Inflation Expels Runaways

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JHEP12(2016)155 Springer October 31, 2016 : December 16, 2016 December 30, 2016 : : 10.1007/JHEP12(2016)155 Received Accepted Published doi: Published for SISSA by . 3 1608.07576 The Authors. c Cosmology of Theories beyond the SM, Flux compactiﬁcations, Superstring

We argue that moduli stabilization generically restricts the evolution follow- , [email protected] E-mail: Department of Physics, ColumbiaNew University, York, NY 10027, U.S.A. Open Access Article funded by SCOAP ArXiv ePrint: originating from the reheating surface,universe. which High leads scale to inflation anstudying is parameter approximately vastly fine-tuning flat favored. and and Our inflationary isotropic initial results conditions point in towards flux a compactifications. frameworkKeywords: for Vacua bias towards inflationary cosmologies.typically The destabilizes energy density Kählermoduli of anddecompactification domain triggers phase walls a can between runaway collapse vacua towards theof large new inflation de volume. occurs Sitter after This regionthe the unless inflationary a transition. expansion minimum generates amount A overlapping stable past light vacuum cones transition for is all guaranteed observable only modes if Abstract: ing transitions between weakly coupled de Sitter vacua and can induce a strong selection Thomas C. Bachlechner Inflation expels runaways JHEP12(2016)155 38 39 9 39 ]. Despite the marvelous success of the 6 12 4 – 1 24 18 26 15 34 – 1 – 23 13 29 37 25 9 32 30 30 36 5 1 33 A.1 The junction conditions B.1 Gibbons-Hawking andB.2 Kruskal-Szekeres coordinates Schwarzschild/de Sitter domain wall dynamcis 5.2 Cosmology in the landscape 4.1 The trouble4.2 with the bubble No trouble4.3 with the double Wormholes in bubble quantum gravity 5.1 Inflation expels runaways 3.3 Dynamics of3.4 false vacuum bubbles Semiclassical transition3.5 probabilities An explicit tunneling trajectory 2.1 Runaway domain2.2 walls Moduli stabilization in IIB string theory 3.1 The Hamiltonian3.2 description of domain Classical walls domain wall dynamics 1 Introduction On large scales theexpansion universe that is evolved into extremely an well approximatelywith described flat Friedmann-Robertson-Walker a cosmology by small, an positive earlyΛCDM cosmological period model constant in of describing [ accelerated cosmological observations, the associated parameters and initial C Alternate expression for the tunneling rate B Classical domain wall dynamics 6 Conclusions A First-order action of domain walls 5 Inflation in the landscape 4 Gravitational effects on viable vacua 3 Thin wall vacuum transitions Contents 1 Introduction 2 Domain walls and moduli stabilization JHEP12(2016)155 ]. 17 – 13 – 2 – ]. It is conceivable that vacuum transitions at very 21 ]. 23 – 18 ]. 28 – 24 ]. Stable and controlled vacua generically require an accidental cancelation of 18 positive vacuum energy [ abilities [ Domain walls between stable de Sitter vacua triggerThe an semiclassical mini-superspace instability approximation applies towards to non- vacuum transition prob- In this work we explore the consequences of moduli instabilities by assuming the ab- The first assumption is motivated by generic instabilities of weakly coupled de Sitter In this work we will discuss and employ further assumptions about the fundamental 1. 2. sence of stablemeans domain that walls any between vacuum dethe transition runaway triggers Sitter phase, an vacua expanding regions indomain true of wall the vacuum remains the landscape. bubble outside new containing the de Although Hubble Sitter this horizon. vacuum are The cosmology stable is at at late risk times of if extinction the dynamics become increasingly relevant. The relevanceis of familiar moduli from stabilization models for of cosmology stringthe inflation. flatness In of that context, anEither couplings inflationary to way, moduli it potential at is and best crucial at spoilobservation to is worst carefully well destabilize appreciated account the in forignored inflationary entire the when model configuration. presence studying building, of moduli vacuum moduli. decay. stabilization is Even often though this multiple large terms in the perturbative expansionrelatively that renders small. the barrier towards runaway Thevacua can large spoil energy theobstacle density to delicate within populating cancelation the a landscape andlow domain [ destabilize energies wall the decouple between moduli. from cosmological potentially This unstable represents moduli, an but at high energies the moduli to describe well behaved lowexpansions energy such physics. as In the particular, string coupling modulito or controlling stabilize the perturbative in compactification a volume are controlledweakly famously regime. coupled difficult This vacua are observation easily is susceptible knowncoupling to as [ a the runaway Dine-Seiberg instability towards problem: strong or weak vacua. While flux compactificationsaccommodate exhibit a a landscape vacuum solution structure tohow that the the may cosmological landscape constant well is problem, be populated.many it able zero-energy The is to deformations, not low referred obvious energy to theory as of moduli, flux compactifications that contains need to be stabilized in order a vast landscape ofunnaturally approximately small stable observed and vacuum populated energy density vacua, [ selection bias leadstheory. to In an orderdynamics to we go consider beyond the following static two parameters assumptions and in turn: attempt to constrain cosmological and the early phaserevert of to accelerated a expansion more bothtuning in appear fundamental the rather description effective theory unnatural. inparameters is. that order We Constraints would have imposed to appear to by evaluate theexample surprisingly underlying how of tuned theory this significant from can mechanism lead the a might to low be fine- energy the perspective. value of A the good cosmological constant. If we assume conditions cannot be explained solely within that model. The small vacuum energy density JHEP12(2016)155 (1.2) (1.1) ], we use the WKB ansatz 26 , 25 ]. , 29 . Inf 2 Inf Late H H π/GH ]. Following [ − – 3 – e log 24 ]. The vacuum transition rate Γ within an initial 1 2 ∼ 28 Γ & e N are the inflationary and late time Hubble parameters, respectively. Late ]. It may be interesting to assume the absence of such tunneling config- H 31 , and leads to a flat, isotropic universe. The lower bound on the required and 30 65 e Inf H Predictions of physical observables are not possible purely within this simple frame- We will not impose any further assumptions about quantum gravity that are some- The second assumption concerns the probability of quantum tunneling in a gravita- work. A prediction woulduum require a transitions detailed in understanding quantum ofcosmological gravity, dynamics the the before relevant probability and landscape, after measure vac- reheating.weak in assumptions However, eternal by about combining inflation some fundamental and simplethe physics and the resulting we cosmology: arrive assuming at a an rich landscape educated of guess vacua, concerning the absence of stable domain geometries that violate localarise energy classically conditions. from Even non-singular thoughconditions these [ initial configurations conditions, cannot quantumurations, fields but violate this local wouldevidence constitute energy we a elect third to assumption, avoid it and here. given the lack of supporting These transitions to high energydomain states walls, are consistent most with likely to our trigger first moduli assumption. instabilities at the times implicit in the literature, such as conjecturing the absence of bubble nucleation potential barriers. In the presenta work thorough we derivation treat is the transition presentedasymptotically rate in flat as [ spacetime an is assumption, then while containing maximized the by highest the inflationary nucleation scale, of a single small region instructive to consider itssuperspace potential description implications. is We a assumesymmetric good that quantum the gravity. guide semiclassical We to employtional mini- compute the theory transition canonical of formalism probabilities spatial tofor in geometries, quantize the spherically see the tunneling gravita- [ wave function to approximate transition probabilities through classical roughly expansion is independent of the bubble nucleation(CDL) process transitions and applies in to theories Coleman-de Luccia with dynamical gravity [ tional theory. Even though our understanding of quantum gravity is tentative, it will be the number of efolds is set roughly by where For high scale inflation late time stability requires inflationary expansion by a factor of enter the Hubble sphere.lapse We if will sufficient inflationary find expansionfrom that the occurs a end such vacuum of that transition inflation allobservational is have observable constraint overlapping protected modes past on against light originating inflation, cones. col- butvia This arises inflation finding independently. and coincides radiation with In the domination a to universe that a evolves late time de Sitter phase the lower bound on via domain wall collapse if the horizon expands and allows the runaway domain wall to JHEP12(2016)155 C and B , A . Appendices 6 . In order for computable, we review the Dine-Seiberg ρ 2 ]. On the other hand, the existence of 32 – 4 – we apply the theory of vacuum transitions to ], and provides for a plethora of potential inflaton 4 35 – 33 , or the volume modulus 1 − s g we speculate about the cosmological implications for large 5 ]: most vacua are expected in strongly coupled regions of mod- 18 ]. It is therefore imperative to carefully consider the effects of moduli 37 , 36 we review the gravitational theory of thin, spherically symmetric domain walls. We Some of the most delicate fields are moduli controlling perturbative expansions, such The organization of this paper is as follows. In section 3 Unfortunately, this can signalaccidentally a much larger breakdown than ofDine-Seiberg all the problem consecutive expansion [ terms. unlessuli one space. This of phenomenon Despite the the iscompactifications terms challenges known it in is as is constructing still the well worthwhile controlled to meta-stable investigate vacua the in cosmology flux of these models. as the inverse stringleading coupling order effects toishes in accurately the describe weak relevant couplinginduces physics, limit. a the runaway The instability effective potential towardsvacuum can potential can weak approach only van- or zero arise strong from when coupling. multiple either terms side, A in which stable, the weakly perturbative coupled expansion are competitive. typically generates a largehundreds slow of roll moduli parameter can produce [ the a observed cosmological complex constant landscape [ thatcandidates accommodates [ the smallness of stabilization when studying cosmological models within the string theory framework. news: on theLight one moduli hand, can all mediate fieldssequent fifth need expansion forces, of to spoil the be the universe.crucial stabilized inflationary Even importance to evolution when for describe and stabilized some at affect astring high low the inflation, realistic energy, where energy sub- cosmology. moduli a phenomena. remain coupling of between This the is volume modulus very and apparent the in inflaton models candidate of It is instructive to recallcorresponding some to of weakly the basic coupledpactified fields compactifications on present of in a the string Riccimoduli low theory. energy flat that effective manifold String are theory without theory massless fluxes com- at contains tree a level. potentially large The number existence of of moduli is both good and bad transition. Finally, in section universes that are stable atprovide late details times. and further We explanations conclude that in would section have distracted from2 the main text. Domain walls and moduli stabilization problem of moduli stabilizationtion and the consequencesdiscuss for the domain classical wall dynamicsthe stability. and Hamiltonian the In framework. semiclassical sec- the description In question of section at vacuum transitions hand in and demonstrate how gravity can stabilize an otherwise forbidden function, we expect to findlogical most constant observers that in experienced an anstable isotropic, extended vacua flat can period universe have of with energylations high small densities shall scale cosmo- well inflation. not below be Note theto confused that scale high with all of scale predictions, inflation. inflation an would observation While be of our consistent signatures specu- with corresponding our assumptions. walls between de Sitter vacua and a vacuum transition rate described by the tunneling wave JHEP12(2016)155 (2.1) ]. 23 – 19 containing a φ , and assume a vanishing wall is a proxy for parametrizing a V . φ τ pl that are decoupled at low energies wall τ M V + and 2 φ τ 2 τ 2 m . Solving the Klein-Gordon equation for the – 5 – + barrier wall V V remains in its original vacuum, we denote the potential ≈ τ )

