

0 @

Baby

(3) ∝ R R ⌦ R ✓ ⌧ ⇢

2 d ⌧

1 =2 . A large domain wall causing the birth of a baby universe. d

d ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect end R R t =0 ↵ ↵

R H 2 + (4) GM = i GM, T 1 metric Schwarzschild the Consider S t dR 2

1 Spherical domain= wall in dust cosmology =4

GM t =2 . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 T ± =2 =0 R GM◆

2 Figure 6 (5)

R t R

1 . , the repulsive ﬁeld can be ignored. In this case, for 2 .

⇢ 0 Consider a spherical domain wall embedded in a spatially ﬂat matter dominated universe. Two as a function of t cosmology. dust a and

, the repulsive ﬁeld can be ignored. In this case, for =0 ⌦ =0 eoew osdrtee the consider we are deﬁned by the relations Before erent time-scales) are relevant to the dynamics of such a wall. One is the cosmological scale 1 2 d is simply the total mass of matter inside the domain wall, which is also the mass T ↵ t

i R

⌧ R ⇢ G t

di .

( R ect of the r domainGM wall, let us ﬁrst2 discuss✓+ the matching of Schwarzschild

1 + 1

↵ 2 bh

and the other is the acceleration time-scale due to the repulsive gravitational ﬁeld of the domain wall⌧ dR, = vacuum. of of region spherical S

=2 ⇢

⇠ ) For ⇢ dT t

t d and

(6) M given in (5). A second integration constant has been absorbed by a shift in the R

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant GM hole black a with left are we universe 1 FRW ambient the in and ⌧ universe, baby a

G GMGM ◆ 2 GM◆ dR.

R 2 coordinate. The ( R 2 2 We conventionally choose the constants

conformally stretched by cosmological expansion. Eventually, when into the wall goes falls wall within the of the horizon, interior the in it originally was which dust The wormhole. a of formation the to leads

=0 R R 8 Before we consider the e Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two Here, we will be primarily interested in the opposite limit, where ⇢

Here, we will be primarily interested in the opposite limit, where erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale t = +

quickly shrinks under its tension and forms a black hole of radius ⇠ 2 =2

↵ ⇢

h atraon twieissz rw atrta h min xaso ae sw hl e,this 0 see, shall we As rate. ⌧ expansion ambient the than faster grows size its while it around matter the

1 ◆

d where limit, 1 opposite the in GM interested primarily be will we Here, For geodesics with the escape velocity, we have This has the solution tial matter density isFriedmann equation the leads same to inside the conclusion and that outside the wall, comparison of (3.18) with the Here, which is excised fromexcised the region exterior crosses FRW, the and horizon. hole singularity move to thefigure6). right, from Note region that Iseparate towards we Schwarzschild region should solutions. II think invacuum the In of layer extended is the figure6 Schwarzshild on diagrams twothese the (see empty are left, layers depicted and on the side both solution by sides for side. the of exterior the The vacuum wall solution layer as is for on segments the the of interior right. two matching the behaviour ofof the the flat two FRW expansion infinitesimal factor. empty The layers geodesics initially at coincide the at boundaries The equation of motion (3.16) then takes the form Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and with Subtracting (3) from (4), we have where an integrationexpression constant for has been absorbed by a shift in the origin of the where spherical region of of vacuum. and a dust cosmology. 1.1 Matching SchwarzschildConsider to the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form the matter around itleads while to the its formation sizea of grows a baby faster wormhole. universe, than The and the dust in ambient which expansion the was rate. originally ambient in FRW As the universe we interior we shall of are the see, left wall this goes with into a black hole remnant cysted in a R ⇢ di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by cosmological fieldquickly expansion. of shrinks the under Eventually, when its domain the tension wall and wall forms falls a within black the hole horizon, of it radius

1 Spherical domain wall in dust cosmology 2 is the

!1 R

the matter around it while its size grows faster than the ambient expansion rate. Aswe for shall case, see, this In this ignored. be can ﬁeld repulsive the ,

✓ ✓ coordinate.⌧ t

2 = GM ⌧

leads to the formation of a wormhole. The dust which2 = was originally in the interior of the wall goes2 into

ucl hik ne t eso n om lc oeo radius of R hole black a 1 forms and tension its under shrinks quickly , T

a baby universe, and in the ambient FRW universe we are left with ads black hole remnant cysted in a 3 , ⇢ R

/ .

Before we consider the e ds 1 M T GM, ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

spherical region of of vacuum. s R ✓

1 )

) For

=2 1 G GM ⇠

( dT R R

and a dust cosmology. 2 GM

!1 3 A

± s+ / (2 GM 2 t 2 = 1

1.1 Matching Schwarzschild to a dust cosmology dT s

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational ⌧ repulsive the to due time-scale ) acceleration the is other the and

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One d wall. a such of dynamics the to ⇢ relevant are time-scales erent

⌥ T GM,

Consider the Schwarzschild metric + ↵

= tanhdi =2

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall 3 (⌧ domain spherical a Consider

⇢ R

d 2 R are deﬁned by the relations =0

= ⇢ R GM In Lemaitre coordinates, this takes the form R 2 and 2 / s+

⌧ can be found3 from (3) as:

⇢ peia oanwl nds cosmology dust in wall and domain R ◆ 1 Spherical where ⌧ GM 1 2 ✓ 3 0 GM@ as a function of=4 Subtracting (3) from (4), we haveT T ±

where an integration constant has been absorbed by a shift in the origin of the expression for

given in (5). A second integration constant has been absorbed by a shift in the R

with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and JCAP02(2016)064 , i t σ and = 1 > t (3.22) σ , with ∗ (3.21) (3.20) ± t ∗ ensures R t C at some σ σ t and . The exact ∗ > t t ∗ σ t t at the time ∗ R σ t . ∗ t   σ t 2 ∗ ∗ t 9 R 4 . Using pl 3 ∗ R m ) , the effect of this interaction 4 2 ∗ . σ t t 9 ∗ bh ∗ / t R . 3 , 2 1 after the time when the first term t σ GM = (4 R t − σ − 1 2 bh − 2 ∗ 2 σ = 2 τ/t t t e 9 4 1 σ GM R t − ∝ bh H 1 + when the empty cavity containing the black R R + s GM H   t 2 . As we have seen, the condition 9 bh – 17 – ∗ the four-velocity of matter at the edge of the cavity and by t 2 ∗ 1. − 2 ∗ R t µ R , the wall moves faster than the dust particle at the 1 9 . Moreover, it is easy to show that for GM 4 σ 1. U ∗ 2 t ) − σ ± 1 = ) /t σ 2 = R > t ∗ ˙ /t t = R ˙ ∗ ( T t ν ( ∼ W . In this case, the wall may initially expand a bit slower than the ∼ µ v σ v U µν < t g i ), or − 2 σ R in (3.20) refers to the embedding of the domain wall as seen from the = /t 2 ∗ w t ± ( γ eq. (3.21) has no turning points and the radius continues to expand forever. , we see that, for O σ bh . This coincides with the time . M σ bh = 1 + t R > t = w , the wall will be approximately co-moving with the Hubble flow at the time . is the proper time on the worldsheet. This expansion is of course much faster than γ ± σ GM ∗ τ t Nonetheless, in order to make contact with the setup of our interest, we must consider In the above discussion, we did not make any assumptions about the time For The dynamics of the bubble wall in between the two empty layers is determined from M the four-velocity of the domain wall, the relative Lorentz factor is given by ∼ This can be seen as follows. Denoting by > t 9 µ t ∗ will be small, andsolution we expect discussed the above qualitative provided behaviourtime that to be the very size similar of to the that of wall the becomes exact larger than a relative velocity of order when inflation ends, matter inside. Consequentlysince some superhorizon matter walls are may approximately come co-moving for into contact with the wall. However, a broader set of initial conditions such that the wall can be smaller than solution illustrating the formationcan of be a constructed wormhole for any is initial obtained value by of a matching procedure, and R hole crosses the horizon.the An baby observer universe, at but the not edge after of the the time cavity can when initially send signals into that the wall willof not matter remains run undisturbed into after matter. The wall evolves in vacuum, and the Hubble flow we have W in the rightHere hand side ofthat (3.21) of can the Hubble bespeaking, flow, neglected a and wormhole and takes in the place the∆ extended in second Schwarzschild solution the forms one baby and universe becomes closes (see on dominant. a figures6 time-scale and7). Loosely where we have used (3.15), (3.16), (3.20) and (3.21), with 2 interior and exterior Schwarzshildthere solutions has respectively. to be Theof embedding a the jump is wall in different, to the since the extrinsic other. curvature Comparing of the (3.20) worldsheet and as (3.21) we with go (3.15) from and one (3.16) side with The double sign and The expansion reaches exponential behaviour edge of the expandingthe interior direction ball opposite of tointo matter. the matter exterior and As FRW its seen Hubble motion from flow. is the well Consequently, the outside, described wall by the never (3.21) wall runs thereafter. moves in Israel’s matching conditions, which lead to the equations [9] and JCAP02(2016)064 1. 3 (3.23) (3.24) (3.25) (3.26) / 2 ) σ /t H t ( ∼ . H t . . From (3.23) we then find that the 3 4 , / ) 1 2 H H /R σt t . and becomes negligible compared with the σ H r 3 H t t / t 2 σ R 1 ) between the inside and the outside: ) )( πC /t ∼ H 2 H p H 16 G H t p /t – 18 – t σ ≈ ∆ /Gt t ∼ ( (1 bh ∼ bag M ∼ M p bag R continue to be dragged by the Hubble flow even after they have crossed the σ t . A space-like slice of a baby universe with an expanding ball of matter in it, connected by a H t Next, we may consider the case of “permeable” walls. Like in the case of dust, the Figure 7 wormhole to an asymptotic dustgravitational dominated field FRW of universe. an The inflating wormhole domain is wall, created depicted by as the repulsive a3.2 red line. Walls surrounded3.2.1 by radiation Small walls Let us start by discussingwith the situation where the domainhorizon. wall is Unlike “impermeable”. dust,initially Small radiation walls its outside effect cannot the betension wall neglected. balances may the The continue difference radius exerting of in pressure the pressure on wall ∆ will it, stop and growing once the wall After that, the pressure outsidepressure decays inside, as which ( isequilibrium given radius by is of order dynamics of small walls isof well the described resulting by black the holes Nambu action can in be a estimated FRW universe. as The mass The mass of the bag of radiation is of order Therefore the gravitational potential at the surface of the bag is very small Φ JCAP02(2016)064 (4.1) (4.2) (3.27) (3.28) (3.29) for this case bh ; then they are 1 M − i H is the nucleation time, ≈ n t R is , and some of the radiation may , where . 4 σ 1 t − )] i − i n t H H − . t ( . . . i H n t t S H H G 62 which we found in the previous subsection t − . − 0 t , so wall gravity can be neglected during inflation. 10 ∼ σ Ae t is determined from the condition ) = 2 exp[ bag ≈ 1 – 19 – H = bh r H M t − i t ( C λ 1 M H − i R H and ≈ ) t bh ( R M . Continuity suggests that the estimate (3.29) may be valid also G 2 / H t ≤ the estimates for a wormhole will start forming near the time bh σ σ t t M ∼ H H t t As in previous sections, we assume that 10 where in a radiation dominated universe The numerical coefficient is foundand by we using find the numerical approach described in appendixA, Note that, although permeableradiation dominated walls eras, do their masses form arebags black much of smaller holes, than matter the both trapped masses in by of the impermeable the corresponding walls. dust3.2.2 dominated and Large domainFor walls surrounded byconverge radiation to the value For follow the domain wallhole into embedded the in babytherefore the universe. FRW Nonetheless, universe the cannot size be of larger the thanrequires resulting a the black numerical cosmological simulation horizon, and is and left for further4 research. Mass distribution ofIn black the holes earlier sectionsformed by we domain have walls or outlineddistribution by a vacuum of bubbles number black nucleated during of holes inflation.and depends scenarios bubbles The on whereby resulting and the black mass microphysics onpossibilities holes parameters here their can and characterizing interaction will be the focus withweakly walls instead matter. with on one matter We specific (soparticle case: will that physics not domain matter walls example attempt particles which interact toModel could can very particles explore freely be are pass all axionic suppressed through the originate domain by the from the walls, walls). large a whose Peccei-Quinn “shadow” Agravitationally. energy couplings sector specific scale. to of The the the walls theory, Standard could which also couples4.1 to the Standard Size Model distribution only Spherical of domain domain walls wallsstretched nucleate by the during expansion inflation ofthe the having radius universe. of radius At the wall is well approximated by for large bubbles in a radiation dominated universe. A determination of The wall nucleation rate per Hubble spacetime volume JCAP02(2016)064 , is 1 − i n (4.9) (4.7) (4.8) (4.3) (4.4) (4.5) (4.6) H dt (4.11) (4.10) , where 1 − ). The size i H t ( R ) = H = t i ( , when the repulsive R R 1 − ) , when 1 in about a Hubble time. and in a time interval ) πGσ is the number of inflationary H ∼ H x t t 3 ( C d inf = (2 N 1 . − σ n M with H , dt ) , 4 x . . H i . Once the wall comes within the Hubble H x ∼ 3 3 i t 2 t R dR 3 R d ( σ d 1 ) λ . n ), where 3 ) t i t GH i t CM 3 t i i i t 2 M ( ( = inf σ/H H H H a = 2 3 a σ 3 H N t e t e σ/H π & 4 i ) = dV M – 20 – dN t = , we find the distribution of domain wall radii [4], ∼ σ ( ) = ∼ = 2 t exp( R 2 λH ≡ ( 1 − M S dV A R − 1, or = i dn H πσH dN ∼ 4 S corresponds to the end of inflation and i ≈ t max in terms of H R n t M to is given by [4] 1 − i S H ∼ R is far greater than the present horizon. is the Hubble parameter, and the time ) is the scale factor, have little effect on the surrounding matter and can be treated as test walls in the max t ( σ R a t a/a After inflation, the walls are initially stretched by the expansion of the universe, The number of walls that form in a coordinate interval = ˙ H gravity of the wallt becomes dynamically important.FLRW background. Walls that The crossenergy) mass the is of Hubble such radius walls at at Hubble crossing (disregarding their kinetic H distribution of the walls during this period4.2 is still given by Black eq. holeAs (4.7). mass we discussed distribution in section3,magnitude the of ultimate two fate time of parameters: a given the domain Hubble wall depends crossing on time the relative is the mass within a sphereradius, of it Hubble collapses radius at to time a black hole of mass e-foldings. For models of inflationsize that of solve the horizon and flatness problems, the comoving where where where is the physical volume element.of At scales the from end of inflation, the distribution (4.7) spans the range In the thin-wall regime, when the wall thickness is small compared to the Hubble radius the tunneling action This estimate is valid as long as With this assumption, the prefactor has been estimated in [19] as Using eq. (4.1) to express JCAP02(2016)064 , , σ t M ∼ (4.13) (4.14) (4.15) (4.16) (4.17) . , when eq H . eq t t t

M 17 in terms of 10 , the critical mass R > is ) remains constant is the critical mass σ cr H M ( 1 GeV, which is rather cr M f . 1 GeV to the GUT scale . M 4 t > t / η ∼ 1 is the time of equal matter 4 / M 3 cr eq 2 t / M 1 eq . , where ) . cr . M g . eq 4 4 4 M . at Hubble crossing. For such domain / , R dR 7 Bλ < t varies from , where 2 ∼ dM ) (4.12) σ dn / ∼ M η DM dR to 10 dM t eq 3 H ρ t 4 4 σ t 2

/ / + 2 ( 1 3 DM t & R M M R ρ /G M 2 σ 17 σ 1 )( t 2 t ≡ / – 21 – 2, and their density at λ ∼ M H , which can be fully described analytically. Note, 1 eq ) t and & ∼ λ ( , then, as eq ∼ R/ . The boundary between the two regimes, M Gt 3 M ) R 4 H η ( = H t M / t = M t ( > t f 3 , the wall starts expanding faster than the background ) ∼ . H σ σ dn ∼ dn σ t t t ( σ 10, we have g M λ is the dark matter mass within a Hubble radius at r ∼ as ∼

η H B ) t M M 17 ( f 10 ∼ , where G 2 eq 2 / Gt 1 eq eq , and substituting in eq. (4.14): t − 2 , and have radii between / B 1 ∼ eq > t and develops a wormhole, which is seen as a black hole from the FRW region. ) ∼ σ eq σ t < t ) , we could only obtain an upper bound, t GeV, the critical mass varies from 10 M eq eq M/σ H t We shall start with the case The estimate (4.12) applies for In the opposite limit, ∼ ( 16 can take a wide range of values, including values of astrophysical interest. If we define t ( t < t 10 ∼ cr DM σ however, that this case requires the energy scale of the wall to be walls, the Hubble crossing time is Let us first consider black holesera, resulting from domain walls that collapse during the radiation small by particle physics standards. The critical4.2.1 mass in this case is and radiation densities. Fort walls that start developing wormholes during the radiation era, R in time. We shall evaluate it at the time of equal matter and radiation densities, ∼ The mass distribution of black holes can now be found by expressing A useful characteristic ofmass this interval distribution in is units the of mass the density dark of matter black density holes per logarithmic Since black hole and matter densities areeq. diluted (4.14) in the derivedρ same for way, the radiation era should still apply by order of magnitude. With Here, and its Schwarzschild radiuscorresponds is to blackintroduced holes in of section3. mass M We note thatthe depending energy on scale the of the wall wall tension at Assuming that the universeof is this dominated black by hole nonrelativistic is matter, we found that the mass JCAP02(2016)064 (4.21) (4.22) (4.18) (4.19) (4.20) ), eqs. (4.14) eq M > t H logM , let us assume for the . in eq. (4.14), we obtain eq > t eq σ t > t ( GM σ σ t ∼ t < R < t R σ , t . . (We shall see that observational 2 2 / , / eq 1 1 4 cr R dR M R/G. < t eq cr 2 M M ∼ σ Bλ. M M t ) t R ∼ H t ) eq at horizon crossing collapse to black holes with ( 2 t – 22 – Bλ λ 1 Bλ M σ > M ( ∼ = ∼ Σ , the mass is f t ) σ ∼ M ) dn M eq M R < t M ( ( M f t > t f . Mass distribution function for 4 1 M Figure 8 L M H eq logf < t . The size distribution of such walls and the mass distribution of the black holes σ cr t Domain walls that have radii For black holes forming in the matter era, but before We now turn to the more interesting case of The resulting mass distribution function is plotted in4.2.2 figure8. constraints on our model parametersstart come developing only wormholes during from the thisfor radiation regime.) era, the and In mass we this of have case, only the domain an resulting walls upper black bound holes. (4.13) M < M are still given by eqs.moment (4.15) that and the (4.17), bound respectively. (4.13) For is saturated. Then, using Finally, for black holes formed at Substituting this in eq. (4.18), we find the mass distribution and4.17) ( are replaced by JCAP02(2016)064 . cr (5.2) (5.1) (5.3) (4.23) M < M appears to be a . With the same M eq cr logM M . eq < t M > M , this implies σ < M < t cr M cr . . 6 M = 8 − − bh eq 10 M 10 M . . 1 . ) Bλ. ) <

g ) ∼ M 2 cr eq 15 ) 1 t 3 - – 23 – , the strongest bound is due to distortions of the M 10 M < M ( 10 (

Σ f t ∼ f M 3 M cr M > ( 10 we find ( M f f eq . Mass distribution function for M > ) in this case should await numerical simulations of supercritical R > t 4 M 1 ( -ray background resulting from black hole evaporation: f M γ Figure 9 L M H logf 10, as before. This distribution applies for ∼ B The assumption that the bound (4.13) is saturated for reasonable guess. We knowit that yields it a mass is distribution indeed that saturated joins in smoothly a with the matter-dominated distribution universe, we and found for Of course, the total massSince density the of mass black holes distribution cannot in exceed figure9 theis density of peaked the at dark matter. CMB spectrum produced by the radiation emitted by gas accreted onto the black holes [21]: where assumption, for walls with The resulting mass distributionthe function assumed is saturation plotted of in the figure9, mass with bound the shown parts by depending dashed lines. on A reliable calculation of domain walls in a radiation-dominated universe.here For provides the an time being, upper the bound distribution we for found the black hole5 mass function. Observational bounds Observational bounds on the massstudied; spectrum for of an primordial upbound black to holes comes have from date been the review, extensively see, e.g., [20]. For small black holes, theFor most massive stringent black holes with JCAP02(2016)064 i ∼ H cr (5.4) M depends pl , where and pl M depends only 3 ) i λ /M evap i M /H H pl ≡ M g g ( and 1 15 -6 3 i − 15 . The corresponding con- ξ = 10 λ 10 cr ∼ σ/H M is shown in figure 10, with red -8 ∼ cr } = pl M cr g ξ M 24 /M -10 i = 10 cr ξ ξ, H i M 2 Pl H { π M -12 2 10 − is too small to be interesting, so we only show ⊙ e 2 M Log λ ξ 3 – 24 – -14 1, where the semiclassical tunneling calculation is ∼ = 10 & λ cr M ξ -16 -18 -20 1 the nucleation rate .

1.5 1.0 3.5 3.0 2.5 2.0 4.0 i

ξ H ξ ) = 1, so the parameter space below this line is excluded by eq. (5.3). cr . These lines, which mark the transitions between the subcritical and super- few. M ( , where the mass function (4.22) depends only on ∼ f accr cr ξ . Observational constraints on the distribution of black holes produced by domain walls. M ≡

The horizontal upper boundaries of the red and purple regions correspond to the regime The model is fully characterized by the parameters The dotted lines in the figure indicate the values , 1 in the entire range shown in the figure, and therefore M ξ M > M 3 ∼ of essentially only on critical regimes in theξ excluded red and purple regions, are nearly vertical. This is because 10 Red and purple regionsration mark and the parameter by values gasallowing excluded, accretion the respectively, onto by formation large smallvalues of black black where hole holes. superheavy evapo- black The hole green seeds.The region indicates solid the straight parameter line values marks the parameter Figure 10 These bounds cananalyzed now in be section4. usedbound to (4.13) As impose is before, saturated. constraints on we the shall domain proceed wall under model the that assumption we that the mass on (see eqs. (4.2), (4.3), (4.5)). The parameter space is the expansion rate during inflation. The nucleation rate of domain walls and purple regions indicatingrespectively. parameter We values show excluded only byjustified. the the range Also, constraints for (5.1)the and values (5.2), JCAP02(2016)064 . Most remains

) = 1, so i M cr H 5 M ( 10 f . exponentially, so i bh H M . 60 e-foldings of inﬂation. from being the dominant

g M 23 2 N ∼ 10 ) at present should be comparable . M bh ( 0 M n . g for the red region (evaporation bound) and 17 . The region of the parameter space satisfying 3 15 − − – 25 – 10 the black holes would have evaporated by now, and Mpc 1 g . . 0 by 20% would suppress the nucleation rate by 10 orders 15 λ i ∼ 10 H G < n ) = 1 in ﬁgure 10 enters the purple region excluded by the gas cr bh M M is the dimensionless nucleation rate per unit Hubble volume. We ( f λ We note that ref. [23] suggested an interesting possibility that dark [22] and their number density . For

11 may result in a strong suppression of small wall nucleation rate. The g M 2, a change in i are produced towards the very end of inﬂation, when the expansion rate 24 3 the line H ≈ , where 10