τ Landscape ( V

plays the roll of a modulus controlling perturbative expansions, Potential Perturbative Control τ is pushed beyond its local minimum when the domain wall tension τ = 0. We are interested in a thin domain wall that does not destabilize V . The instability towards a runaway phase leads to the collapse of the interior . Illustration of a potential landscape as a function of a field parametrizing stabilized vacua 1 →∞ τ One immediate consequence of the Dine-Seiberg problem is that weakly coupled de To estimate whether the modulus barrier of its confiningmodulus minimum we find by that the modulus. Forplanar simplicity, domain wall we of consider approximately constantpotential a energy energy density background in configuration each of domain of wall the and vacua. thethe modulus, With domain we wall only as expand gravitational the interactions effective between potential the for the modulus within 2.1 Runaway domain walls Consider two canonically normalized scalar fields and interact only via Planck suppressed operators.landscape of The vacua, field while so lim in moduli space, exacerbatingmain the wall Dine-Seiberg between metastable problem. vacua will Therefore,in generically the de-stabilize figure some presence moduli, of as a illustrated de do- Sitter vacuum and naively poses an obstacle to populating the landscape [ away. This observation has severeConsider consequences a for transitions landscape between withphysics. metastable distinct vacua. All vacua moduli that arebut describe stabilized there significantly in is different each no lowvacua: of reason energy the the to contributions minima expect to by the a effective some stable potential accidental domain change cancelation, significantly wall along that the interpolates trajectory between any two and a modulus controllingdomain the wall interpolating perturbative between expansion. vacua of Moduli the couplings landscape. classically destabilize a Sitter vacua — if they exist — have a relatively small barrier protecting them from a run- Figure 1 JHEP12(2016)155 , φ ∆ (2.3) (2.2) wall V √ ∼ σ that is classically , the axio-dilaton, B ]. α ξ 21 and = 1 supergravity theory A 0 N ) implies a sufficient condi- > A 2.2 is very small compared to the Λ . is its spatial thickness. For a domain , barrier V a R The nucleation of a true vacuum bubble wall barrier V 0 2 barrier V V > & B & – 6 – 1 Λ 4 pl 2 pl 2 wall M σ M V =0 8 C s . We neglect any gravitational effects. Λ C 0 , approximate stability requires a low vacuum decay rate, > A 1. Combining this bound with ( wall

Λ V away from the vacua, the modulus will be classically destabilized t − wall V is the domain wall tension, and ]. At low energies this setup corresponds to a 3 wall 42 /V wall – 4 σ 38 aV is roughly the canonical length of the trajectory interpolating between the two . = . Illustration of the evolution of a domain wall between vacua φ 2 ∼ − σ If the vacuum energy differences between metastable minima are set by the typical A slightly different argumentThere arriving at may the be same accidental conclusion cancelations is or presented other in effects [ that reduce the domain wall tension below the 1 2 orientifold [ in four dimensions. The moduli include the complex structure moduli naive expectation and leave the modulus stabilized at the domain wall. in figure 2.2 Moduli stabilization inTo see IIB the string Dine-Seiberg theory the problem particularly in action, well we studied now context briefly of review IIB moduli string stabilization theory in compactified on a Calabi-Yau effective potential by the presence of acoupled domain vacua wall. of This string isinterpolating precisely compactifications between the and distinct situation de we presentmay Sitter in might trigger vacua. generic not the weakly runaway expect instability and stable could domain lead walls to a time-like singularity, as illustrated tion for destabilizing the modulus, Therefore, if the height of the confining barrier where ∆ vacua in field space. scale of thelog(Γ) potential, where wall of a scalar field theory we can approximate the domain wall tension as Figure 2 unstable towards the runaway phase in Planck units exceeds the barrier height, JHEP12(2016)155 (2.6) (2.7) (2.4) (2.5) 1. In j T run over all j i ]. I πq 35 depends on the 0 W ) may depend on the , . Let us consider a case 2.6 0 , W 3 πqρ ¯ Ω 2 1. Famously, the superpotential , ∧ , Ω 2 1 + j | 0 T V j Z i to its minimum. For typical values of i W | πq πqρ χ − 2 3 2 ] . Demanding weak coupling requires that − − − e i 2 44 Ae | , A log F + – 7 – | 43 i − 0 X ) 0 W V K = e curvature correction arising via a four-loop correction 3 np 0 = πqρ α 2 log( W 2 F ]. 2 − − V ρ 33 = , and set the axion 0 iχ πqAe K . The resulting F-term potential is given by . The tree-level holomorphic superpotential i + are rational. The terms that appear in ( is the Kählercovariant derivative and the indices 2 = T / ρ j i I 3 V q ) D = ¯ T -model leads to a contribution to the Kählerpotential that scales with , T 0 are one-loop determinants independent of the Kählermoduli, and the σ + ]. The leading non-perturbative contributions to the superpotential are W 3 pl T I M D 39 and string loop corrections to the Kählerpotential break the no-scale structure is the only remaining dynamical field. The scalar potential in this simple , = ( 0 ∼ = α 38 T V is the compactification volume measured in string units and Ω is the holomor- i I A V F One way to arrive at well controlled vacua where all moduli are stabilized is the KKLT Both the scalar potential has a minimum at small volume which is beyond the regime of 0 case becomes where we defined W perturbative control. This is precisely the Dine-Seiberg problem at work. However, for scenario [ accidentally competitive with the treewhere level the flux fluxes superpotential are fixedmodulus at a high energy scale such that at low energies a single Kähler where entries of the matrix choice of fluxes. Again, thethe contributions same are way only that smallmoduli complex at generically large structure give volumes, moduli rise 2 gave to rise a vast to axion the landscape flux for landscape, each flux the choice Kähler [ receives no perturbativeinternal corrections, manifold but give rise non-perturbative to effects superpotential terms on of four the form cycles of the of the potential. For example, the of the worldsheet the inverse of the compactification volume [ the compactification volume is large in string units, choice of fluxes, but isform independent of and the the Kählermoduli, so F-terms thecal potential of compactification is the manifolds of Kählermoduli the therecomplex no-scale do are structure not vastly moduli, enter more such the that possibleis we scalar known flux expect potential. as choices a the than vast Forsmall flux number there typi- vacuum of landscape are energy distinct density and flux [ can vacua. might This be able to accommodate an exponentially where phic three-form. For thegiven case by of a single Kählermodulus the compactification volume is where moduli. The leadinggiven by Kählerpotential for the Kählerand complex structure moduli is and the Kählermoduli JHEP12(2016)155 , 21 ] for (2.8) (2.9) 42 , that (2.10) (2.11) φ that would , the barrier σ ∗ ). Canonically ρ 2.8 ]. While the scale in ( 35 ∗ ρ V . . . ∗ ) φ πqρ ( πqρ 4 2 φ , − is required to be extremely small. Under − V α e e α D α 2 1 ∗ V ) that a highly tuned cancelation of the tree- ρ ∗ ρ V A ρ 2 qA ∼ √ 2.7 q – 8 – D/ is exponentially small, the Kählermodulus will ) = & δV σ ∼ − pl ρ, φ ( σ ∗ ), we find the critical domain wall tension M ρ V V ]. Consider a landscape parametrized by a field via its energy density, 2.2 the potential has a minimum at large volume with small, 21 ρ 0 W are both positive. This coupling can induce a decompactification insta- α 3 and = 3 this would correspond to the form of the F-term potential for the complex D ]. For an axion landscape the situation is more complicated [ α Finally, we are in a position to evaluate the impact of domain walls on moduli stabi- The validity of de Sitter vacua in string theory has been critically examined in recent years, see [ 45 3 , help to maintain stabilizedis Kählermoduli, required as in in orderadditional to this instabilities obtain scenario so a a a more rich large sophisticated vacuum number analysis structure. of is required. The axions higha review dimensionality of induces the related subtleties. generically be destabilized byvacua a with different domain fluxes, wall. theof relevant In a scale five-brane particular, for wrapping for the22 a domain domain three wall walls cycle tension separating and isof generically given the leads by domain to that moduli wall tension destabilization is [ set by non-perturbative effects, it is not clear that this will normalizing the fields and using ( de-stabilize the modulus, Thus, unless the domain wall tension With structure moduli. Assumingprotecting the that vacuum from the decompactfication is modulus given roughly is by at its stabilized value ble moduli. lization in the KKLTcouples scenario to [ the Kählermodulus of the form where bility when the sourcethese becomes conditions large, we so find a weakly coupled vacuum with positive energy density and sta- to arise at largeto volume. find If a we stable aresupersymmetry vacuum interested breaking with in in positive describing a energy wayHowever, a any that density. realistic source is Ideally, cosmology, of decoupled we one positive fromthe need might energy the overall hope localized volume Kählermoduli stabilization. of to in the four achieve compactification. dimensions At interacts large at volume, least this with corresponds to a coupling negative vacuum energy We can see from the formlevel of and the non-perturbative potential terms ( in the superpotential was required for a stable vacuum accidentally small values of JHEP12(2016)155 ) t, r (3.1) (3.2) ( R . From now r . 2 2 and Ω t d ) t, r ( , 2 1 spherical domain walls 2 2 R Ω − d + 2 N 2 4 ] R dt + ) 2 t, r is the coordinate of the outer boundary of ( dR ) r α and Ω by r R N ( R 1 + , − α T A dr )[ – 9 – + 2 , such that ˆ t, r ( dT 2 ) L ) are non-dynamical and set the gauge, while R → ∞ ( + α t, r N 2 ( r A r dt − ) N ] for related works). Consider = t, r 0 and ˆ 55 ( 2 α 2 – t ≡ ds 48 0 N r − = ) and shift 2 . The metric within each of the patches, defined by a radial coordinate patches of static Schwarzschild-(anti) de Sitter spaces labeled by the in- t, r ds , is given in static coordinates ( t α ˆ r is the metric on a unit two-sphere. While the spacetime is static within each N ...N N ] (see also [ 2 2 Ω 47 . , = 1 d α < r < 46 α Thin wall vacuum transitions in gravitational theories have been the subject of intense 1 We implicitly define ˆ 4 − α ˆ The lapse corresponds to the radius ofon curvature we of will the mostly two-sphere omit at the coordinates explicit dependence on the coordinates toregion simplify the notation. where region, the domain walls are dynamic,metric so in let 3+1 us consider dimensions, the most general spatially isotropic dex r 3.1 The Hamiltonian descriptionLet of us domain review walls ing the [ Hamiltonian formulationseparating of gravity with spherical symmetry, follow- literature. However, some ofwe present the a results self-contained discussion arewalls of not the in immediately leading the semiclassical intuitive, presence dynamics sowith of of negative in gravity. thin energy Although this domain density, we most section avoidmostly of an on explicit our review de discussion of Sitter readily the vacua. applies related subtleties to and vacua focus theory as a proxy forbe the very more interesting complex tobeyond problem study the of vacuum scope dynamical decay of compactification. the in It present the would work. higher dimensionalinvestigation theory, and but most this results is presented in this section have previously appeared in the vacuum transitions in 3+1ones dimensions. relevant These in vacuum the transitionsthe context are of runaway not phase string necessarily theory. connectfour the phases Domain dimensional of walls Newton between different constant stabilized dimensionality.theory approaches vacua terminates. In and zero, Despite the and this runaway the observation, phase we validity the will of consider the the four effective dimensional field effective In the previous section welating discussed between a de generic Sitter runaway vacua. instabilitywith of Naively, thin this domain represents walls domain an interpo- walls. obstaclebetween two for However, de vacuum it Sitter transitions phases is inthis interesting the specific to absence of ask problem, any if direct we there domain now walls. exist review Before stable considering the transitions general theory of spatially isotropic thin wall 3 Thin wall vacuum transitions JHEP12(2016)155 , 2 α (3.5) (3.6) (3.3) (3.4) , the m , A +   2 2 α ˆ p , +1 α L ˆ α . The energy A ˆ r with time-like A b.t. ) µν α S α h s ˆ r M denotes boundary , respectively, and + +1 α − , m ˆ α M r A ˆ b.t. ( L ∂ A 2 α , α δ g S 2 α ˆ L 1 ˆ , ] √ L − =1 ) away from domain walls. − Shell α X α x 47 N r − ρ S [ 4 ( − 1 d + 2 ρ 6 0 µν α, − 2 =1 ) + 0 α, g ˆ R X α M N r ˆ R ( Z ≈ − =1 q N ρ + X α q 2 + m − arises due to a possible discontinuity − r − L K − ) h H + ] α , α 0 α, πLR r ˆ r 0 α, √ ˆ with primes and dots, respectively. . We denote all quantities evaluated at a R 56 N ˆ t R α y − + 4 Shell 3 − r S   d ( L t and δ H M r − . The last contribution – 10 – log t ∂ µν α 2 0 Z π h N L M η R , α 5 2 pl G α σ − α G ˆ − 1 p R M ˙ 2 0 R )ˆ − =1 α α R + X α N 0 ˆ r ˆ π R + 1. We denote the metric at the shell by R d − L − RR + α g 2 ]. The Hamiltonian densities are given by r = ˙ Z √ ( L δ L + x 53 µν 1 and is the reduced Planck mass. π + , 4 α − T =1 . The domain wall separates two (approximate) vacua of a scalar d ˙ ˆ α r are the is the surface tension and radial coordinate of the domain R X α ξ N 47 πG α π , α M ˆ p L − r Z π 46 = 8 ), and the second term in 0 dtdr dt L 2 2 pl L R G are the induced metric and extrinsic curvature on 2 − pl π Z Z and ˆ Lπ M − K M . The action is given by α = = − = 2 L ξ 2 M R S is only non-vanishing within region , α S ∂ and and tension R π = = sgn( 0 α 2 α GLπ σ ij ρ π R Shell σ η h at the shells [ S = = 0 is the four dimensional metric on t r In this work We denote partial derivatives with respect to 5 6 R H H µν where of domain wall with a hatwe and bring an index the specifying actiondynamical the terms domain into are wall. first-order given To simplify in form. the terms analysis, of Deferring canonical the variables as derivation to appendix where wall separating regions density We further assume that differentdensity phases are separatedfield by a theory, thin such wall thatThe with the energy surface surface momentum energy tension tensor is equals then the given energy by density [ of the domain wall. g terms that may be necessarywill in assume order that for any Hamilton’smuch intrinsic shorter equations than dynamics to any of hold. scale the relevant Forwalls, for simplicity, so matter the we we dynamics Lagrangian can of occur approximate the the over spherically matter symmetric time Lagrangian domain scales boundary where Consider a scalar field theory coupled to gravity on a spacetime region JHEP12(2016)155 r by N (3.7) (3.8) (3.9) α . We (3.10) (3.13) (3.14) (3.11) (3.12) 2 R , 2 , while 2 H ) r r α . − ) ˆ N are continuous α ˙ ˆ and ˆ r + L = 1 + R α α ˙ ˆ r r , α A 0 A ( 2 ˆ L and 2 α N R ( L ˆ L 2 α R . √ ˆ L − . α 2 0 L,α ) with t α 2 α ) in the vicinity of the shell π m ˆ ˆ 0 N L 3.1 L 2 α 3.8 R q , m arccosh = 0 = , α + R . α 2 α ˆ ) R,α p ˆ p − . Shell α S , , q Hr 1 ) A α α , π α = 0 = 1. The constraint equation then has the − 2 =1 G ˆ ˆ ˆ p L R α r t X α N sin( L LR A − − 1 ( H N t + − − − ∂ – 11 – = = 2 = t 0 H 2 t 2 0 − − − R . 0 L = Space H R 0 α, ) GN 3 p ˆ S L,α, R R r ) on the system, we find an explicit expression for the ˆ π = 0 and LR = − π R r G r − 3.9 S π + G N N η Rη + ( 0 α, ˆ R = = dr L,α, ˆ π = 1, R + L,α 1 − L π α − ˆ r α . The action does not contain kinetic terms for the shift and lapse, ˆ r 2 α R , π Z ˆ R ˙ are momenta conjugate to the canonical variables . Using this notation we can integrate ( R α =1 N p t X α − πσ α 0 to find r = R =ˆ GN and ˆ α r r = 4 r | 0 R N α π R Space appear as non-dynamical Lagrange multipliers that impose the secondary Hamil- , 1) equations that determine the classical domain wall dynamics, in addition to the S = m t L = L − N π We now consider the junction conditions at the domain wall locations by integrating Let us pause for a moment and consider a single domain of positive energy density constraints that determine the spatial geometry between domain walls. For constant π N 0 α, N ˆ located at ˆ three-geometry is illustrated in figure the constraint equations acrossacross the the defects walls. and noting WeR denote that the both derivatives just inside and outside the domain wall and vanishing mass parameter,pick so coordinates the where metric issolution given by ( As expected, this simply corresponds to a spatial slice of de Sitter space. The corresponding where After imposing the constraintsfull ( action, 2( 2 energy densities within the domains we can rewrite the constraint equations as where which implies the secondary constraints By considering these constraints at the location of each of the shells, we obtain the required and tonian constraints. The conjugate momenta are related to the velocities by and JHEP12(2016)155 (3.19) (3.18) (3.15) (3.17) (3.16) , 4 α = 0, and we 2 ˆ gives a simple R α G 2 α p 4 L κ π + that determines the ], 0 2 α is the physical radius of . ˆ R R 56 ) . These junction condi- R ) α G α r , r α (ˆ ) (ˆ α α πσ A r +1 + (ˆ α + 0 α, 2 determines whether the domain A +1 = 4 ˆ 0 , 2 α R G α ˙ ˆ α ˆ 2 R α R A ]. Evaluating an angular component κ ) + ˆ R α = 57 α r , ˆ (ˆ L − ) = r α r 2 κ 0 0 α, ( + 1 2 ˆ A ˆ R R R α, ˙ ∓ − ˆ ∝ T , α 2 ) ˆ , L α – 12 – )) r ) ) α θθ, at the wall, (ˆ α r α − = 1. The constraint equation for ˆ t r (ˆ r K (ˆ +1 corresponds to the proper time of an observer and we (ˆ 0 α, L α α +1 t ˆ and the more abstract quantity 2 α R +1 α A A κ α ˙ α A R 2 + A ˆ R G − parametrizes the spatial slice. 2 α κ 4 − = ˙ ) ˆ 2 r R ) α − α r is continuous across the shell: it corresponds to the velocity of such that = r (ˆ α, (ˆ ˙ ˙ α ˆ ˆ r T R + α A 2 0 α, A ( ˆ R = = + α, 0 2 L,α, ˆ R π ) and the normalization of the four-velocity to find the static coordinate time ) we see that = ˆ 3.18 ) in terms of the proper time . Illustration of a spatial slice of de Sitter space. The quantity − 3.15 3.1 2 L,α, In the rest frame the coordinate ˆ π of ( the wall and should agreecan use for ( observers traveling onT either side, but close to the wall. We extrinsic curvature, With ( Remember that the definition ofnot “interior” be and surprised “exterior” is to quite find arbitrary both and possible we curvatures. should now pick the coordinate relationship between the velocity of the wall’s extrinsic curvaturewall shows is that curved the towards sign what of we called the interior, or the exterior [ so that sign will be important to develop a good intuition for the domain wall dynamics. where we introduced the rescaledtions domain determine wall the tension dynamics of a given domain wall [ In the rest framefind of the a simple junction domain conditions wall the momentum of the wall vanishes, ˆ Figure 3 curvature, while the coordinate JHEP12(2016)155 . α , the r (3.22) (3.20) (3.21) , α in region M ) and can be 2 pl + 2 α M ˙ 3.15 ˆ 2 α R H + . α α = 3 2 α ˆ for physically identical κ R α GM ρ 2 4 . To illustrate this feature, ∓ 0 − R +1 2 α 2 α . ˆ ) to find the asymptotic mass R H α 2 α H − 3.18 R 2 α − GM 2 1 H . In this case the contribution from the 2 α r s − ˆ ) 3 α R at the location of the wall, the situation 2 ˆ α − ) and ( R r R κ α, 2 α 2 0 ˆ ) + H 3.15 R α – 13 – − M sgn( . The domain wall tension always seeks to decrease 0 − α R ) = 1 m at the domain wall is determined by ( +1 0 R α + ( 3 α R α M ˆ ( R A G . This problem has been intensely studied in the literature, 2 α G 2 , the sign of this term depends on κ 2 A ) with r , . When the physical radius increases with the coordinate − = B R 3 α 3.1 ˆ R ) and recover the spatial geometry at a classical turning point, where is set by convention. In the next subsection we will use the rela- ) to obtain the domain wall dynamics both in terms of static coordi- = 0 α, . Each of these terms has a simple and intuitive interpretation. The ˆ ˙ ˆ +1 R 2 α α T 2 α ˆ 3.15 3.19 R H G 2 πσ − ]. 2 α H 58 = 4 ) and ( , α = 56 m 3.18 +1 α = 0. The resulting spatial geometries are shown in figure M L sufficient to consider thediscussion walls to independently. a single In domain the walllabeled separating following two by we Schwarzschild-(anti) de indices therefore Sitter constrain spacetimes, so the we only presentdetails some [ of the main results in this section and refer to the references for asymptotic mass. The quantity written as The constraint equations determine the classicalspacetimes, dynamics of so domain walls separating in static the absence of collisions there are no dynamic interactions and it is the radius of curvature tension forces the domain walldomain to smaller wall values tension of to thethe asymptotic radius outside of mass the isis three positive. reversed sphere On and decreases the the other with domain hand, when wall tension contributes with a negative sign to the outside a smaller radius of curvature.with Since the radial the coordinate radius ofwe curvature can can integrate either ( increase orπ decrease configurations but different signs of where first term is the contribution duethe to gravitational the surface-surface vacuum energy interaction density, term, theof second and the term the constitutes shell. third As term is one due would to expect, the the energy latter term always drives the domain wall towards We can rewrite theparameter in constraint the equations exterior ( region, The generalization to multiplestraint domain equations walls is a straightforward, different butof gauge Schwarzschild-(anti) to is de obtain Sitter needed simple spacetimes for withThe con- each energy metric wall. density is given We in consider ( spherical regions tions ( nate time and proper time along a3.2 trajectory. Classical domainWe wall now dynamics proceed to discuss the classical dynamics of a single domain wall in its rest frame. where the sign of JHEP12(2016)155 by z (3.24) (3.25) (3.26) (3.23) . ), 2 z κ ( A A V . M M 2 A = 0. , + − H ˆ R 2 B ) where B B H 4 M M 1 3.1 4 z 0, such that there exist 2. The dynamics within + − + ≤ 2 ˆ 2 r , A ˆ r , z γ ≈ − ) H 2 A and has the form of an energy γ + . The effective potential has a r > 2 H − max r < 2 ˙ ˆ E z R 2 z B + ( for H for 2 B V are given by − H , = ) , γ + A A B ) = 2 M z κ max R ( R V . For fixed asymptotic masses the parameter GM − GM 2 2 2 + = 5 B H 4 − – 14 – + − M 2 E,V 2 R R , where ,H 2 2 B A ) = 3 2 sgn( H H z ˆ max R ( | z = − − V A + M = 1 = 1 2 , γ − 2 + B A 4 + B A H A H ∂t separating two static metrics of the form ( ∂z | M | B r G 2 M 4 + 3 2 κ κ H 4 − 8 ≡ A 3 z ) to find an expression that is quadratic in M | denote the inside and outside regions, respectively. We can solve the constraint G A 2 3.21 . Spatial slices of two de Sitter phases joined by a domain wall. The interior and exterior = and 2 / To set the notation, we discuss the dynamics of a spherically symmetric domain wall 3 B | determines whether the gravitational interaction of the domain wall tension or the energy E | wall, the small tension orthe weak potential gravity limit are corresponds determinedmaximum to by at the a constant finite ofboth coordinate motion bound andmasses unbound vanish solution the when only “bound” the solutions masses are are static configurations finite. at When both asymptotic We illustrate the effective potentialγ in figure density are dominant in the dynamics. If the mass parameter vanishes on either side of equation for a particle moving under the influence of an effective potential where the real, negative energy and the parameter conservation equation. To simplify the expression we define a new radial coordinate This leaves the constraint equation in a particularly simple form of an energy conservation and equation ( are physically identical and related by a coordinate change. at a radial coordinate ˆ Figure 4 geometries are shown intrue light vacuum and interior, while dark the lines, right respectively. shows a The bubble left with figure a shows false a vacuum interior. bubble with The geometries a JHEP12(2016)155 . The (3.27) (3.29) (3.30) (3.28) | V | for varying = z | U | , 2 2 Ω d . 2 2 2 . R 4 Ω ) . t d ( 2 . We illustrate both spacetimes ) 0 ) + 2 R 0 R R B ( R S dU ) + A 3 ∝ 2 + = T dU t S dV ˙ is related to the change in coordinate time T + − 0 A ∂ ( 2 R , ∝ 2 z , dV ) GM r 2 – 15 – − R 0 ) ) ( ( t t R/ 2 R ( ( − dS e U V A 3 1 ) − HR H = ) as a function of the rescaled radial coordinate R z GM ( dS 1 + ˙ arctan V T 32( t 0 ∂ ) by picking opposite signs for the de Sitter and Schwarzschild = = 0 2 4 6 8 − − − − 2 KS 3.19 2 GH that are smooth everywhere, and implicitly denote Gibbons-Hawking V 2. ds ds V ≤ γ and ≤ 2 U − . In these coordinates light travels along 45 degree angles, and the spacetimes ) the change in polar angle along a given trajectory is directly proportional to . Effective potential 6 3.19 One convenient feature of Kruskal-Szekeres and Gibbons-Hawking coordinates is that In order to illustrate the causal structure of the trajectories we can change to new regions, This coordinate definitionappears implies as that propagating a in the light same ray direction crossing in a either diagram. given time-like The opposite trajectory sign choice the change of the radius of curvature with We can chose a convenient convention forwith how proper time in ( with ( region. The coordinates are explicitlyin defined figure in appendix are divided into four regionsmetrics by the in Schwarzschild the and new de coordinates Sitter horizons become at energy density densities is conceptuallyof straightforward, this most particular interest special to case our will remaining be discussion and iscoordinates simple to illustrate. coordinates for the de Sitter region and Kruskal-Szekeres coordinates for the Schwarzschild 3.3 Dynamics ofWe false now vacuum turn bubbles tospace the from discussion a of Schwarzschild domain spacetime. walls While separating the a generalization spherical to patch arbitrary of masses de and Sitter Figure 5 parameters JHEP12(2016)155 3 is 4 T (left)