10 , and the number of objects within our observable universe would be of the g ξ N i when inflation ends, the sizes of these objects would be distributed in the range can be easily found: 3 t M < 24 i t N λ 15 λe for the purple region (accretion bound). As we have emphasized, the mass func- e 10 bh M > 10 ∼ does not yield any additional constraints on model parameters. > 26 . − ∼ i N eq bh We note also the recent paper by Pani and Loeb [26] suggesing that small black holes The solid straight line in figure 10 marks the parameter values where A very interesting possibility is that the primordial black holes could serve as seeds for The black holes that form dark matter in our scenario can have masses in the range R 10 g < M The allowed mass range of black hole dark matter is likely to be wider in models where black holes are M > t . 15 evap . 11 i σ dark matter constituents. This conclusion, however, was questioned in6 ref. [27]. Summary and discussion We have explored theand cosmological vacuum bubbles consequences which of may haveAt a nucleated the population during time the of last spherical domain walls have shown that suchwide walls spectrum and of masses. bubbles Black may holes result having mass in above the a certain formation critical of value would black have a holes with a t order could be captured by neutron starsmechanism and excludes could black cause the holes stars with to masses implode. 10 They argue that this of this range, however, appears to be excluded by the gas accretion constraint (5.2). formed by vacuum bubbles. matter could consist of intermediate-mass black holes with 10 to the density of large galaxies, this condition is markedthis in region green may in overestimate figure the10 . black (As hole before, massthe function.) the parameter horizontal space upper below boundarythe this of line red is and excluded purpleadditional by excluded eq. constraints (5.3). regions, on suggesting This theally line that excluded model lies the regions completely parameters. bound within are (5.3) plotted We does in note, not figure however, impose10 thatassuming any the that observation- the expansion rate for accretion constraint. straints on λ tion (4.22) that we obtainedblack for hole super-critical mass domain walls function, representsthe and an actual thus upper size the bound of red on the the t and excluded purple part regions of in the figure parameter10 space.may We overestimate havesupermassive verified black that holes the observed regime atshould the be galactic centers. The mass of such seed black holes nearly constant duringM inflation. Smallmay domain be walls substantially reduced. thatany form decrease The in black tunneling action holes (4.3) with depends masses on evaporation bound onthat the could model form parameters dark matterexample, can or with then seed the be supermassive evaded, black holes while may larger still be black produced. holes For 10 of magnitude. JCAP02(2016)064 , ∼ /G b (6.1) (6.2) t H t crosses i . the black R bh cr M M , and assuming σ bh < t M H t or vice-versa. σ t , i M /G. i } σ M is smaller than , t b b t t i σ t t { 0 is the vacuum energy inside the bubble and + – 26 – Min 2 i 2 b > t t the bubble avoids the singularity and starts growing ∼ b ρ cr cr ∼ M is the matter density. This energy is mostly in the form bh > M m M . At the end of inflation, the energy of a bubble is equiva- ρ σ bh , where , t 1 M b − t ) this may be important, since some radiation may follow the bubble cr Gσ , where ( i M 3 i t R ∼ ) i . Alternatively, if the wall is impermeable to matter, a pressure supported t σ ( t , the bubble collapses into a Schwarzschild singularity, as in the collapse of bh 2 H m cr M ρ σt 3) and ∼ is the time when the co-moving region corresponding to the initial size < M 2 π/ is relevant. If the wall is small, with a horizon crossing time / bh H 1 t bh σ − M t The case of domain walls is somewhat simpler than the case of bubbles, since only the The evolution of vacuum bubbles after inflation depends on two parameters, If the bubble is surrounded by radiation, rather than pressureless dust, there is one If the bubble is surrounded by pressureless matter, it subsequently decouples from the ) M b = (4 is the bubble wall tension. For simplicity, throughout this paper we have assumed the i Gρ scale which is sitting in theits middle mass of is an larger empty cavity. or The smaller fate than of a the certain bubble critical depends mass, on whether which can be estimated as hole mass, given by eq.However, (2.44), for is not significantly affected by the pressure of ambient matter. where the horizon. that the only interaction withmass matter is gravitational, thebag wall of will matter collapse will to form, a with black a hole substantially of larger mass which is given in eqs. (3.6) or (3.25). more step to consider.matter After the and bubble comes wall to haschange transfered rest the its mass with momentum of respect to the the to surounding resulting the black Hubble hole. flow, We the have effects argued that ofinto the for pressure baby may universe, thus still seems affecting to the require value a of numerical thegrounds, study resulting however, which we black is pointed hole beyond out mass. the that scope This the of mass process the of present the work. black hole On is general bounded by nontrivial spacetime structure, with aFRW baby region. universe connected by a wormhole to the exterior ( exponentially fast insidewormhole. of a This process babyslice is universe, is represented which in depicted is the ininternal connected vacuum causal figure4. energy to diagram of the of the Thewall. bubble, figure3, parent exponential or The of FRW due growth dominant to which by of effect the a a the depends repulsive spatial gravitational on bubble field whether of size the can bubble be due to the usual matter. However, for σ separation of scales lent to the “excluded”M mass of matter which would fit the volume occupied by the bubble, For Hubble flow and collapses to form a black hole of mass of kinetic energy offactor. the bubble Assuming walls,from which that the are particles bubble expanding of wall, intosurrounding this matter matter matter kinetic with on cannot energy a a is penetrate time-scale highrest quickly much with the Lorentz dissipated shorter respect bubble than by to the the momentum andshell Hubble Hubble transfer of flow. time, are to matter so In the reflected the is this process, wall released, comes a while highly to the relativistic energy and dense of exploding the bubble is significantly reduced. JCAP02(2016)064 1, ∼ is now 3 i H t σ/H will begin to = σ ξ , where > t /G -ray background, and H H t γ t ∼ , lead to black holes which . bh eq the universe is dominated by t /G M σ H . t t H ∼ . t t . bh . Under these assumptions, we have σ M t , as well as bubbles of all other vacua /G i H H t ∼ bh – 27 – M . The narrowness of this range is related to the fact that 3 i H ∼ σ 12 , developing a wormhole structure. If at σ t ∼ is the expansion rate during inflation. This calls for a rather special choice of t i in the radiation era, the estimate of the mass is more difficult to obtain, since some H Our preliminary analysis provides a strong incentive to consider the scenario of black On the other hand, the bubble nucleation rate is highly model-dependent, and the Super-critical black holes produced by vacuum bubbles have inflating vacuum regions A systematic analysis of all possible scenarios would require numerical simulations, and σ t On the other hand, this relation may turn out to be natural in certain particle physics scenarios, once 12 hole formation by vacuum bubbles,hole which density has a resulting larger from parameter domain space. wall Note that nucleation the is black appreciable only if parameter space yielding annote appreciable also density that of string blackcan theory holes expect suggests is a the likelyrates. to existence large be of number a increased. of vast We bubble landscape types, of vacua some [25],inside. of so These them one regions withenergy will relatively inflate parent high eternally vacuum nucleation and with will the inevitably nucleate expansion bubbles rate of the high- environmental selection effects are takenthis into out account, to see us. e.g. [24]. We thank Tsutomo Yanagida for pointing shown that black holes producedcosmology. by Hawking nucleated evaporation domain of wallsradiation can small emitted have black a by holes significant gas can impactspectrum. accreted on produce onto A a large substantial portion blackpresent of holes observational the can constraints, parameter induce but space distortionslarge there of in black is the holes the a model in CMB range iscenters numbers of already of sufficient parameter excluded to galaxies. values by seed thatdark the For would matter. certain supermassive yield black parameter holes values observed the at black the holes can also play the role of where model parameters, with the nucleation rate of domain walls is exponentially sensitive to the wall tension. saturate the upper bound on the mass, allowed by the underlying particleend, physics and [2,3, bubbles12, nucleated28].universes prior inside. In to some A the similar of structure the endby would domain bubbles of arise walls. inflation within inflation the will Indeed, will supercritical inflatingthe form black domain coupling holes black walls produced is can holes only excite withto gravitational. any inflating fields jump In coupled back particular, to to they them, theaccessible even can inflationary if neighboring excite plateau vacua the [29, [31]. inflaton30], field, or We they causing thus may it cause conclude transitions that to the any eternally of the inflating multiverse These estimates apply respectivelyat to the the case time where matter when is the non-relativistic or wallnonrelativistic relativistic crosses matter, the the mass horizon.the of the time resulting Larger when black wallsFor the hole with is comoving regionradiation affected may by flow wall into nucleationnumerical the study comes baby and within universe is the followingwere left able the horizon. for to domain further find wall. only research. This an As upper issue in bound requires the on a case the of mass, bubbles, for this case we we have not attempted to dothe it case in this of paper. aAlso, Nonetheless, awaiting distribution for confirmation illustration, from of a we more have domainwalls considered detailed entering walls numerical study, the which we horizon have interact assumed in that with the large matter radiation era, only with gravitationally. inflate at JCAP02(2016)064 – 28 – This is reminiscent of a well known illustration of the 13 If a black hole population with the predicted mass spectrum is 14 . The eternally inflating multiverse generally has a very nontrivial spacetime structure, We would like to conclude with two comments concerning the global structure of super- Black hole formation in the very early universe has been extensively studied in the A similar spacetime structureIt was has discussed been in recently a suggested different [33] context that by black holes Kodama with et a al. relatively [7]. wide mass spectrum could be formed 13 14 after hybrid inflation [34]. However,and the spectrum its does shape not is typically span rather more different than from a few that orders in of magnitude, our model. critical black holes, which(see may ref. be [35] for relevant aamount to recent of the review). information so-called First carried black of by hole all, Hawking given information that radiation paradox the as black the hole has black a hole finite evaporates mass, into the the discovered, it could be regarded as evidence for inflation and for the existence of a multiverse. with a multitudewell of known eternally illustration inflating ofcourtesy the of regions the inflationary connected author multiversethe (see by by also balloon Andrei wormholes. [28] Linde, shaped andbubbles which references This regions or therein). we is should domain reproduce In walls. reminiscent be the here of present interpreted by context as a the wormholes links between which aregenerally created has by a supercritical veryregions nontrivial connected spacetime by structure, wormholes. with a multitude of eternally inflating literature, and a number ofBlack possible holes scenarios could have beenAll be proposed these formed (for scenarios a at typicallycharacteristic review cosmological give see mass [18, a phase comparable32]). sharply to transitionsscenario the peaked or predicts horizon mass at a mass spectrum blackmany the at of orders hole the end black of relevant distribution magnitude. of holes, epoch. with with inflation. In a the contrast, very our wide spectrum of masses, scanning Figure 11 inflationary multiverse, by Andreicontext the Linde links between which the balloonnote we shaped that reproduce regions the should mass in be distributionsbubbles figure interpreted of are as black11. wormholes. expected holes We resulting to In from be domain the different walls and present and can from in vacuum principle be distinguished observationally. JCAP02(2016)064 15 (A.1) (A.3) (A.2) . , could be described by a closed chart 2 ) i 2 Ω t = 1 for the radiation dominated d Ω 2 = d p r , and the worldsheet metric can be t 2 η r + + 2 , 2 p dη dr i 2 η 0 η + defined in section3. r 2 m – 29 – = 1. dη 1 + C i ) = − − η H ( )( a η and ) ( r 2 η ( a C 2 a = , so that 2 = p 2 3 ds = ds i η = 2 for the dust dominated background and is the conformal time. The scale factor is given by p η The metric for a FRW universe is The domain wall worldsheet is parametrized by This does not contradict the result of Farhi and Guth [36], that a baby universe cannot be created by 15 where written as classical evolution in theinflating laboratory. region That of result thesurface. follows baby from universe, With the the existence theseThis null of is assumptions, energy an a Penrose’s condition, anti-trapped problem theorem andcomplete. surface if the in [38] our However, existence the laboratory implies it of isInflation that is a embedded itself some not non-compact in is geodesics an a Cauchy bounded geodesically asymptotically are problem below. Minkowski incomplete past in space, to a incomplete. In whichuniverses, is cosmological the the but geodesically present setting past it context, which is [37],the such results just if infinite geodesic from a we Cauchy incompleteness feature an surface requireexample, is of inflationary in the the not the phase. spacetime the related inflationary inflationary to FRW to phase Hubble universe the a that past rate of precedes pathology of interest to them. of the is be end-of-inflation baby not It surface should necessarily also a be global noted Cauchy that surface. For Acknowledgments This work is partially supportedCPAN CSD2007-00042 by Consolider-Ingenio 2010 MEC (JG) FPA2013-46570-C2-2-P, and AGAUR(AV by 2014-SGR-1474, and the JZ). National J.Z. Science was Foundation alsois supported grateful by to the Burlingame Tsutomu Fellowshipour Yanagida at for scenario Tufts University. a for A.V. useful dark discussion, matter suggesting and the to possible Abi relevance Loeb of forA several useful discussions. Evolution of testIn domain this walls appendix, in weand solve FRW the find evolution the of numerical test coefficient domain walls in a FRW universe numerically where background. We choose of de Sitter space,not in be which applicable. case all global Cauchy surfaces would be compact and Penrose’s theorem would asymptotic FRW universe will betum finite. state For that of reason, thethe it infinite cannot supercritical possibly multiverse collapse encode which leads the developswhich to quan- survives beyond topology arbitrarily the change, far black andtion into hole should to the horizon. not the future. be formation Second, In expectedin of this in the a the context, baby baby parent it universes. universe, universe appearstopology once Finally, that we change we trace unitary should and over evolu- stress the toany that degrees formation exotic of the physics. of freedom formation baby It of occurs multiverses wormholes, within by leading completely black to natural holes, causes does from not regular require initial conditions. JCAP02(2016)064 15 . and i 0 R i i (A.4) (A.5) (A.7) (A.6) (A.9) (A.8) H t . The ' 60 4 η − m C 10 50 r < takes the form 40 η . t 10, we find > are defined as 30 i r R i C . H 20 and , 10. . ) = 0 2 m 2 , r . For 0 > . i 3 C r i 10 , 2 H 2 a R 0 2 bh R H r 0 − ) = 0 2 i t i r = r πσt M η p − H 2 (1 ( 4 i − 0 r a 1 r 2 0 p C 2 R 1 i ) 0.0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 √ 1 + p H p 2 ) + i p R πσ 2 i 62 for 0 . – 30 – H dτ on , r ) = r 0 i 60 (1 + is shown in figure 12. We see that the coefficients H r Z R t − ' i C ( ) = 4 = r a η R (1 πσ i 0 ( ,r C ) = satisfies r 50 i R m and = 0 η M a a = 4 C ( H i m t r R S C i + 3 40 H 15 and 00 . r 0 on ' ,r 30 m m C C 20 62. 10 . . The dependence of 0 ' r m We regard a black hole as formed when the radial coordinate drops to According to section3, the numerical coefficients 0 C C 0.2 0.1 0.0 0.4 0.3 0.5 and the equation of motion is We solve this equation with the initial conditions determine its mass from eq. (3.8), which in terms of the conformal time This mass grows whileconstant the during wall the is late being stages stretched of by the Hubble collapse. expansion, but remains nearly The solutions are plotted in5 figure in terms of the cosmological time The dependence of approach the values and In this appendix,domain prime wall indicates action a is proportional derivative with to the respect worldsheet to area, the conformal time where the Hubble-crossing time Figure 12 JCAP02(2016)064 T (B.2) (B.6) (B.4) (B.1) (B.3) (B.5) (B.7) has the 0 ρ coordinate. = ρ , ρ corresponds to ± ρ T . 2 − Ω ! d 2 R R GM + 2 2 dR. r 1 dR, . 1 dR − 2 1 − 1 , − Ω 3 − d / 1 2 ) tanh R R coordinate have a similar behaviour, GM R 2 − ρ . GM R + GM 3 2 GM 2 − / 2 2 (2 1 ) − dρ R − 3 0 GM 1 / ρ 2 1 2 . We shall consider geodesics which start from can be found from (B.4) as: R + ± r ) GM ρ ˆ R t ρ ρ 2 GM + – 31 – ( + R 2 2 GM + + 2 ∝ 2 / ˆ t and 2 r 3 ( dT ˆ t R ˆ t 2 3 r d + − − dT = R = GM R pl GM dT 2 2 R 2 ± m ds − 3 1 = = 1