=0 2 =0

R S ∞ A =0

κ < H

= dS T 0 A

1 z) /H (

< V , for

=1 ′ +

z R 0 R > 0 2 z

= const. ′ −

R R =0 T 1 −

=0 0

2 R > − ′ + R 0 = 0 surface. Static coordinate time 3 − 0 4 6 8 2 T 2 2 1 0 1 − − − − − − V 3 4 – 16 –

=0 =0 2

T S

=0 3 A 0

0 dS ) as a function of rescaled radial coordinate < A z) z

< (

=0 ′ −

(

/H =0

R V

∞ R ′

R +

V =1

= R R T 2 z 0

= const.

R 1 − 1 0 0 In both diagrams the trajectory moves upwards with increasing proper time. > 2 > 7 ′ − (right) for a de Sitter/Schwarzschild domain wall. The horizons are shown by dashed − ′ + R . Spacetime diagrams of Kruskal-Szekeres coordinates for Schwarzschild (left) and . Effective potential R ) corresponds to an increase (decrease) in the radial coordinate R in region I/III | 0 3 V 0 6 8 2 4 | − κ > H ( − − − − As we shall see, this kind of trajectory could correspond to an expanding true vacuum bubble. 2 0 2 1 1 7 | − − V U on both sides wouldpolar appear angle, in while region it I wouldpolar appear angle. of in the region Schwarzschild III diagram, of with the an de increasing Sitter diagram with a decreasing stems from the| fact that for(II/IV), the while the reverse Schwarzschild is diagram trueconsequences for an the it de allows increase Sitter for diagram. in aimmediately While simpler this the determine choice interpretation has coordinate the of no physical the signcoordinate coordinate diagram. of diagrams For the and example, a we extrinsic trajectory can curvature where the of domain the wall curvature domain is positive wall from the Figure 7 and and dotted lines in thewalls diagram is and illustrated the on sign the of top. the extrinsic curvature on both sides of the domain Figure 6 Gibbons-Hawking coordinates of deradius Sitter surfaces, space while (right). theincreasing upwards/downwards dashed in The line regions solid illustrates I/III. gray the lines illustrate constant JHEP12(2016)155 . 7 (3.32) (3.31) is small . . 3 γ | πGσ / 8 + E | H κ < H 2 + 3 = 4 / , 2 2 = κ ) κ 4 A max max max max =0 A A < V < V GM (2 =0 =0 , z − 0 0 − + < E < V < E < V 4 ˆ ˆ R R / < E = 2 | =0 =0 γ E 0 0 − + | ˆ ˆ R max R − ,E E < V V V V E < V 1 2 2 B B s H H ] it was shown that the unbound solutions 0), the extrinsic curvature on the de Sitter = − + 51 2 2 γ > =0 κ κ B – 17 – A = 2 , z 2 3 , γ z | z γ B E − | − H as 1 ) that the wall’s extrinsic curvature is positive for small radii. 4 p 2 z simplifies and we have and the initial conditions the domain wall trajectories are bound, or = = z . 2 when the rescaled domain wall tension 3.17 2 0 + + E 1 ˆ R ≈ − , ,H , and the location of horizons. We can express these quantities in terms 0 2 3 γ 3 ˆ z ˆ R R | γz A E | 2 + p 2 + H GM 2 Unbound IV - I - II III Unbound IV - III - II III Unbound III-II IV - III Bound III IV - III - II Bound III IV - I - II Type de Sitter Schwarzschild Conditions 2 = . Summary of all possible trajectories of a de Sitter/Schwarzschild domain wall. Depending = 0 − 3 ˆ z R Note that low mass, unbound solutions have negative extrinsic curvature when they For the case of a de Sitter (inside)/Schwarzschild (outside) domain wall the new, come to rest andthe start de expanding, Sitter i.e. toSchwarzschild the the diagram radius Schwarzschild and region. of region I curvaturecontains These of decreases more solutions the as than pass de we half through Sitter passde of diagram. region Sitter from a III This region spatial means is of slice that causally the of protected the from bubble de extinction. Sitter To space an when observer it in comes the Schwarzschild to rest, so the approach a finite radius and expandare again. not In buildable [ byHowever, classical it dynamics: is possible theybarrier that always and a emerges contain non-singular, as a an boundare singularity unbound solution summarized solution. in tunnels in All their through table possible past. the classical potential domain wall trajectories When the domain wall tension isinterior dominant is ( always positive. ToFrom illustrate this the full figure dynamics we we cansolutions show determine the and potential all in unbound possible figure solutions. domaincollapse wall after The trajectories. bouncing bound off There the solutions are potential. bound emerge The from unbound vanishing solutions size, recede and from infinite size, of the new radial coordinate We immediately see with ( rescaled radial coordinate Here we seecompared that to the Hubblewhen scale. considering the There domain areradial wall coordinate two dynamics: main the quantities change that of we the are radius interested of in curvature with Table 1 on the constant ofunbound, motion and pass through different regions of the conformal diagrams. JHEP12(2016)155 3 =0 2 T ], and by 29 1

=0 =0

R R ) the trajectories 0 3.30 = 0. A natural question . 1 L − 8 π ]. Recall that our deﬁnition of 2 52 , − 51 3 − 2 2 1 0 1 − − 3 – 18 – , which determines the direction from inside to r change sign, such that with ( r

=0 2 R we provide a brief explicit discussion of the Schwarzschild 1 B . In particular, as we reverse the deﬁnition of inside and 4 ), and is immediately clear when recalling the spatial geometry ˆ r 0 r > 1

− =0

2 R − . Illustration of the unbound trajectory of a de Sitter/Schwarzschild domain wall that tra- ) and outside ( 3 ˆ r The dynamics of a bubble containing a de Sitter phase inside a Schwarzschild region − 2 1 0 2 1 − − r < question was first askedFarhi by and Coleman Guth for and disconnected de falsethe Luccia vacuum bubbles bubble for [ interior true is vacuum arbitraryfor for bubbles true de [ Sitter and vacua, false so vacuumlation the bubbles. which classical dynamics This allows are invariance to equivalent is approach manifest the in nucleation the Hamiltonian of formu- true and false vacuum bubbles within 3.4 Semiclassical transitionStudying probabilities dynamics revealed botha bound classical and potential unbound barrier. trajectoriesbut A gets that classical reflected are solution at separatedto the cannot by ask turning penetrate is point the whether of effective the transmission potential, potential through where the barrier is possible in a quantum theory. This outside all derivatives with respectwill to appear on oppositepresent signs discussion of the of Kruskal-Szekeres avacuum coordinate bubbles. false diagram. In vacuum appendix Therefore, bubble(inside) the - already de captures Sitter the (outside) domain full wall dynamics dynamics of for reference. true are identical to the dynamicswith of the a exception bubble that ofoutside, Schwarzschild is the phase redefined. coordinate inside This a reflects( the de arbitrariness Sitter of region, our definition ofof what we the called inside two cases in figure appear reversed: both extrinsic curvaturesincreases are as positive they for approach them, the i.e. domain theknown wall radius CDL from of domain their curvature wall inside, evolution. whichthe corresponds An Schwarzschild to example and the of de well an Sitter unbound coordinates, trajectory, is as shown seen in both figure in Figure 8 verses the regions IV-I-II indiagram. the de The Sitter spacetime diagram and diagram is is contained not in applicable region III in of the the fadedphase, Schwarzschild region. who experiences a reversed definition of inside and outside, the situation would JHEP12(2016)155 (3.33) (3.34) (3.35) to a final state I 8 . ). dS 3.5 with fixed sign and leave the yet F . I B Z . , ) to estimate the leading exponential i ) 2 = 0 0 , see ( ~ − ( ] S ψ ∼ P 3.35 O as a transition rate Γ, the bound turning r,t 24 + ~ H P iS/ = – 19 – − = exp e ψ r B π = η ,N t − ψ N e π = the exponential dependence of the wave function is related ~ I→F P . Remember that this prescription only holds for the leading S ]. 53 , = 0, is the bound classical turning point an empty de Sitter space. In 47 ˆ , R 46 in the exponent of what we will interpret as a probability is potentially am- , and depends only on the spatial geometry. In order to avoid the ambiguities of π r η N At first sight the situation is somewhat precarious: we obtained the transmission coef- We should emphasize at this point that the initial state is not a pure de Sitter phase, In order to employ Dirac quantization, we impose the constraints on the wave function and Remember that in the framework of canonical quantization the states correspond to spatial geometries. 8 t . The transition probability, derived from the transmission coefficient, is given by , which yields the Wheeler-DeWitt equation [ ficient for a masslessmechanics, energy eigenstate, this which dilemma of isthe course resolved energy is by due time-independent. considering to In the all quantum non-perturbative possible tunneling contribution trajectories, to which yields a small imaginary part a thermal fluctuation. Notevanishes that at even the though bound theremains finite radius turning everywhere of during point the curvature bubble inwe of nucleation the process. can the therefore massless domain In the use wall configuration, vanishingdependence the mass the limit of transition the curvature probability the scalar ( vacuum transition rate, Γ Only in the masslesscorresponds limit, to where the turningorder point to of interpret the thepoint bound tunneling of domain probability the wall trajectory trajectoryphase constantly should satisfies arise the required at initial some conditions, while frequency. the massive In case may the require massless limit a de Sitter The sign biguous. We proceed toundetermined define sign a in the tunneling definition exponent of the action but corresponds to a classical turning point of a bound or unbound domain wall trajectory. and perturbations around the leadingwill contribution mostly to the be path interested integralF in are the prohibited. transition We rate from some initial state such that to leading orderto in the classicalsemiclassical action approximation. In this work we restrict ourselves to the leading contribution, The primary constraints demand that theN wave function is independent of thequantizing gauge a choices gravitational theory wethe impose wave function the in classical the equations WKB of approximation motion as and expand walls in the Hamiltonianprobabilities framework [ to find explicit tunneling trajectories andψ transition a unified framework. We proceed to review the leading semiclassical evolution of domain JHEP12(2016)155 , , ]. π π η ≡ 26 , ), (3.37) (3.36) 1) 25 , where − 0 limit, = 0, we . While that are 2 2 3.35 1 h A ˆ B R ˆ → R . Using the or r G < R and 1 A ). At a classical ) 1 2 h A ˆ , and arccos( ˆ R R R , 3.11 iπ − ) ≡ 1 2 is negative between the h 2 A h 2 B 1) 9 . The contribution to the 0 R R − B R )( − 0 + ). At these points the domain , and noting that 1 , and we label the initial and , ˆ 2 R 2 R h 2 B ] 0 ( ˆ ˆ R R − R R A ( B < − = N 1 2 Θ[ ]. ˆ R ) + Θ( 28 2 /L ) + h 2 B 2 2 0 h 2 A R dR R R . R − Z 9 2 − , where ˆ – 20 – 2 π 1 R , π 1 G h 2 A )( η ˆ we can immediately evaluate the integral for the R R 0 − ( h 1 ˆ = R A R − , so the action simply becomes N α ], and merely present a heuristic argument in this work. Θ( + Space A 2 and 28 iS π L G h 2 π ) between the two classical turning points 2 η = R count the number of spacetime regions in phase 3.5 2 0 = ]. Naively we might expect that the decay rate is small whenever B . We call the horizons of the exterior spacetime R ˆ / R , ) = 0, and similarly for the interior region F A 60 , 2 N h A ]. In the presence of causally disconnected regions this limiting case may ) as the integration proceeds along a path of increasing Space 59 ] the sign convention appears to differ. and R , 29 ( iS 47 I A 3.36 26 , A ) and coincides with the tunneling wave function proposal by Vilenkin [ 25 B = +1 [ ) = π 1 η h A To obtain the contribution to the action from non-trivial variations at the shells we We consider transitions between a bound and an unbound turning point of a classical R The alert reader shall not be confused by signs when comparing to the literature: it is important to ( = sgn( 9 A π this is a hardvanishing asymptotic integral masses. in general, Dropping we the subscript can of compute it analyticallyuse for consistent sign the conventions. special Inwhile this case in work parts we of of use [ the convention arccosh( where the integers disconnected from the domainis wall, illustrated and by vanishes a for specific anti example de in Sitter figure spaces.evaluate This the notation integral in ( spatial contribution to the action away from the wall and find where Θ is thethe Heaviside integral step ( function.Schwarzschild-(anti) de Care Sitter has metric,outer to and and be remembering inner taken that horizons in picking the limits for action from the spacetimeturning regions point between we domain have walls is given by ( trajectory, where the spatial geometry satisfies wall radius of curvaturefinal is states labeled by by A where not be instructive, andcontexts, there [ have beenthe multiple action proposals of for a fixingη classically the forbidden sign trajectory inA becomes various careful large, derivation of which this fixes result the is sign presented to in [ be to a subsequent publicationIn [ particular, the signis of yet the undetermined, action and alongthe potentially the WKB matching ambiguous. relevant conditions, tunneling but In trajectorywhether since principle, in we a dropped the ( given all sign time-dependence mode is it is determined is not ingoing by obvious or outgoing. The sign is known in the for the energy of the metastable state. We defer a thorough derivation of the decay rate JHEP12(2016)155 for . 0 r ˆ R (3.39) (3.40) (3.41) (3.38) ,

1 2 scales with , , , − 0 0 0 B F > < < , 2 B B 0 0 0 + + + N ˆ ˆ ˆ R R R GH − , , , I , where , 0 0 0 | B . B . > < > N = 2 disconnected regions in 4 ) −| I 0 0 0 κ − − − e B + , ˆ ˆ ˆ R R R ) N + ∼ 0 + ˆ , and switches the sign of R , 2 2 B Space B ) S , , 2 H B sgn( 2 2 H − ) ) + + + 2 2 and A A 2 A 2 2 A A F H H BA , H , A H − − A 2 2 B B . Right: the corresponding spatial geometry. GH 2 2)( N 2 H H A κ − Shell ( ( κ 4 ˆ − ˆ ˆ R S R R κ I ( + + , + ) ) + 2 κ κ – 21 – A 2 2 A A + 2 κ κ 2 A F 2 N 2 2 , H H A A H ˆ 2 R B + + 2 H H B + 2 B 2 2 B B 2 2 B B H 2 H H H ) is an important and beautiful result, so let us pause Space GH κ 2 A ( ( − 8 ) GH GH S ˆ ˆ R R 8 8 2 ( B 2 2 2 A κH i κ κ 3.41 H H + + +4 ) ) + ˆ R 2 = 2 B 2 B 2 A 2 2 A ) = H H H B 2 B H − 2 2 − A π H ) as prescribed by the matching conditions for the WKB tunneling 2 2 A B ˆ η R − +( H H B 2 ) we then obtain the transition rate Γ A 2 ( ( ) κ κ H κGH 2 ( 2 2 B 4 π π π 3.38 H π π π = 1 disconnected region in phase η η η = sgn( − 2 A A π            η N H = ( ) and ( ˆ R . Left: radius of curvature over the coordinate parametrizing a spatial geometry