ˆ t d dρ − as a function of GM = 0 at the bottom of the Kruskal diagram, see figure 13. In the region = T 2 R both grow towards the future. A sphere of test particles at = 4 coordinate may grow or decrease depending on which spatial direction we ds is the proper time along them. The double sign in front of R T T ˆ t are defined by the relations ± ρ the and ˆ t and given in (B.5). A second integration constant has been absorbed by a shift in the ˆ t GM R 2 where with Adding (B.3) and ( B.4), we have coordinate. Since the metricgeodesics, (B.2) is and synchronous, the lines of constant spatial coordinate are with a shifted value of proper time. B Large domain wallIn in this dust appendix cosmology wefigure6, describe which in corresponds moregravitational to detail field the a of construction large theshall of wall domain start the creates wall by solution a cosidering embedded representedmatching wormhole in the of structure in matching leading two a of Schwarzschild toare dust metrics Schwarzschild a put cosmology. accross to baby together. a a universe. The domain dust We wall. cosmology, andB.1 Finally, then the the different Matching pieces SchwarzschildConsider to the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form where an integration constant hasThe been expression absorbed for by a shift in the origin of the two possible choices of geodesicthe coordinates singularity at R < Thus, co-moving spheres with different values of the go, while physical radius given by

sthe is

⌧

(6) coordinate.

sthe is

⌧

sthe is

(5)

(6)

coordinate.

⌧

⇢ T

1 (6)

coordinate.

oriae The coordinate.

GM ⇢

(4) s

(5)

⇢

(5) (3) 1

tanh

A oriae The coordinate.

⇢

⇢ 2

s oriae The coordinate. GM

(4)

⇢

GM R

s 2 1 (4)

(2) s

tanh

(3)

◆

) (3)

GM 3

tanh

dR.

s (2

✓

◆ (1) @

(2)

) (2)

◆

a efudfo 3 as: (3) from found be can 0

2 s

⇢ 1

dR, ) 1 ⇢

◆ +

⌧

2 2 1

(  )

2 ◆ GM 1

and

✓

R dR. @ 2

sthe is

⌧ /

◆ 3

3 3

⌧

(1)

dR.

(1)

a efudfo 3 as: (3) from found be can

T GM ± /

)

◆

✓

GM 2 ⇢

R R R

⇢

2 @ 1

✓

dR,

+ 3

(6)

◆

s )

coordinate.

⌧

2 GM dR, GM a efudfo 3 as: (3) from found be can

⇢ 

(

and

2 ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given

◆

⇢ ⌦ ⌦ T

✓

⌧

d d

=4 +

1 3

R 1

⌧

(

 2 2

= ± T

✓

and

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

GM dT

⌥

R safnto of function a as

GM 3 2

⌧

+ +

with ✓

2 1 2 ⇢

ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given GM

d R 2 ✓ =

. sthe is

. 1

◆ R

⌧ ± s ⌦ R 2

⌦ ⌦ 2

(5) d

⌦ ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

⇢

d ,

safnto of function a as

⌥

R 2 ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given ⇢

2 GM GM s

(6) with

2 R R

R R 2

2 coordinate.

R GM

+ +

s dT ⌥

1 +

⇢

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since 2 = safnto of function a as ⌧

GM 2

⌧ 2

oriae The coordinate.

d 2

⇢

xrsinfor expression

± dT

with

R ✓

⇢

◆

2 1

R d

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where dT

1 (4)

⇢ ⌧ GM

GM + R

(5) 1

R 2 +

xrsinfor expression

1 ⌧ 2

⌧

⇠

⇢

◆

d GM GM

⌧

2 2

⌧ hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where

(3)

. ✓

1 2

d ✓

t ,

⇠

oriae The coordinate. 1

tanh

⌧

xrsinfor expression =

nti ae h alrepels wall the case, this . In

1 R

⇢

h oanwl is wall domain , the

utatn 3 rm() ehave we (4), from (3) Subtracting

(4) t t

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where

+ H

⌧

⇠

R ds

JCAP02(2016)0642

⌧ /t

2 2

2 1

◆

2 ◆ t

2 dT

✓

⇠

R R

H 1

(3)

r endb h relations the by deﬁned are 1

⇠ /t

nti ae h alrepels wall the case, this . In

R h oanwl is wall domain , the

utatn 3 rm() ehave we (4), from (3) Subtracting

⇢

2 R

S t (2) ⌧ ⇠

tanh ,

nti ae h alrepels wall the case, this . In

R ⌧

GM h oanwl is wall domain , the

3 utatn 3 rm() ehave we (4), from (3) Subtracting

= GM

and

◆

)

✓

⌧ ⌧ R = GM r endb h relations the by deﬁned are =2

2 ✓ GM

1 2 /

(2) 1

⌧ ds

⇢

R 3

r endb h relations the by deﬁned are

⇢

dR.

2 m

⌧

✓

and

+ 3

R ◆

(1)

where

(B.8) = (B.9)

@ /

= R

) ρ , 1

⌧

◆

and

t 1

1 (B.10) (B.13) (B.11) (B.12)

2 3 2

⌧

ds /

(1)

) (5) (6) (1) (2) (3) (4)

ds a efudfo 3 as: (3) from found be can 0 takes a 3

2 dR.

R ⇢ 1

dR,

GM (2 ⇢

◆ H

2 is the

◆ +

where

✓

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect ν

GM @

⌧

3 ;

where

⇠ ↵ ⌧

3 2 2

( 

R )

GM dR,

and 2

a efudfo 3 as: (3) from found be can

⇢ ✓ µ

0 ◆

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In

R 1

⌧ ⇢

3 + GM

=4 n 2 . Continuity

coordinate.

⌧ run over the c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect 2

(

 ✓

R 0 T GM ± and 1

↵

GM c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect

GM T 2

R . 3

⌧

↵ r

t 2 ✓

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In

= =4

1 β nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In

GM .

. s

✓ 2 T

R ± coordinate. The

R ⌧

2 ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given has only angular

2 =

⌦

⌦ s

⌦ 2

⌦ = const. has only

d R

osdrteShazcidmetric Schwarzschild the Consider

d ⇢ d

R ⇢ GM

R µν

2 , the domain wall is

R r +

2 2 , 2

=0 /t R R

the in shift a by absorbed been has constant integration second A (5). in given r

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

dT ⇢

⌥

2 s

t GM ⇢

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 αβ

s safnto of function a as

2 and

⌥ t

2 R K +

+ .

2 =

osdrteShazcidmetric Schwarzschild the Consider

GM with

osdrteShazcidmetric Schwarzschild the Consider

2 2

⇢

safnto of function a as T

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

2 dR

⇢

2 d ⇢

2 ⇠ ⌧

⌦

◆

T d

= 1 A

with

R α

eoew osdrtee the consider we Before . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 d

± 1 . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 n utcosmology. dust a and

!1 dT ◆ R ±

= R . In this case, the wall repels 2

⇢

⇢ GM

⇢ d

2 R

1 b" d

GM !1 R

+ , where the two solutions

= GM

⌧ e the consider we Before

n utcosmology. dust a and GM

2 +

eoew osdrtee the consider we Before 2

peia eino fvacuum. of of region spherical

⌧ = 2 n utcosmology. dust a and .

, t t

d ⌧

✓ 2

dR, dR.

d ⌧ 1

!1 s

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

⌧

GM 1 1 ⌧

M T GM,

S for expression

peia eino fvacuum. of of region spherical 1 d = 2

xrsinfor expression ⇠

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the

M T GM,

✓ ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

peia eino fvacuum. of of region spherical

hr nitgaincntn a enasre yasiti h rgno the of origin the in a shift by absorbed been has constant integration an where

/t ,

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

t 1 =2 1 ◆ ◆

⌦

. 2

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant 3 integration an where h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the

2 .

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here,

t ⌧ ◆

⇠ d

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the /

dT =2

, = const. or R ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

⌧

t ) nti ae h alrepels wall the case, this . In 2

h oanwl is wall domain , the ◆

+ M T GM,

1 )

⌧

. R ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, 2

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the

t ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

t tanh

R R 2

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the R

ds GM GM

R ρ

⌧

H .

GM 2 2 ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, =2

⇠

◆

) h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the ⌧

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly +

/t Ω R dT

1 R

utatn 3 rm() ehave we (4), from (3) Subtracting GM

R GM 2

) R

=2 2 t

2 For

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

t d .

⇠ G 2 ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

✓

⌧

S ⇢

(

R (2

nti ae h alrepels wall the case, this . In For

⇠ R 1 1

1 2

R d r endb h relations the by deﬁned are

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally R

h oanwl is wall domain , the

!1 ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

(

) 3

have we (4), from (3) Subtracting GM

⇠

t ✓ ✓

⇢ r

t 2

For

M T GM, 2 1

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

⌧ t = M T GM,

GM (

2 R

⇠ ✓

t . Noteand that such geodesic coordinates only s

)

✓

2 .

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent

=2 R , R R n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the 1and

⌧ ⇢ ds 2

r endb h relations the by deﬁned are =2 1

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent ρ

GM GM

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and ↵ +

=2 3

R +

↵

M T GM, 2 2

t ⇢

osdrashrcldmi alebde nasailyfltmte oiae nvre Two universe. dominated matter flat spatially a in embedded wall domain spherical a Consider 2 R , S osdrashrcldmi alebde nasailyfltmte oiae nvre Two universe. dominated matter flat spatially a in embedded wall domain spherical a Consider ⌧ αβ

2 di +

2 H +

di /R.