) that determines the leading contribution to the tunneling rate, , and π 3.37 Shell B = iS Finally, we are in a position to evaluate the probability for vacuum transitions. We | 3.35 B | and we used wave function. The exponentto ( appreciate some of itsthe features. domain wall The first tension,each term while of in the the the phases. latter tunneling We two exponent wrote terms the depend tunneling rate only such on that the the statistical Hubble nature scales of de in Sitter With ( of insides and outside, which reverses the roles of add the contributions fromin the ( spatial, and shell parts of the action to find the exponent radius of curvature As expected, the contribution to the action due to the shell is invariant under the exchange where the turning point corresponding to an unbound domain wall trajectory occurs at a multiple Schwarzschild-de Sitter spaces.phase In this example we have find the action contribution at the shell location Figure 9 JHEP12(2016)155 ), G (4 (3.42) (3.43) / in a de ] dS = 0. These 62 B A = 0 because F F , are reversed. , B A N B N = . I 1 2 and , B − A N = 1 and = I 1 2 , we rewrite the transition A − N C F , B Remember that the replacement N . In both cases 11 − A 0 we have , I , < A in this case. For example in the case of a true . This case corresponds to the massless B 0 + N −S R 10 B does not affect the transition probability S ]. In appendix ,N = 1 because the tunneling trajectory does not cross e 0, prohibiting the transition entirely. This 1 2 61 F = , → ) holds, this replacement always maintains the A – 22 – = N ) A I ↔ F ρ = 0 + 3.42 A→B B→A | | ˆ I R ). , 2 Γ Γ A decreases. We can easily understand this by noting that N 3.36 10 sgn( A + ρ F ). This is what one expects in quantum mechanical tunneling , ) is then identical to the generalization of the CDL result to A is the entropy of a de Sitter space occupied by either phase. This 0 we have N 3.35 B > / 3.41 − 2 is the horizon area. A 0 + I , 2 R ) in ( A − correspond to an exchange of initial and final states, but the nucleation of a , and compare this rate to that for the process where B N π/GH A πH not = = 4 B / = sgn( , so the contribution to the decay rate at the wall cancels and we find [ dS A does B π A S η Let us now consider the nucleation of a false vacuum bubble in more detail. In the Let us first recover the scenario considered by Coleman and de Luccia, where there Some care has to be taken in determining the integers Globally this is more complicated as the number of disconnected regions is measure dependent. 10 11 vacuum bubble where any horizon. In contrast, forthe a false false vacuum vacuum bubble bubble where nucleatesobservations beyond follow the immediately horizon from of ( region energy density. However, wespacetime. can Instead, imagine consider a a transition transitiona that that causally maintains disconnected does the phase entire not initial that region, terminate containstotal the nucleating horizon the new area initial vacuum. is In independenttransition this is case of illustrated the the in change initial the in Hubble lower the part scale. of figure The spatial geometry of this as the outside energythe density extrinsic curvature on thespacetime outside disappears of the during shell theinverse is transition. negative, Hubble so The scale the change majority andcorresponds of in diverges to the the the as initial action well increases known with result that the Minkowski space cannot transition to a higher the initial and finalwhen spatial geometry process through a widethe potential probability barrier for in a the single WKB incident approximation wave because packet to weCDL be consider scenario, transmitted where through wormhole formation the is barrier. prohibited, the transition rate approaches zero where means that semiclassicalbalance vacuum in transitions the absence appear ofA causally to ↔ disconnected satisfy B regions. thetrue principle vacuum bubble of and the detailed nucleation of a false vacuum bubble, respectively. Exchanging In the absence ofsign wormholes, of where ( The transition ratearbitrary ( extrinsic domain wallrate curvatures slightly [ to make thisof equivalence these manifest. transitions, To consider gainSitter the some vacuum intuition nucleation for rate the of probability bubbles occupied by phase where exist no disconnected spacetimes space is manifest: the last two terms are proportional to the de Sitter entropy JHEP12(2016)155 = 0, (3.45) (3.44) F , B . Again, 2 N ˆ R = , I , and 2 B 1 N 2 ˆ R ! . ˆ 2 B R R

) ) 2 π/GH ˆ h 2 2 − R e R − . ∼ − 2 2 h 2 2 2 ) 2 κ R R κ )( )( 2 + + 2 ˆ ], for which we have h 2 1 R 2 2 B B = 1 and take the following ansatz for the R 52 H − H L ( − 2 h 2 1 π G R R − ( ( – 23 – becomes negative, so the tunneling wave function = ) = B B 0 R ( → A lim ] and we denote the momentum evaluated at the domain 1. As expected, the transition rate remains finite in the 2 H , f − ˆ ) R , R 1 = ). ( (exterior). The transition is parametrized by the domain wall ˆ ˆ R f R π [ ( L A η L π ∈ π ˆ R ≡ ) = ˆ L R ( π L = 1. Here the sign of π F , A . Illustration of continuous tunneling trajectories for the Coleman-de Luccia nucleation (interior) and N B = To obtain an explicit solution we demand I , A momentum along the tunneling trajectory transition process we nowterpolates solve between for the an boundwe explicit, and consider continuous a unbound tunneling single classicalphases trajectory thin turning that domain points inwall thatradius connects of curvature, two distinctwall Schwarzschild-de with Sitter a hat, ˆ pressed by the horizon area of the new de3.5 Sitter space, Γ An explicit tunnelingTo gain trajectory some more intuition for the evolution of the spatial geometry during the vacuum limit of vanishing initial energy density The limit of vanishing domain wall tension corresponds to a nucleation rate that is sup- (lower row). limit of the Farhi-Guth-Guven (FGG)N process [ proposal indicates that Figure 10 of a true vacuum bubble (upper row) and the Farhi-Guth-Guven nucleation of a false vacuum bubble JHEP12(2016)155 , ), r 2 ( R . The (3.46) h 2 < R h 1 ) then yields the R 3.22 ) are satisfied: the might naively appear to 3.15 , and the negative sign . A 16 B / A | ) and ( ) R 0. In this limit ( 3.9 ( 2 → f 2 L A ˆ π 2 L < M 2 B G M ) + along the tunneling trajectory. The boundary R – 24 – ) to obtain the three geometry specified by ( ˆ R B / 3.9 A , while the formation of a true vacuum bubble in the absence of A B 2 L so the domain wall does not have to cross any horizons. < M = ) and the boundary conditions at the horizons fix the spatial A the Schwarzschild horizon is crossed between the second and B 12 / . The horizons of each spacetime are labeled as M A B 10 | 3.46 2 0 R shows two specific trajectories for the quantum nucleation of true and false ) is chosen such that the constraint equations ( 10 3.45 Figure Some care has to be taken in evaluating the extrinsic curvature. Figure 12 face this much more difficult problem, inimply that this the work extrinsicthat curvature we figure does displays model switch the sign case theany for 0 mass = runaway the parameter nucleation phase corresponds ofexpected to as a the result true opposite of a vacuum same limit, bubble. sign extrinsic However, curvatures for the bound and unbound turning points. weakly coupled vacua populatedthere in generically the do presence not of existInstead, stable, runaways? all thin domain As domain walls discussed connect wallsdimensional to separating in a vacua effective section runaway in phase. theory the In breaks landscape.to this down revert hostile to phase as the the the simple higher four spacetime dimensional decompactifies, theory so to we model ought this transition. To avoid having to 4 Gravitational effects onIn viable vacua the previousfinally section in we a carefully position to reviewed address thin the question wall relevant vacuum to the transitions landscape and population: we how are are third steps. Thiscorrespond simply to corresponds false to vacuum nucleation the throughparametrized the FGG by Hubble transition. a horizon, and monotonously The is increasingprocesses not CDL radius of continuously transition of false would curvature. vacuum nucleationto Both are different the allowed final CDL at geometries and the and FGG semiclassical have level, different but transition correspond rates. simply corresponds to thefalse CDL vacuum transition. bubble the On wall’s exterior thethat extrinsic other curvature the does hand, switch wall during sign, crosses the whichmonotonously indicates at nucleation increasing of least radius a one of horizon.false curvature vacuum at The bubble the through continuous domain the trajectoryin wall Schwarzschild the horizon. corresponding yields lower the to For panel the formation of a nucleation of figure process a shown vacuum bubbles. During thevature nucleation does of a not true switchTherefore, vacuum sign, bubble a the tunneling wall’scurvature trajectory extrinsic is cur- parametrized continuous by if the a bubble monotonously nucleates increasing within radius a of causally connected region. This The differential equation ( geometry for any domainconditions wall at position horizons aretinues necessary behind to the specify horizon, whether and the if spatial there geometry are ends any or domain con- walls in that patch. applies in the inner region ansatz ( momentum vanishes at the horizons andcan takes now on solve the the constraint correct equation value at ( the domain wall. We where the positive sign in the exponent applies in the outer region JHEP12(2016)155 A / C (4.2) (4.1) 0 gives and . Because → can occur = 0, while C A G B B / B in the absence M B = and ) simply as A to A M A 3.1 . i 2 . The constraint equation for ˙ ˆ R , C 2 2 ), which in the limit Ω 1 + domain walls are prohibited, so any d 2 3.21 q B ]. In this work we do not impose any R / i 2 . This question was first discussed by , after the tunneling event there exists ˆ + A C 58 R B , i 2 . B πσ ]. 52 dR , and 23 + 29 + 4 – 25 – is given in ( and A 2 . We take the initial and final states to be in a i i 3 B ˆ A ˆ R dT R B − ρ and = π 3 . The spatial configuration before and after the vacuum 4 2 A r = ds . Initially the entire space is occupied by vacuum = are positive, but vanishes in C 0. In this limit we can write the metric ( 11 R B M may experience a non-vanishing mass parameter. The vacuum ]. They argued that even in the absence of a tunneling instanton → C is an asymptotically flat region. We will consider the dynamics of 23 and G C in between A C = 1 and L transition between de Sitter vacua contains a double bubble with To summarize, in our model of the landscape In order for a transition to persist at late times a horizon has to be crossed by the the intermediate phase energy densities of the inner wall with radius of curvature or equivalently transition is shown in figure of the absence ofa domain region walls of between phase pure vacuum configuration, such that the asymptotic masses vanish, As a warmup exercise, letof us dynamical first gravity, consider a vacuum transition from domain walls, where these configurations and present thein geometry a and rate stable of vacuum a transition nucleation between process that results 4.1 The trouble with the bubble on that solution. However,is remember identical that the in cosmological bothregardless evolution of cases, after which the so mechanism transition populates any the constraints vacua. on theA cosmology → of B the new phase apply the original de SitterCDL phase or or the a FGG wormholeconstraints transition, horizon respectively beyond is [ traversed, obeying which thetransitions. corresponds Hamiltonian In to the the constraint limit equations, ofmore a likely so small to we initial occur vacuum allow energy because for density it the both preserves FGG the process is initial vastly spacetime, so we focus our discussion any de Sitter vacuumthe transition Hilbert will space. eventually Even occurdo though due not our to explicitly approach the substantially invokecompatible the differs finite with from thermodynamic dimensionality and that properties of extend work of the and de results we of Sitter [ space, ourdomain results wall are during the bubble nucleation process. Either the cosmological horizon of wall the runaway instability isprovides merely a triggered, good but approximation the towhether four local and dimensional physics. how description In stable still thisif vacuum section all transitions we domain between address the de wallsBrown question Sitter connect and of Dahlen vacua to in a [ runaway phase stable vacuum with vanishing energy density. It is conceivable that close to the domain JHEP12(2016)155 (4.3) ) domain C ) = 1. The / B , r . For simplicity = 0 B t domain walls exist. ( L C / , contains a Schwarzschild B 0 ], but in the presence of gravity C in the presence of gravity, 63 < and B ! ) A can be negligible. On the other . As anticipated, in the absence = 0 where / B i 2 C B t ˙ and ˆ i R ˆ R A . Note that the extrinsic curvature of . We can obtain the geometry along a 7 (1 + B + 2 i 2 ˙ ˆ R – 26 – will never occupy all of the new de Sitter region 1 + C q i B σ ρ