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent s s g

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and 3

⌧ 2 0 =2

where 3

↵

t ( =

=0 t dT =0 ⇠ r

⌧ and

dr /

n ◆ R

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider +

di 3 2

2 ⌧

R R ◆ ) 2

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect ∂ 

ds ΩΩ can be found from (3) as: )( ↵ t ρ =0 t dT dT

R g ( 1 2 = ⇢

R ( π ± ⌥

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In GM 2

a 3 ∝

2 where 2 R

2 peia oanwl nds cosmology dust in wall domain 1 Spherical 4 ✓ peia oanwl nds cosmology dust in wall domain 1 Spherical = = ) a and ds – 32 – 3 1 = = = t 0

⌧

. 1 ( 0 @ =

peia oanwl nds cosmology dust in wall domain , the domain1 Spherical t wall is

✓

⇢

⌧ c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect αβ osdrteShazcidmetric Schwarzschild the Consider t a

R d d ↵ ΩΩ

⌧ ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild 2

↵ . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 is the normal derivative, and 2 /t form the takes this coordinates, Lemaitre In GM

⌧ K

R t M K = !1 ), we have dt

2 ρ

eoew osdrtee the consider we Before =4 n utcosmology. dust a and ⇠

t ⌧ in the Schwarzschild solution corresponds to the total ?? , it is then straightforward to show that II" − ∂ GM ds T . In this case, the wall repels 0

peia eino fvacuum. of of region spherical t will be glued to a co-moving sphere with ) is synchronous, the lines of constant spatial coordinate are geodesics, and M T GM, ± ρ =2 / ). A second integration constant has been absorbed by a shift in the

1 ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a S 1

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the M

osdrteShazcidmetric Schwarzschild 0 the Consider

R ??

(1) ?? ) t R

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads 2

as a function of

R = ρ

=0 ⌧

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, ) from ( , the repulsive ﬁeld can be ignored. In this case, for

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the are deﬁned by the relations ⇢ . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1

. T

!1 ρ R ds 1 ??

⇢

) !1 . ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

1 1

= 0, for diﬀerent values of

For GM

⌧

G (2) ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally e the consider we Before , which corresponds to radial geodesics traveling towards region II from

( ρ n utcosmology. dust a and !1 ⇠ ) = 2

GM, T and

given in ( R

to the the matching hypersurface M T GM, R

−

_" T ⌧

=2 t

=2 R

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent (

R vacuum. of of region spherical µ

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and 2 . R/ ↵

R Here, we will be primarily interested in the opposite limit, where Before we consider the e Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat a spatially in embedded wall domain a spherical Consider erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale

di I ⌦ ⇠ n ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted =0 remnant hole black a with left are we universe FRW ambient the in and universe, baby a (3)

M T GM, 2 III"

↵ d

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the R ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

with Since the metric ( Subtracting ( where an integrationexpression constant for has been absorbed by a shift in the origin of the quickly shrinks under its tension and forms athe black matter hole around of it radius leads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust in region ambient which of expansion the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant1.1 cysted in a Matching Schwarzschild toConsider the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form where

!1 1 Spherical domain wall in dustR cosmology di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by ﬁeld cosmological expansion. of the Eventually, when domain the wall wall falls within the horizon, it

R t

=2 2 + = ( ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here,

⌧ (4) h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

R 1

GM, T +" dR

. r

peia oanwl nds cosmology dust in wall domain 1 Spherical 1

1 Spherical domain wall in dust cosmology !1 ∂ )

R radius of hole black a forms and tension its under shrinks quickly

GM◆ 1

For

G 2

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally , and (5)

=0 ( R , the repulsive ﬁeld can be ignored. In this case, for 2 . − 1 . ⇠ R Consider a spherical domaint wall embedded in a spatially ﬂat matter dominated universe. Two 0

⌦ a

erent time-scales) are relevant to the dynamics of such a wall. One is the cosmological scale 1 2 d ρ

↵ T GM,

di G ⌧ t + R

( R ect of the domain wall, let us ﬁrst discuss2 ✓ the matching of Schwarzschild+ rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent =2 2 1

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due ↵ time-scale acceleration the is other the and +

and the other is the acceleration time-scale due to the repulsive gravitational ﬁeld of the domain wall =

. Lemaitre coordinates in the extended Schwarzschild diagram. The left panel shows the dR,

⇠ For ↵ dT ⇢ R ). Using (B.5) with

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider ˆ t d t di (6) 2 = 0 for the two possible choices of the double sign in eq. (B.6). In the left panel, we adopt GM n

IV" GM◆ 2 ◆ 1 dR. =0 2 R GM

conformally stretched by cosmological expansion. Eventually, when the wall falls within the horizon, it ∂

R 2 = coordinate. The

Here, we will be primarily interested in the opposite limit,ρ where R quickly shrinks under its tension and forms a black hole of radius R 2 + ⇢ t Since the normal 1 ⌧ The metric (B.2) can be matched to a ﬂat FRW metric 1 ◆ GM πGt d 2 the matter around it while its size grows faster than the ambient expansion rate. As we shall see, this R is the ✓ ✓

leads to the formation of a wormhole. The dust which was originally in the interior2 = of the wall goes intoGM ⌧

2 = 2 (6 coordinate. a baby cosmology universe, dust in and wall in the ambient domain FRW universe1 Spherical we are left with a black hole remnant cysted inR a 1 T ds , / ds 3 , ⇢ sphericalBefore region we of consider of vacuum. the e / corresponding to a1 dust cosmology. Such cosmology has a matter density given by components, and that these are the same on both sides of the matching hypersurface: The co-moving sphere with with temporal and angular coordinates. It is straightforward to check that of the temporal and angular components of the metric accross that surface requires Therefore, the massmass parameter of dust that wouldare be matched. contained The within relation thethe sphere (B.11) edge of can of radius also theexplained be FRW in metric derived subsection by should3.1.2. imposing be that a the geodesic dust ofradial particle both component, at FRW and and Schwarzschild. the This metric is is diagonal, the extrinsic curvature where simple form given by Figure 13 curves cover half or the extended Schwarzschild solution. the lower sign, indicated by the white hole singularty at s R ✓ 1 ) and a dust cosmology. dT GM 1 2 GM R 3 ± s / A + 2 (2 GM = 2 1.1 Matching Schwarzschild to a dust cosmology 1 dT s ⌧ ) Consider the Schwarzschild metric d ⌥ ⇢ = + ⌧ tanh ⇢ 3 ( d are deﬁned by the relations 2 R ⇢ = In Lemaitre coordinates, this takes the form GM R 2 and 2 ⌧ can be found/ s from+ (3) as: ⇢ 3

where and R ⌧ ◆ GM 1 2 ✓ 3 0 GM@ as a function of=4 Subtracting (3) from (4), we haveT T ±

where an integration constant has been absorbed by a shift in the origin of the expression for

given in (5). A second integration constant has been absorbed by a shift in the R

with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and

sthe is

⌧

(6) coordinate.

sthe is

⌧

sthe is

(5)

(6)

coordinate.

⌧

⇢ T

1 (6)

coordinate.

oriae The coordinate.

sthe is T

GM ⇢

⌧

sthe is

(4)

s ⌧

(5)

coordinate. ⇢ (6)

(5) T (3) 1

coordinate. (6)

tanh

A oriae The coordinate.

⇢

⇢ 2

s oriae The coordinate. GM

(4)

⇢

GM R

s 2 1 (4)

(2) s

tanh

, (3)

(5)

⇢ 1

◆

) (3)

, 1

(5) GM

⇢

tanh

dR.

oriae The coordinate. 2

s (2 GM

✓

R ◆

(1) 2

⇢

+ s 1

3 oriae The coordinate.

(2)

) (2)

◆

a efudfo 3 as: (3) from found be can 0

2 2 s

⇢

⇢ /

dR, )

⇢

2 (4)

, 1

s /

◆ +

⌧

2 2 1

(  )

2 ◆ GM 1

and

✓

(4)

✓

tanh

R dR. 1 @ 2

sthe is

⌧ /

◆ 3

3 3

⌧

(1)

(3) dR.

(1)

a efudfo 3 as: (3) from found be can

GM ± / )

◆

tanh

, 1 GM

✓

GM 2 ⇢

R R R

⇢

1 2 2 @ 1

✓

dR,

1 R

(3) (1) (2) (3) (4) (5) (6)

(6)

◆

s )

coordinate.

⌧

2 GM dR, GM a efudfo 3 as: (3) from found be can

⇢ 

(

and

2 ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given 2

◆ H

⇢ ⌦ ⌦ T

✓

2 is the

⌧

d d

+ +

1 3

2 R

⌧

⇠ , 2

◆

( ⌧

 2 2

= ± T

✓

and

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

GM dT

t ⌥ s

R safnto of function a as

GM 3 2

2 ⌧

+ +

(2)

)

=0 2

with ✓

3 coordinate.

1 GM

2 t ◆ ⇢

ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given

. d R

2 ✓

✓

. sthe is

1 T 0

. @

R 2

/ 2

(2)

◆ R

⌧ ±

s ⌦ R 2

t )

⇢ /

(2 1

⌦ ⌦ 2

(5)

⌦ ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since ⇢ 3

d dR.

d 1

◆

safnto of function a as 2

sthe is coordinate. The ⌥ dT

2 = ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given 2 ⇢

2 GM

GM s

(6)

⌧ with

2 ✓ R R

GM a efudfo 3 as: (3) from found be can

⌧ 1 R

2 3

2 coordinate.

R )

⇢ R

+ + ⇢

s ⇢ dT 2 (2 ⌥ ⇢ 1 +

(1)

, the domain wall is dR. R

⇢

2 0

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

2 = safnto of function a as ⌧ ,

⌧ 2

oriae The coordinate.

/t +

◆ d

⇢

d 0 2 r

xrsinfor expression ⇢ dR,

dR dR

d 2

± dT t

T ⌧

and

with a efudfo 3 as: (3) from found be can

coordinate. GM R ✓ 1

2 3

⇢



◆

t ⇢

( ◆

)

◆ .

2 GM 1

R d 1

⌧ = T ⇢

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where 2 dT

⇢

± 2

(1) 1 ✓ (4)

⇢ ⇠

3 ⌧ GM ⌧ (6)

r ±

sthe is T

⌦ +

1 d

R GM

(5) 1 A 1 R R

2 + dR,

2 d

⌧

and

 +

◆

R ( . In this case, the wall repels 2 xrsinfor expression

⌧ 2

2 R ⌧

⇠

⇢

◆

⌧

GM =4

d /t

2 GM

⌧

R GM

2 2 ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given

=0 ✓

⌧ GM

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where

1 R

(3)

✓ t

2 GM

d ✓

1 t

✓ t ⌧ ⇠

oriae The coordinate. 1 R

tanh

s 2 +

⌧

xrsinfor expression =

coordinate.

1 nti ae h alrepels wall the case, this . In

(6) ⇢ t

h oanwl is wall domain , the

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

GM 2

utatn 3 rm() ehave we (4), from (3) Subtracting dR, dR.

(4) t t

. hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where 2 T

+ H

⌧ GM

⇠

R ds ,

GM 1 1

+ 2

, ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given

R R

. .

s S

⌧

1 /t

2 with ⇢ 2

safnto of function a as

2 ⌦ GM

1 2

✓ ◆ dR

◆ t

d 1 R

2 dT

R dT

✓ R ⌦ s

⌥ ,

⇠

R 1 ◆ ◆

1 (3) ⌦

r endb h relations the by deﬁned are 1

⇠ const. r /t

GM 3

2 nti ae h alrepels wall the case, this . In

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

h oanwl is wall domain , the d

utatn 3 rm() ehave we (4), from (3) Subtracting

⇢ t

2 2

S t (2) ⌧

⇠

R 2

1 R

+ t

+ ◆ R

tanh ,

nti ae h alrepels wall the case, this . In

s )

with (5)

R ⌧

safnto of function a as

⌦ tanh

R R

R + R

h oanwl is wall domain , the GM GM

d 3

⇢ have we (4), from (3) Subtracting

t GM ⇢ =

GM 2

⌦ and

◆

⌥

oriae The coordinate.

T t

2 2 2

JCAP02(2016)064d

)

1 dT

+ ⇢ ✓

⌧ ⌧

R ⇢ GM r endb h relations the by deﬁned are

=2 1

2 R ✓ R GM

R d

dR GM

(2)

1 GM

⌧ A 2 ds

S 2

⇢

R ,

r endb h relations the by deﬁned are R ◆

2 +

s sthe is

tanh

⇢ ⇢ R

(5)

⇢ (2

, 1 1 +

dR.

d ⌧

GM R

GM (2

⌧

⇢ ✓ xrsinfor expression

2 3

and

1 GM

GM ◆

(1) ✓ ✓

where

d @ /

2 =

± + /

) 2 R

, 1

⇢ 2

⌧

◆

and

⌧ 2

1 s 1

d hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where

2 3

⌧

/ =

R R

ds /

◆ ✓ (1)

) (5) (6) (1) (2) (3) (4) oriae The coordinate.

a efudfo 3 as: (3) from found be can 0

3 s

(6) GM

)

GM d coordinate.

dR.