, there always exist unbound solutions where the extrinsic . − 2 pl B = M i 4 ¨ ˆ / R 2 BC σ 3 We are interested in transitions for which at least the inner ( < 13 B ρ . The spatial geometry of the formation of a double bubble in the absence of gravity. . In the presence of gravity the radius of curvature of the inner shell can exceed the 12 This disappearance of the instanton without gravity was discussed in [ 13 Figure 11 the shell, and the gravitationalwithout self-interaction of bound, the the shell isbecause runaway negligible. phase of Despite the expanding cosmological horizon in vacuum some of the instantons reappear. gravitational self-interaction and inflationhand, inside when phase the domain walldomain tension walls is small have compared a tois negative the the extrinsic energy curvature familiar density, the on situation unbound horizon. both of sides In a of this true the case vacuum domain the bubble wall. domain that wall has This expands nucleated due behind to a the wormhole different energy density across domain wall dynamics canthe be inner read shell off is negative from inhorizon figure the for exterior any region, unbound such solution. thatWhen phase There the are two domain qualitatively wall differentinterior, unbound tension solutions. i.e. in Planck unitscurvature changes dominates sign over across the the energy shell. density These in domain the walls expand due to their repulsive figure radius of the outer domain wall,does while not in exist. the absence ofwall gravity the grows corresponding without instanton bound,we leaving pick a coordinates part such of that the the transition spacetime occurs in at vacuum but again in theGravity absence has of a direct dramaticwithout domain gravity, impact walls: the on initial only the andthe final energy possible spatial densities, transition. geometries mass are parameters,spatial In domain no geometry wall longer contrast changes. tensions fixed: to and Two depending the boundary possible on conditions situation geometries the after the transition are illustrated in of horizons there doesoccupied not exist by the a new vacuum transition phase that would create4.2 a persistent region No troubleLet with the us double now bubble turn to the transition between vacua For an initially staticinner configuration domain wall we immediately find the equation of motion for the inevitably leading to a collapse of the region in vacuum JHEP12(2016)155 . 13 (4.4) (4.5) (4.6) . , o ) at the domain ˆ 0 R < 2 CA 3.22 0 i + κ ˆ CA R − κ 2 2 A , i H . ˆ R = 2 B t 0 H o 2 + B ˆ BC R H − κ BC 2 + κ 2 BC , are illustrated with dark blue, light blue 2 κ 0 2 BC C κ = > 0 0 i − o − and ˆ ˆ R R B cosh , – 27 – , , A 2 B 1 1 H − BC − + κ o 2 i 2 2 ˆ 2 BC ˆ R R κ 2 2 . The continuous tunneling solution is shown in figure ) ) 2 A 2 = B i 3.5 H H ˆ 2 CA R 2 BC + κ + κ 4 4 2 BC 2 CA κ κ ( ( domain wall with small (left) and large (right) tension. C = = / B i 2 o 2 ˙ ˆ ˙ ˆ R R . Two possible spatial configurations of a double bubble that leads to an unbound . A continuous tunneling trajectory for the formation of a double bubble in the presence To give a concrete example in which we can easily understand both the initial geometry Note that after theis tunneling causally event the disconnected spacetime from inwall original are which spacetime irrelevant. the and We inner show the the domain dynamics classical wall of domain evolves the wall evolution exterior in domain each of the three regions These are just the equationswalls governing collapse two expanding into true a vacuum singularity. bubbles, For so the none inner of domain the wall we have the solution after tunneling and themass subsequent parameter classical in dynamics, each we ofwalls the consider become three solutions regions. of The vanishing junction conditions ( and dashed gray lines, respectively. tunneling trajectory bydouble solving bubble, the as constraint in section equation and junction conditions for the Figure 13 of gravity. The regions occupied by phases Figure 12 trajectory of the JHEP12(2016)155 ), ).

=0 B R (4.7) (4.8) 3.35 = 0. Phase T : de Sitter | . ) as ) is small compared to A I 0 0 , κ < > 3.36 0 0

i i .

Space =0

ˆ ˆ R R R S 2 − 2 BC for for F 2 BC κ , so the outside domain wall has , κ , contains a closed universe. A , 2 + + 2 B − B i 2 Space 2

B B =0 ˆ

=0 < H H R R H H R ), and we ﬁnd the total tunneling expo- 1. In the limit of a small outside domain π π + π π G G =0 η η ∼ CA T . Furthermore, the tunneling rate decreases . For simplicity we will only discuss the − 3.38 κ BC 2 B − − A , 2 has a number of interesting properties. Let us ( | decreases. This feature is similar to the familiar contains open FRW cosmologies on both sides of a – 28 – Shell B 2 CA C A κ S ) = : Schwarzschild −| π/GH I e C , + + = 4 CA ∼ 2 | A κ 0. Again, the tunneling exponent is deﬁned in ( contains two copies of open universes on opposite sides CA B , H ) patches. The bubble is nucleated at time Space | C > S A 0 2 A o Shell − ˆ H S =0 R

(

R F i , 2 π G | = = Space S B ( i π ]. 2 η 64 ), and de Sitter ( , C : de Sitter 52 , B . The region in phase . Illustration of the trajectories of a double bubble connecting de Sitter ( 29

=0 R We now evaluate the tunneling rate to form a double bubble configuration from an contains a closed FRW cosmology, while as the Hubble scale of thenucleation initial of vacuum a trueshell vacuum become bubble: equal as the the radiusare vacuum of two energy the competing densities initial terms bubble ontunneling in increases both rate the without becomes sides tunneling bound. surprisingly of large, rate. Finally, the i.e. there Γ When the two terms are competitive the The corresponding transition rate Γ begin with the limiting casethe of Hubble weak gravity, scales or involved. small The tension,the where tunneling newly created rate de is Sitter simply phase, suppressed by the horizon area of The contributions from the shellnent are given in ( The spatial contribution to the tunneling exponent is given with ( of a wormhole, while the region occupied by vacuum initial de Sittercase space valid occupied in by thepositive limit vacuum extrinsic of curvature, weak gravity, where Figure 14 Schwarzschild ( B wormhole [ in figure JHEP12(2016)155 (4.9) (4.10) the vacuum . For small B 4 pl ], and yet their M 81 – × 1 65 . 0 ∼ B ρ ]. , . 89 B B – H H 82 , , 64 , BC 2 BC B 58 ) implies that transitions among the highest , κ πk/GH 4.8 2 − √ e – 29 – . Note that this result is crucially dependent / , κ k ∼ × B BC Γ κ H 2000 2 CA 2 CA − disconnected de Sitter phases in vacuum . Abandoning the tunneling wave function and choosing κ κ e π k ( η ∼ ∼ 3 A H the transition appears to be unsuppressed when A H CA κ ]. 47 In discussions that employ a semiclassical approximation such that ambiguities about The particular final geometry considered above is the leading transition channel for = +1 instead would have given a divergent rate as the number of disconnected universes π treated on equal footing.disregard some Much of care thegravity. has The classical Einstein-Hilbert to solutions action is be baseddescription well of taken understood, on wormholes remains while when presumed elusive. the attempting full knowledge Vaguegravity arguments quantum to ought and about mechanical beliefs to constrain about quantum behave how or may quantum be misleading. which nucleates a wormholesuch geometry. an event is Much admissible effort inRather quantum has gravity, than but been providing no definite spent a conclusion comprehensive ona has literature been studying number review, reached. of whether we relevant refer works the on interested the reader subject, to seethe [ quantization of gravity are irrelevant all classical solutions to the Hamilton-Jacobi are Wormholes have been the focusrole of intense in research nature for is manystable far decades domain [ from walls clear. between We cosmologicalhorizon saw vacua that in bubble stabilizes the the nucleation transition. occurs previous subsection across An an example that of event in such the a absence transition of is the FGG process, nucleation rate ison roughly the Γ sign choice weη made for grows [ 4.3 Wormholes in quantum gravity domain wall tensions the nucleation rate is roughly given by which greatly suppresses the nucleation ofof disconnected a universes. metastable For vacuum example, with in almost the Planckian case energy density stable configuration at late time, but other thick-wall transitionsthe are population possible. of atension. false vacuum There with a are highnucleation many Hubble other of scale possible not and final just a states. one, small outer but For domain example, wall we could consider the one might bethat concerned about there unsuppressed maypopulated. not transition Instead, rates, be the tunneling but any exponent(approximately) ( we metastable stable should vacua vacua are remember at exponentiallyone arbitrarily preferred. particularly high simple In this thin energy work wall that we process merely could to present be illustrate the mechanism that gives rise to a In either case theoriginal energy vacuum scale and of the tunneling new towards phase high is energy larger configurations than is the scales favored. involved in Naively, the wall tension, JHEP12(2016)155 (5.1) 1. We − = p/ρ , such that the is described by a τ = B w by an expanding domain C . 2 2 . So far we only considered a Ω B d 2 R + 2 + const. After the nucleation event the τ dR ≈ + 2 1 – 30 – − Rad dτ ) − aH 2 a + const, while during radiation domination the horizon τ = 2 ≈ − ds . Given sufficient expansion to overcome the initial curvature 1 + − Inf ) Late H aH 0, so the domain wall is hidden outside the Hubble horizon as in the left only for a finite amount of expansion and is followed by reheating, < Inf . At the end of inflation the equation of state changes. The precise evolution 0 − H R 12 During inflation and late time energy domination the comoving Hubble sphere shrinks domain wall separating theoutwards new along cosmology an from approximatelySitter null the space. trajectory runaway For phase domain in wallscale initially a tensions we accelerates causally have that are disconnected small regionpart compared of of to figure the de inflationary Hubble metric takes the form with conformal time, ( expands and allows modes to enter, ( Hubble scale radiation domination and finallylate a time classical Hubble evolution scale towardsdomination, a we can (potentially approximate much the lower) metricis by a most flat obvious FRW cosmology. when The expressing causal structure the metric in terms of conformal time closed FRW universe and iswall. separated from The region the occupied runaway phase byby the an runaway phase open with vanishing FRW energythe cosmology density cosmology with is described of a the (verystationary, energy small) newly dominated black created interior hole. with universe anow in constant We relax equation vacuum are this of most condition, state and curious consider about an evolution that maintains the initial inflationary We now turn to the cosmologycosmology after that the undergoes tunneling a event. finite To periodfinally be of specific, settles inflation, we into followed discuss by energy an radiation FRW relevant domination domination, spacetime and at is divided late into times. two regions: Shortly the after new phase the in tunneling vacuum event the the cosmological horizon, butsphere if the of evolution a allows late thecosmology this domain time means wall observer that to the the enterthe initial new the domain inflationary Hubble wall universe phase from should is the last at long late enough risk time to5.1 horizon. of remove extinction. Inflation In expels an runaways evolving the transition, the newrunaway phase phase is remains outside causally the protectedthe Hubble from cosmological horizon. domain evolution We wall in nowplicit collapse the consider tunneling only the new trajectory implications if vacuum. and for the applies Thisbasic for idea discussion general is is transitions simple: independent with of the thin the new domain walls. phase ex- is The causally protected from collapse of the shell due to In the previous sectionsitions we between discussed two a deillustrated tunneling Sitter one geometry vacua particular that in trajectoryother facilitates the that vacuum ways absence tran- induces to of stable populateare a transitions, beyond the direct there the landscape, may domain reach such be of wall. as many our dynamical While analysis. and we However, thick-wall regardless solutions of that the precise geometry after 5 Inflation in the landscape JHEP12(2016)155 , 14 , so 15 κ (5.2) Reheating CMB Observer . This scenario appears likely 12 , is small compared to the tension B ]. During inflation a runaway domain wall Inf