⇢

GM R R

dR,

GM = (2 ⇢ 2

⇢ ◆ + ◆ ⇢

2 GM GM

H =

(4)

T GM

+ R

µ is the s

✓ 1 ◆ xrsinfor expression +

where +

✓

R 1

2 2

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect ⇢

GM @

2 +

⌧ 3

R u

where

⇠ ↵ ⌧

3 2

2 1

(  ⌧ R

)

dR,

and / 2

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where

a efudfo 3 as: (3) from found be can

⇢

✓ s s

(B.14) (B.15) d (B.17) (B.18) (B.16)

/ ⌧ 2 t

✓

⌧ 2 2

1 3 ◆

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In

R 1

⌧ ds + ⇢ (

(4) dT

+ GM ⌧

=4 ◆

2 coordinate.

+ 3

⌧ 3 2

d R

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect 2 =

( ◆ 2

 hypersur-

✓

✓ ◆

R : T 0

GM ± ⌧

and tanh

1 

↵

(3)

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect

dT can be found from (3) as:

utatn 3 rm() ehave we (4), from (3) Subtracting

GM T 2

= 3

R . 3

)

⌧ ↵ , / τ

t 1 1 ✓

GM dT dT

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In R

= = ⇢

1 !1

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In GM

= 2

GM R

(5)

. R

✓ A ± ⌥

1 as: (3) from found be can 2

+ GM ds

R GM (3) 2 ±

coordinate. The GM

/t 1

⇢

R ⌧

H (2

2 ⇢ 2

ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given

2 R (5) (6) (1) (2) (3) (4)

⇠ 2 ⌦

⌦ s

. ⌦

⌦ = ✓

d R =4

t , 2

osdrteShazcidmetric Schwarzschild the Consider

⇠

d ⇢

+ R

◆ d

⌧

⇢ GM

R 1 H and

nti ae h alrepels wall the case, this . In ds

, the domain wall is T

and

dT is the

) utatn 3 rm() ehave we (4), from (3) Subtracting ie n() eoditgaincntn a enasre yasiti the in shift a by absorbed been has constant integration second A (5). in given R 1 3 r endb h relations the by deﬁned are

+ t

= = 2 2 , 2 2

=0 ρ

/t 3 R ⇢

✓

the in shift a by absorbed been has constant integration second A (5). in given

(6) (1) (2) (3) (4) (5)

⌧

/ t

oriae The coordinate.

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

h oanwl is wall domain , the

dT ⇢

⌥

2 ⇢

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since

2 ⌧

⇢ ⇠

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 ⌧ 1

+ s

safnto of function a as +

dT 0 @ 2



⌥ t ◆ 2

R +

. dR. ⇢

GM, T t

2 R

⌧

= t

osdrteShazcidmetric Schwarzschild the Consider

◆ ✓ GM with osdrteShazcidmetric Schwarzschild the Consider

( ⇢

⌧

2 2

⌧ /t ⇢

GM is the safnto of function a as T

R ,

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since H

2 dR

⇢

2 d ⇠ ⇢ 2 with

=0 s "FRW"""

⇠

⌧

ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild and R

t coordinate. ⌦

◆

T d

⇠ R

(2) = (4)

1 A 2

with

R t

⇠

1 ⌧

eoew osdrtee the consider we Before

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 d

GM /

=2 / dT

↵ ⌧

± nti ae h alrepels wall the case, this . In 1

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1

n utcosmology. dust a and )

GM !1

dT 3 ◆ R

± 1 =

R relations the by deﬁned are . In this case, the wall repels

1 T 0 t

S GM

⇢ .

(2) ✓

⇢

GM ⌧ t

h oanwl is wall domain , the

R . ⇢

d t 2

0 ⇢ ✓

⇢

R 2 2 R ✓ 1

b" d

t GM 1

=0 R

GM !1 R

ds +

⇢ 2

◆

t GM coordinate. = GM safnto of function a as

eoew osdrtee the consider we Before R

⌧ dR,

1 =4

n utcosmology. dust a and

GM coordinate. The

R t

tanh

2 = ⌧ GM

2 + eoew osdrtee the consider we Before

3 (3) peia eino fvacuum. of of region spherical ⌧ ⌧ = 2 T

n utcosmology. dust a and (2 .

where

, IV Dust

t t d ds ⌧

and

✓ R

2 GM

dR, dR.

⇢

T 0 imply that the extrinsic d

⌧

1 ⇢ T ⇢

dR. 0

. R 1

s 2

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

◆ ⌧

=2 1 1

⌧

M T GM, t

, the domain wall is ⌧

= .

⇢ 0

R 1

. as: (3) from found be can

xrsinfor expression S s

H ,

peia eino fvacuum. of of region spherical 1

R d = 2

± d GM /t 3

xrsinfor expression

⇠ !1

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the S

0 r )

dR R

/ ⇢ 2

M T GM,

⇢ ⌧

✓

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

peia eino fvacuum. of of region spherical 2

R 0

hr nitgaincntn a enasre yasiti h rgno the of origin the in a shift by absorbed been has constant integration an where

✓

⇢ /t

, 2 2

1 coordinate. The

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

t t 1 GM =2 1 ◆ ◆

= R ⌦ R

t .

. 2

⌧ ds

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration an where

⇢ 3 s 2 h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the t

+ 2

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, =

t GM

⌧

◆ (1)

= ⇠

2 +

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a 1

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the /

r as a function of

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

⌥ ⌧

t ΩΩ

t ⇢

(1)

nti ae h alrepels wall the case, this . In ⇠ 2

⌧

and

h oanwl is wall domain , the ⇢

⌧ r where

◆ dT

⌦ 2

, the repulsive ﬁeld can be ignored. In this case, for  xrsinfor expression

+ R M T GM, dR,

1 )

⌧

R (

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, 2 1 A

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the + g t

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads are deﬁned by the relations

t tanh

R R

, the domain wall is ,

◆

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

R ⌧

⇢ 0 T

ds ,

(2) GM GM

⌧ /t

◆

t R

. In this case, the wall repels

2 =0

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In 0 r 2

GM ±

2 2 s

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, T =2

⇠ ⇢

⇢ ✓

◆

3 ) h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

2 =

the of origin the in shift a by absorbed been has constant integration an where ⌧ /

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly t

GM, T

+ ↵

R dT

. ) !1

R 1

GM 1

=0 GM

R GM

1 R

utatn 3 rm() ehave we (4), from (3) Subtracting GM

. = R

GM / 2

2 ˙

) 1

R .

=2 2 t

R ˆ GM ⇢ t 2

For

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

= ⌧

. ⇠ =

G 2

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally ⌧ 2 ✓ d

⌧

S ⇢

2 +

(

⌦

R 1

2 (2

nti ae h alrepels wall the case, this . In

For GM

⇠ the in shift a by absorbed been has constant integration second A (5). in given ⇠ R 1 1 ±

1 ⌧

G t

) 2 =2

R d r endb h relations the by deﬁned are ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally r

h oanwl is wall domain , the

⌦ ✓

2 dT

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly dR, dR. and

( =2

) 3 t

have we (4), from (3) Subtracting

+ 1 A (2

R ⇠

t given in (5). A second integration constant has been absorbed by a shift in the

✓ ✓ @ s

R dR. GM

⇢

2 0 2 /

d R

os.r const. 1 1

. 2

◆

For S

M T GM, 2

R 3 1

⌧

✓ ˙ ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally +

GM R . In this case, the wall repels 2

G ,

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric the Since ⌧

t =

M T GM,

. 2

s , dR

ρ G =0

GM c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect

(

R form the takes this coordinates, Lemaitre In

(1)

R 1

⇠ ✓

R 3

0 t

and , ( ⇢

s 1

)

GM 0

a efudfo 3 as: (3) from found be can 2

↵ ◆ ◆

⌦

)

✓ ⌦ 2

2 R

=0 1

⇢ t

const. r rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent 3

=2 GM R R R

d , 2

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the 1and

⌧

R ˙

⇢ ⌧

ds 2 ⇢

r endb h relations the by deﬁned are =2 1

Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two Here, we will be primarily interested in the opposite limit, where Before we consider the e with

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent d

GM GM ⇢

erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale ρ

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and ↵

+ metric Schwarzschild the Consider d =2 .

⌧ + R

+ of function a as ˙

⇠ ↵ 2 R

R ˆ dR, t,

M T GM, =0

2 2 2 R

t ⇢ + osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider

R 2 +

◆

↵

S ΩΩ osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider

⌧ 2

di ) +

◆ 2 +

2 H .

+ tanh R R ⌦

t R

R ⌧

requires that

/ ⌥ 2

GM GM

2  g

and

R dR, ( dR.

⇢ GM

=2 d utatn 3 rm() ehave we (4), from (3) Subtracting rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent ˙

s s

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and GM

⌧

2 Subtracting (3) from (4), we have where an integrationexpression constant for has been absorbed by a shift in the origin of the with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and di and the other ist the acceleration time-scale due to theconformally repulsive stretched gravitational by ﬁeld cosmologicalquickly of expansion. shrinks the under Eventually, domain when its the tension wall wall and falls forms a withinthe black the matter hole horizon, around of it radius itleads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust ambient in region which expansion of the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant1.1 cysted in a Matching Schwarzschild toConsider the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form where 1 Spherical domain wall in dustR cosmology

GM =2

where 3

2 ρ,

s s 2 2

⌦

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1

2 ↵ os.r const. GM

t (

= ✓

◆ 2

=0 t dT

=0 ⇠ ⌧

and .

1 1 ⌧

dR +

2 d σ

R R

. =4 2 ⌧ ◆ R osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider µ R R

S 3 1

+ R 2 1

3 2 GM GM d

d t ⌧ 2 =

R GM 2

R ◆

−

± T

2 2 =

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect  + u

⇢ = R

ds can be found from (3) as:

⇢ =

↵ 2 (2

◆ ◆

1 1

⌦

dρ

dτ 1 =

=0 d GM

const. r R

3 R

GM dT dT GM

+ −

osdrteShazcidmetric Schwarzschild the Consider

⇢

⇢ 3

2 GM = ⇢

✓ ✓

GM 2

± /

eoew osdrtee the consider we Before

dT ⇢

2 cosmology. dust a and

R d

✓

2 2

± ⌥ ds

2 nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In 1

2 d

R = 2

✓ 2 1

◆ the in shift a by absorbed been has constant integration second A (5). in given = s where GM

2 R

) R

peia oanwl nds cosmology dust in wall domain 1 Spherical =

. R

dR tanh

R R ✓ GM ✓

0 R

s .

peia oanwl nds cosmology dust in wall domain 1 Spherical )

GM 1

◆ GM GM R . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 – 33 –

⇢ 2

1 + for expression

R R

and ˙ ds 2

] = R = ⇢ 2

⌦ =

1 3 + !1

= 2 = GM GM

2 2 +

⌦

ic h erc()i ycrnu,telnso osatsailcodnt r edsc,and geodesics, are coordinate spatial constant of lines the synchronous, is (2) metric 0 the Since GM

d + GM

peia eino fvacuum. of + of region spherical

2 2

d R

⌧

⇢

1 2

1 ◆

. ΩΩ 2

hr nitgaincntn a enasre yasiti h rgno the of + origin the in shift a by absorbed been has constant integration an where

0 @ r endb h relations the by deﬁned are

safnto of function a as

peia oanwl nds cosmology dust in wall domain 1 Spherical

+ 2 2 GM +

t 2 r

⌥ dT , the domain wall is ⌧

= 2 R with

GM R s

2 /

✓ ⇢

⇢ R

⌧ dT

2 d

s s c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a 2 ⌧ 2

osdrteShazcidmetric Schwarzschild the Consider t

3 T

⇢ ΩΩ ⇢

⌧

+ d ( d

↵ dT (2

ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild e the consider we Before ⌧ n utcosmology. dust a and µ + 1 1

⌧ d

◆

r R =

+ GM

3 and 2

↵ 3 ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

GM ⇢ . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1

2 GM

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In ◆ 2 /t =

∂ GM

✓ ✓

⌧ K

2 ⌧ h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the

2 / 

R ˙

M T GM,

[ ˆ t ✓ t

can be found from (3) as:

dR ± dT µ

2 !1

✓ 1 ), we have =

t R ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, GM

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

= ⇢

1 dT dT

2 ◆

!1 = ⇢ eoew osdrtee the consider we Before

⌧

!1 d R GM

✓

n =4

n utcosmology. dust a and ⇠ = R 1

⌧ =2 s

t vacuum. of of region spherical )

± 2 ⌥ ??

GM where

II"

µ ds ds R R

2 GM

= ⇢ 2 R

+ 2 1

GM R

= GM GM T .

. In this case, the wall repels 2

+ ✓

R +

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, n baby a

2 2 peia eino fvacuum. of of region spherical

t have we (4), from (3) Subtracting

and ⇢

) is synchronous, the lines of constant spatial coordinate are geodesics, and

M T GM, ±

2 ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

=2 2 ◆ 1 3

). A second integration constant has been absorbed by a shift in the for expression +

= =

are given by the simple expression

⌧ 2

)

⌧

1 dT

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

S d = /

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the

GM 2 . ⌧

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads d

1 osdrteShazcidmetric Schwarzschild the Consider

⌧ s s ⌧

t =2 0 @ ⌧ ??

2 R 2

For

hr nitgaincntn a enasre yasiti h rgno the of origin the in shift a by absorbed been has constant integration 3 an where ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

(1) for case, this In ignored. be can ﬁeld repulsive , the ?? GM, T

t R

ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads ν

M T GM, 0

✓ as a function of ( dT ✓ =

⇢

⌧

;

R ⌧

=0 ( ⌧ ,

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here,

1 t ◆ ) from ( ⇠

, the repulsive ﬁeld can be ignored. In this case, for where limit, opposite the in interested primarily be will we Here,

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around d matter the

+ h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the d

= 3 2

ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild

µ are deﬁned by the relations

⇢ ⌧

. acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1

◆

T 1

=2 /t

=2 .

↵



R ΩΩ ⇠

1 GM

?? t

2 can be found from (3) as:

nti ae h alrepels wall the case, this In . ⇢

GM r endb h relations the by deﬁned are

)

ds ⌧ . ⇠

t =

!1 t R ,

2 ⇠ . t ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

dT dT

R T GM, R

1 1

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In

✓

2 ⇢

nti ae h alrepels wall the case, this . In

R = ⇢ a"

, K

!1 2

For GM ◆

⌧ 1

G =

(2) R 1 ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

e the consider we Before

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

dT =4

( =

n utcosmology. dust a and ⌥ ± n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent t

!1 GM 2

⇠

) )

h oanwl is wall domain , the

R H !1 GM

and

GM, T ⇠

h oanwl is wall domain the , ds

2 ↵ ⇠

H R

given in ( R

M T GM,

and T

utatn 3 rm() ehave we (4), from (3) Subtracting

For t

t ✓

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally ⌧

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider

R _" 2

=2 ⌧

GM G t ⇠

=2 ⌧

⌧

G t

t µν

± ( and S ds

nti ae h alrepels wall the case, this . In

⇠ =2 R

R 1 3 = rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent ( = =

R vacuum. of of region spherical n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and

2 R h oanwl is wall . domain , the

↵

GM R R

=2 ⌧

Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two Here, we will be primarily interested in the opposite limit, where Before we consider the e

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat a spatially in embedded wall domain a spherical Consider ⌧

erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale =0

t 2 0 @ I

⌦ R ⇠

ayuies,adi h min R nvrew r etwt lc oermatcse na in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

S as a function of =0 (3)

✓

ds M T GM, GM, T 2 relations the by deﬁned are III"

↵ d

⌧

c ftedmi al e sﬁs ics h acigo Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect

GM t

✓ R

⇢ , the repulsive ﬁeld can be ignored. In this case, for

1 ⌧

=0 M T GM, t R ↵ h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the where

R are deﬁned by the relations ⇢ ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads R T where an integrationexpression constant for has been absorbed by a shift in the origin of the with Since the metric ( 1.1 Matching Schwarzschild toConsider the a Schwarzschild dust metric cosmology In Lemaitre coordinates, this takes the form where Subtracting ( t conformally stretched by cosmologicalquickly expansion. shrinks under Eventually, when its the tension and wall forms falls a withinthe black the matter hole horizon, around of it it radius leads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust in region ambient which of expansion the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant cysted in a

!1 1 Spherical domain wall in dustR cosmology di and the other is the acceleration time-scale due to the repulsive gravitational ﬁeld of the domain wall

⇢ ect of the domain wall, let us ﬁrst discuss the matching of Schwarzschild

R t t =2

t + wall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales ⇢ erent osdrteShazcidmetric Schwarzschild the Consider

2 R

run over the angular coordinates. The continuity of the metric and of the

ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, GM, T

⌧

0 (4) . =2 =2 ↵

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

↵

⌧ GM and

R 1

⇢

1 =

GM, T +" R ⇢ 0

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a = Consider dR ⌧ di R

⌧

= R

) t

peia oanwl nds cosmology dust in wall domain 1 Spherical and 1 2 =2

r . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1 peia oanwl nds cosmology dust in wall domain 1 Spherical !1

⇢ given in (5). A second integration constant has been absorbed by a shift in the

1 Spherical domain wall in dust cosmology =4 ) ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

R 0

R =0 ⌧

◆ r

ds GM G

For R where r ( G 2

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally T

(5) R . A Schwarzschild solution can be matched to a dust FRW solution at ta

=0 ( R

, the repulsive ﬁeld can be ignored. In this case, for 2 . form the takes this coordinates, Lemaitre In 1 . ⇠ Consider a spherical domain wall embedded in a spatially ﬂat matterFor dominated universe. Two Here, we will be primarily interested in the opposite limit, where Before we consider the e

Consider a spherical domain wall embedded in a spatially ﬂat matter dominated universe. Two erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale Schwarzschild of matching the discuss ﬁrst us let wall, domain the of ect

R ± ⇠

t !1

⌦ ↵

↵ =0

) 1 2 d cosmology. dust a and erent time-scales are relevant to the dynamics of such a wall. One is the cosmological scale =0

M T GM,

e the consider we Before

↵ 0), where a dot indicates derivative with respect to the wall’s proper time

t with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and where an integrationexpression constant for has been absorbed by a shift in the origin of the where Subtracting (3) from (4), we have In Lemaitre coordinates, this takes the form the matter around itleads while to its the formation sizea of grows a baby faster wormhole. universe, than The and thespherical dust ambient in region which expansion of the was of rate. originally ambient vacuum. in FRW As the universe we interior we shall ofand are the see, a left wall this dust goes with cosmology. into a black hole remnant1.1 cysted in a Matching Schwarzschild toConsider the a Schwarzschild dust metric cosmology conformally stretched by cosmologicalquickly expansion. shrinks under Eventually, when its the tension wall and falls forms a within black the hole horizon, of it radius 1 Spherical domain wall in dustR cosmology di and the other ist the acceleration time-scale due to the repulsive gravitational ﬁeld of the domain wall

di G ⌧ + R =0 ,

( R ect of the domain wall, let us ﬁrst discuss2 ✓ the matching of Schwarzschild+ as a function of rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent

=2 2 1 n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due ↵ time-scale acceleration the is other the and

The matching of Schwarzshild and FRW metrics can be done in two different ways, and the other is the acceleration time-scale due to the repulsive gravitational field of the domain wall , the repulsive field can be ignored. In this case, for Schwarzschild of matching the discuss first us let wall, domain the of ect 0

Mr GM dR,

↵

peia oanwl nds cosmology dust in wall domain 1 Spherical are deﬁned by the relations

⇠ dT ⇢ ↵

For R T

nLmir oriae,ti ae h form the takes this coordinates, Lemaitre In

osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. t dominated matter ﬂat spatially a in embedded wall domain spherical a Consider d di t vacuum. of of region spherical =2

os.r const. (6) ˙ ⇢ ρ,

GM GM,⇢ T

IV" .

◆ r const. 1

GM 2 = ◆ dR. metric Schwarzschild the Consider =0 GM

2 R 1

conformally stretched by cosmological expansion. Eventually, when the wall falls within the horizon, it a in 2 cysted remnant hole black a with left coordinate. are we The universe FRW ambient the in and ˙ universe, baby a ˆ

= t,

R ⇢

R ⌧

Here, we will be primarily interested in the opposite limit, where R

GM which are representedSchwarzschild in exterior figure solution, as14 . indicatedembedded in in An the a left expanding dust panel,hole ball or FRW connects a of exterior, through black a as matter hole wormhole indicated may interior to can in be the be the asymptotically embedded right flat in panel. region a In this case, the black face. This can beto done a in Schwarzschild two interior, different whileexterior. ways. the right The panel left shows panel a shows black an hole expanding interior ball embedded of in dust aHere matched dust Ω FRW and Ω extrinsic curvature means that there is no distributional source at the junction. B.2 Domain wallLet in us Schwarzschild nowLemaitre consider coordinates, the a motion radial of( vector a which domain is wall tangent in to a the spherically worldsheet symmetric is vacuum. given by In The normal to the worldsheet is then given by Since the normalextrinsic is curvature in the radial and temporal directions, the angular components of the Israel’s matching conditions for a thin domain wall of tension The normalization of the radial tangent vector curvature should change by the amount + Figure 14 quickly shrinks under its tension and forms a black hole of radius 2 ⇢ ⇢

2 ) =2

and

1 ⌧ into goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

◆ cosmology dust a to Schwarzschild Matching 1.1 given in (5). A second integration constant has been absorbed by a shift in the 1 R

d GM

2 =

R GM

the matter around it while its size grows faster than the ambient expansion rate. As we shall see, this R is the ⌧ ⇢

osdrteShazcidmetric Schwarzschild the Consider

✓ 2 G GM

✓ this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the = R ⌧ leads to the formation of a wormhole. The dust which was originally in the interior of the wall goes intoGM 2 (

2 = 2 2 where limit, opposite the in coordinate. interested primarily be will we Here,

peia oanwl nds cosmology dust in wall domain 1 Spherical R 1

Before we consider the e For Here, we will be primarily interested in the opposite limit, where a baby universe, and in the ambient FRW universe we are left with a black hole remnant cysted in a Consider a spherical domain wall embedded in a spatially ﬂat matter dominated universe. Two T !1 ,

erent time-scales are relevant to the dynamics of such a wall. One= is the cosmological scale ds for case, this In ignored. be can ﬁeld repulsive the , . acigShazcidt utcosmology dust a to Schwarzschild Matching 1.1

eoew osdrtee the consider we Before

ds cosmology. dust a and 3 , ⇢ ⇠

Before we consider the e = =0 / t spherical region of of vacuum. ⌧

s R ✓ ↵ ⇢ 1

) ⇢

radius of hole black a forms and tension its under shrinks quickly

Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and expression for with Subtracting (3) from (4), we have where an integration constant has been absorbed by a shift in the origin of the where Consider the Schwarzschild metric In Lemaitre coordinates, this takes the form a baby universe, andspherical in region of the of ambient vacuum. FRW universe weand are a left dust with cosmology. a black hole remnant1.1 cysted in a Matching Schwarzschild to a dust cosmology conformally stretched by cosmologicalquickly expansion. shrinks under Eventually, when its the tension wall and falls forms a withinthe black the matter hole horizon, around of it radius itleads while to its the formation size of grows a faster wormhole. than The the dust ambient which expansion was rate. originally in As the we interior shall of the see, wall this goes into R di and the other ist the acceleration time-scale due to the repulsive gravitational ﬁeld of the domain wall

1dT Spherical domain wall in dust cosmology GM 1

and a dust cosmology. 2 R e the consider we Before .

GM cosmology. dust a and

M T GM, peia eino fvacuum. of of region spherical

ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the 3 when Eventually, expansion. cosmological by stretched conformally

± !1 A + / (2 1 )

2 GM For

= =2 2

1.1 Matching Schwarzschild to a dust cosmology a in cysted 1 remnant hole black a with left are we universe FRW ambient the in and universe, baby a dT G ⇠

( s R

⌧ ) vacuum. of of region spherical

d ⌥ ⇢

Consider the Schwarzschild metric !1 ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

+ a in cysted remnant hole black a with left are we universe FRW ambient the in and universe, baby a

= t

⌧ tanh for case, this In ignored. be can ﬁeld repulsive , the

M T GM,

3 ( T GM, ee ewl epiaiyitrse nteopst ii,where limit, opposite the in interested primarily be will we Here, h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the

h eusv edcnb goe.I hscs,for case, this In ignored. be can ﬁeld repulsive , the ed otefraino omoe h utwihwsoiial nteitro ftewl osinto goes wall the of interior the in originally was which dust The wormhole. a of formation the to leads

⇢ t

⌧

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and

are deﬁned by the relations 2 t

R where limit, opposite the in interested primarily be will we Here,

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent

h atraon twieissz rw atrta h min xaso ae sw hl e,this see, shall we As rate. expansion ambient the than faster grows size its while it around matter the =2

= ⌧

⇢  T GM, ↵

. R

In Lemaitre coordinates, this takes the form R GM R

R .

2 =2 osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider

ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly

and R

2 R

/ s ) ucl hik ne t eso n om lc oeo radius of hole black a forms and tension its under shrinks quickly ⌧ can be found from+ (3) as: !1 )

3 !1

For ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

⇢

=0 G For ofral tece ycsooia xaso.Eetal,we h alflswti h oio,it horizon, the within falls wall the when Eventually, expansion. cosmological by stretched conformally

G (

⇠

( R

where and R ⇠

⌧ ◆

GM T GM,

2 t

1 T GM,

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale cosmological the is One wall. a such of dynamics the to relevant are time-scales erent

n h te steaclrto iesaedet h eusv rvttoa edo h oanwall ✓ domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and n h te steaclrto iesaedet h eusv rvttoa edo h oanwall domain the of ﬁeld gravitational repulsive the to due time-scale acceleration the is other the and

rn iesae r eeatt h yaiso uhawl.Oei h omlgclscale 3 cosmological the is One wall. a such of dynamics the to relevant are time-scales erent

↵

0 ↵

R osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider

peia oanwl nds cosmology dust in wall domain 1 Spherical osdrashrcldmi alebde nasailyﬂtmte oiae nvre Two universe. dominated matter ﬂat spatially a in embedded wall domain spherical a Consider di

GM@ R

as a function of=4 =0 =0

Subtracting (3) from (4), we haveT T R

± R

where an integration constant has been absorbed by a shift in the cosmology origin dust of in the wall domain 1 Spherical expression for cosmology dust in wall domain 1 Spherical

given in (5). A second integration constant has been absorbed by a shift in the R

with Since the metric (2) is synchronous, the lines of constant spatial coordinate are geodesics, and JCAP02(2016)064 (B.24) (B.25) (B.20) (B.21) (B.22) (B.23) (B.19) . # 1 − R GM 2 , + σ . R ) have the same functional form, t σ , , 2 R 2 σ − t " σ σ t ˆ t R = 1 ( 2 − − ), should be glued to a mirror image of ˙ . ρ ρ 1 − ˆ t − R > t R < t ( ) R refers to the choice of Lemaitre coordinate R − GM R GM ρ 2 GM 2 ± 2 trajectory ) and for for , " πGσ s ) + − ˆ t 1 + R – 34 – + ( direction. Here 1 ( + wall σ = (2 ρ " r R V t ρ σ , , ± − + t − 0 0 ˙ = 0, and it is straightforward to check that " ρ = = ). The extrinsic curvature of the domain wall will be the 1 ρ ˆ t ). ˙ − 2 ( ˙ ˙ ρ > ρ < T ˙ G R ρ domain R GM 3 2 √ R (3 GM / we have ˙ 2 σ t σ − t = 1 ). Here, the double sign = cr + ˆ t R ( = ), should be the same as calculated from both sides. This requires that the ˙ + ρ τ ρ ( M > M R . The trajectory of a domain wall between two segments of Schwarzschild solutions with Note that for A similar calculation in the Schwarzschild coordinates gives a simpler form for the equations of motion [9], which has the solution The trajectory of a domainin wall figure gluing two15 . segments of the Schwarzschild solution is depicted a property which will be important for the following discussion. By continuity of theproper metric time, accross the wall, the proper radius of the wall as a function of Figure 15 as we go accross the wall in the positive trajectory of the wall as seen from the “inside”, the same mass this trajectory, system in eq. (B.6), see figure 13 , but same from both sides, but with opposite sign. Combining (B.16), (B.17) andB.18), ( we have which we shall simply denote by JCAP02(2016)064 , D D ]. (B.26) (B.28) (B.27) Phys. (1984) (in , ]. SPIRE IN Phys. Rev. Phys. Rev. D 30 , , (1983) 177 SPIRE IN B 133 (1981) 549] [ Phys. Rev. , 33 (1981) 2052[ 66 . Phys. Lett. 2 , , towards infinite size (see figure ). 15 σ R . t m 3 larger (or smaller) than a critical mass / . R 1 Fate of Wormholes Created by First Order − G M 2 σ 3 σ t √ Thin Wall Vacuum Domains Evolution 3 R GM the wall recedes from matter geodesics with the GMt 2 = – 35 – σ Prog. Theor. Phys. − cr , = Quantum creation of topological defects during inflation , where the expansion of the wall radius is unbounded ]. Pisma Zh. Eksp. Teor. Fiz. M m Quantum Fluctuation and Nonsingular Universe cr R > t R ) = 1 Gravitational Effects on and of Vacuum Decay R ]. SPIRE ( Decay of the True Vacuum in Curved Space-time IN V M > M (1981) 532 [ SPIRE ]. ]. IN and recollapse at a black hole singularity. Here we are primarily The Gravitationally Repulsive Domain Wall 33 ). The maximum of this potential is at m R ( SPIRE SPIRE (1991) 340[ V IN IN ) is negative (or positive) for m R < R ]. Gravitational Field of Vacuum Domain Walls (1983) 91[ R ( D 44 , a domain wall which starts its expansion at the white hole singularity will JETP Lett. V ]. cr SPIRE B 120 IN (1987) 1088[ (1980) 3305[ SPIRE IN Phase Transition in the Early Universe Lett. [ 712[ 36 Phys. Rev. 21 Russian), given by: M < M cr [8] V.A. Berezin, V.A. Kuzmin and I.I. Tkachev, [7] H. Kodama, M. Sasaki, K. Sato and K.-i. Maeda, [6] J. Ipser and P. Sikivie, [5] A. Vilenkin, [4] R. Basu, A.H. Guth and A. Vilenkin, [2] S.R. Coleman and F. De Luccia, [3] K.-M. Lee and E.J. Weinberg, [1] V.F. Mukhanov and G.V. Chibisov, bounce at some interested in the opposite case Eq. (B.24) is analogous toin the a equation static of motion potential for a non-relativistic particle of zero energy For and continues past the maximum of the potential at B.3 Domain wallAs in mentioned dust around eq. (B.22), for where The value of escape velocity, on both sideswhere of an the wall. expanding ball Because ofsolution of matter, that, is we described glued can by to constructThis a an a segment in exact vacuum of solution layer turn a represented is flatof by matter glued the a dominated to segment FRW same a ofthe mass the second second on Schwarzschild vacuum layer solution. is the layerrepresented glued also other in to described figure6. side an by of exterior a flat the Schwarzschild matter solution wall, dominated as FRW solution. describedReferences The in result subsection is .B.2 Finally, M JCAP02(2016)064 D 19 Phys. 38 ]. , D 70 ]. (2013) 113 ]. Class. (1998) 4669 (2008) 829 , ]. (1993) 3265 Phys. Rev. , B. Carr ed., (2009) 044034 , 768 SPIRE SPIRE 680 IN SPIRE D 57 [ ]. ]. IN Phys. Fluids IN D 47 [ SPIRE , D 79 ][ IN Phys. Rev. ]. , ][ (1994) 6327 SPIRE SPIRE (2010) 495 IN IN (2013) 037 ][ ][ Phys. Rev. Astrophys. J. SPIRE 10 Int. J. Theor. Phys. (1998) 2230 D 49 , , Phys. Rev. , Astrophys. J. IN 05 [ , Phys. Rev. , , ]. D 57 Universe or multiverse? arXiv:1511.08801 arXiv:1402.4671 JCAP arXiv:0912.5297 , , , SPIRE arXiv:1401.3025 Cosmological Selection of Multi-TeV , in Phys. Rev. New cosmological constraints on IN , ][ hep-th/0302219 Phys. Rev. arXiv:1506.00426 , The Dynamics of False Vacuum Bubbles Quasiopen inflation ]. arXiv:1512.01203 , (2014) 026[ A comment on “Exclusion of the remaining mass – 36 – Effect of Primordial Black Holes on the Cosmic Res. Astron. Astrophys. (2010) 104019[ , ]. 06 SPIRE Cosmological wormholes IN (2015) 298[ D 81 ][ Fluid dynamics of relativistic blast waves arXiv:1411.7479 ]. SPIRE JCAP IN Dynamics of Astrophysical Bubbles and Bubble-Driven Shocks: Spherical Domain Wall Collapse in a Dust Universe , ]. ]. ]. ]. ]. ]. ][ Recycling universe Black holes from nucleating strings Watchers of the multiverse B 749 ]. ]. ]. SPIRE ]. IN SPIRE Phys. Rev. SPIRE SPIRE SPIRE SPIRE SPIRE Tidal capture of a primordial black hole by a neutron star: implications IN , SPIRE SPIRE SPIRE ]. IN IN IN IN IN ][ IN IN IN ][ ][ ][ ][ ][ SPIRE ][ ][ ][ (2015) 155003[ Primordial Black Holes Phys. Lett. IN The Primordial Black Hole Mass Range astro-ph/0406260 , SPIRE Supermassive black holes from primordial black hole seeds Observational consequences of a ‘domain’ structure of the universe 32 IN gr-qc/9904022 The Anthropic landscape of string theory Renormalized stress tensor in one bubble space-times Nucleation rates in flat and curved space (1987) 1747[ A brief history of the multiverse D 35 (1974) 3161[ astro-ph/9707292 arXiv:1210.7540 hep-ph/9208212 arXiv:0801.0116 arXiv:1212.0330 arXiv:0901.1153 hep-ph/9308280 arXiv:0709.0524 hep-ph/9711214 (1976)[ 1130 [ 10 [ [ Quant. Grav. [ [ [ [ primordial black holes Microwave Background and Cosmological Parameter Estimates Basic Theory, Analytical Solutions and Observational Signatures Rev. [ Cambridge University Press (2009), pp. 247–266 [ (2004) 064015[ for constraints on dark matter window for primordial black holes. . . ”, arXiv:1401.3025 Supersymmetry [ (1999) 3091[ [9] S.K. Blau, E.I. Guendelman and A.H. Guth, [12] J. Garriga and A. Vilenkin, [13] A.E. Everett, [14] H. Maeda, T. Harada and B.J. Carr, [16] J. Garriga and A. Vilenkin, [17] N. Tanahashi and C.-M. Yoo, [19] J. Garriga, [11] R.D. Blandford and C.F. McKee, [15] J. Garriga and A. Vilenkin, [20] B.J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, [21] M. Ricotti, J.P. Ostriker and K.J. Mack, [10] M.V. Medvedev and A. Loeb, [22] N. Duechting, [23] P.H. Frampton, [24] K. Harigaya, M. Ibe, K. Schmitz and T.T.[25] Yanagida, L. Susskind, [26] P. Pani and A. Loeb, [27] F. Capela, M. Pshirkov and P. Tinyakov, [28] A. Linde, [29] J. Garc´ıa-Bellido,J. Garriga and X. Montes, [30] X. Montes, [18] M.Yu. Khlopov, JCAP02(2016)064 88 B ] (1965) 57 14 Phys. Lett. , ]. Rev. Mod. Phys. , ]. SPIRE astro-ph/0511743 , IN arXiv:1501.07565 (2012) 331 ][ SPIRE IN Phys. Rev. Lett. , ][ B 713 (2015)[ 023524 Phys. Lett. astro-ph/9605094 , gr-qc/0110012 Density perturbations and black hole formation in D 92 ]. – 37 – Inflationary space-times are incompletein past SPIRE IN (1996) 6040[ Phys. Rev. ][ , Massive Primordial Black Holes from Hybrid Inflation as Dark (2003) 151301[ ]. D 54 90 An Obstacle to Creating a Universe in the Laboratory ]. SPIRE IN ][ SPIRE Phys. Rev. IN , arXiv:1409.1231 Gravitational collapse and space-time singularities Jerusalem Lectures on Black Holes and Quantum Information Phys. Rev. Lett. Primordial black holes: Do they exist and are they useful? A Novel Channel for Vacuum Decay , ]. ]. ]. (1987) 149[ SPIRE SPIRE SPIRE arXiv:1112.1638 IN IN IN [ [ Matter and the seeds of Galaxies [ directions [ hybrid inflation (2016)[ 015002 183 [31] B. Czech, [33] S. Clesse and J. Garc´ıa-Bellido, [32] B.J. Carr, [37] A. Borde, A.H. Guth and A. Vilenkin, [34] J. Garc´ıa-Bellido,A.D. Linde and D. Wands, [35] D. Harlow, [38] R. Penrose, [36] E. Farhi and A.H. Guth,