Collapsing Domain Wall Late 42

H icnetdDmi Wall Domain Disconnected H – 31 – log 1 2 Hubble Sphere & Comoving Distance e

Light Cone N

Energy Radiation

Domination Domination Inflation ofra Time Conformal 0 and the domain wall is at risk of re-collapsing and terminating newly created . Conformal time and comoving distance for a universe undergoing inflation, radiation > 0 − R Even though inflation clearly can exile a collapsing domain wall from the late time We thank Matthew Kleban for discussion on this point. 14 cosmology and save the universecondition. from its Remember demise, that itstrong, when is the the not domain obvious self that wallwall interaction this tension in is bubble is a an wall large, necessary empty is or universe. gravity repulsive sufficiently There and may leads be to concerns an associated expanding with domain strong gravitational which corresponds to aboutThis 65 amount efolds of of inflation expansion, leaves the depending universe on in the a scale surprisingly of flat inflation. and isotropic state. that all null geodesicscones, originating and from is the slightly timeCMB. stronger We of find than reheating a have the rough overlapping requirement lower past bound of light for super-horizon the correlations required in number of the efolds as universe. However, if the lateis time possible cosmology that is even a dominated collapsingbe by domain the a domain case positive wall if energy never the density, re-entersfor bubble it the radius the horizon. exceeds existence the This late of will timewhich a shows Hubble the region scale. evolution of The of sufficient comoving spacetimedomain distance condition to with wall conformal survive to time. indefinitely remain The can out condition be of for the causal seen contact in with figure a late time is precisely the requirement of the bubble will dependit on is the possible dynamics that during after reheatingthe inflation and spatial the radiation geometry energy domination, corresponds density but toif the right we part demand of a figure vacuumcase energy density small enough to allow for galaxy formation. In this Figure 15 domination and a lateexpands phase in of a energy causallyre-enter domination the disconnected [ horizon region. unless all After observable reheating, past light-cones the were domain in wall causal might contact. collapse and JHEP12(2016)155 (5.3) that is typically remains C A κ < H ]. is the axion decay = +1 instead would 99 f π η , where , the bubble may collapse, pl A M . , separated by a single domain f A ). In the limit of weak gravity the . The rate for this process to occur B . 3.41 2 Inf 1. π/GH S > − e – 32 – will trigger an expanding true vacuum bubble ∼ 16 A Γ . However, the nucleation of a double bubble can induce C . This bubble nucleates across a wormhole horizon as in the C . Again, there are two interesting scenarios: either the new phase ]. If the transitions corresponding to strong gravity are indeed B 98 – 90 , 81 ). Again, tunneling towards high energy states are favored. If the energy [ 4.8 15 . This may be a stable ground state of the four dimensional effective theory C Depending on the cosmological evolution in the new phase Let us consider an initial state with an asymptotically flat background geometry in The domain wall dynamics after reheating will depend on the details of the cosmological It may be a curious coincidence that in a simple axion model the low-tension constraint Relaxing our assumption of employing the tunneling wave function and picking 15 16 is classically stable and leads to more eternalcoincides inflation, with a or naiveconstant the and formulation equation we of assumed of the a state weak large instanton gravity changes action, conjecture give a divergent nucleation rate in the semiclassical approximation. We do not discuss this case further. is given in ( density of the domainenergy wall density is inside set the cosmology, approximatelyand the the by domain domain the wall wall same tension isplace scale in in only the as guaranteed Planck new to units the vacuum expand is initial if negligible vacuum an extended period of inflation takes drive eternal inflation, orof result direct in domain a walls viabletransition to late inside time other the cosmology. decontaining de Sitter Because the Sitter of runaway vacua phase phase the witha absence lower vacuum energy transition to density, a any different vacuum de Sitter vacuum The tunneling processcausally disconnected gives from most risemostly of to undisturbed. the original a spacetime. closed The runaway FRW phase cosmology in vacuum wall from the initialFGG phase process, and thetransition nucleation rate rate is is given suppressedenergy by by vacua ( the are horizon exponentially area favored, of the new phase, so transitions to high approximation to vacuum transition rates. a phase or the decompactified runawayfalse phase. vacuum bubble Gravitational effects containing stabilize the the de nucleation Sitter of vacuum a cosmological evolution after reheating is beyond the scope5.2 of this work Cosmology [ inFinally, the we landscape are in awhen position both to speculate our aboutdomain assumptions the walls about cosmological between evolution fundamental de in Sitter physics a vacua, are landscape and met: the tunneling there wave exist function no provides a direct good prohibited, the only wayperiod to of remove inflation. the domain wall from aevolution, late so time even though cosmology this may is notthat seem via halts likely, there a domain could be wall some collapse non-trivial evolution within the cosmological horizon. A detailed study of the interactions JHEP12(2016)155 (6.1) . Demanding that the 1 − Late H . Inf Late H H that allows for galaxy formation. The latter – 33 – log 1 2 Late H & e N ), and is sufficient to create a surprisingly flat and isotropic 5.2 . For low domain wall tensions the nucleation rate is suppressed 12 Our results point towards framework to pursue the generation of inflationary initial In this work we employed two well motivated assumptions about fundamental physics in theories of quantum gravitymake are predictions, important of to further inflationary our parameters understanding, in and the ultimately landscape. conditions in flux compactificationspactification and of string its theory explicit is realization an in important a problem for well the controlled future. com- extended period of cosmic inflationfunction to approach stabilize vacuum to decay, and transitionof that rates these the exponentially tunneling effects wave provide favors potentiallyously high much assumed inflationary stronger measures scales. selection in effects theoryof Both than efolds space, some that such stems of as from the atial. the previ- polynomial requirement Therefore, of suppression small a in slow the detailed roll number parameters understanding in of a moduli random poten- stabilization and vacuum transitions This lower bound on theall inflationary observable expansion modes guarantees overlapping and past results light in cones an for isotropic andand flat arrived universe. at surprisinglyscape. strong We implications saw for that generic the instabilities cosmological of evolution weakly in coupled a string theory land- vacua require an place to permanently exilecosmological the phase shell is causally from protected a froma late a lower possible time bound re-collapse horizon of on the the domain number wall of gives efolds of inflation, Hubble scales. Theseentropy states transitions that can are benew subsequently cosmology interpreted frozen is as by protected small,the from the horizon domain local formation of an wall of fluctuations observer, collapse which aThe to imposes as domain Hubble a long low wall severe horizon. can as constraint re-enter on the the the The shell cosmological horizon evolution. unless does a not sufficient enter amount of expansion has taken the instability of domain walls,the there exist landscape vacuum is transitions between populatedtransitions de by Sitter are quantum vacua mediated and tunneling. viais a In illustrated double the in bubble weak figure configuration gravityby that limit the contains the a de leading wormhole Sitter and horizon area of the nucleated phase, favoring transitions towards high We considered thin wallbetween vacuum metastable transitions de in Sitter the vacua,regions absence and allowing a of only runaway phase domain for with walls vanishing domainby vacuum interpolating energy walls the density. between Dine-Seiberg This the problem setup in is de motivated weakly Sitter coupled compactifications of string theory. Despite case is safe fromhave domain overlapping wall past collapse light-cones. ifinflationary expansion all This set observable requirement by ( modes translates fromuniverse to for the a late time minimum time of amount observers reheating of to occupy. 6 Conclusions and relaxes to a lower Hubble scale JHEP12(2016)155 ]. (A.1) (A.2) (A.3) (A.4) (A.5) 53 , , 47 , 2 . ) ) 0 , 0 46 0 ) , R R r L r r N N L N − − − Gravity r ,, ˙ ˙ + ( 2 R R ) 0 H ( L L )( r R t π LR L t N N ( 2 − LR t 0 ( R − ∂ t t ∂ ) + 0 − t 2 GLN 0 0 2 ) GN L N R RR + Gravity 2 t − − LR R H r ) 2 π t 0 t L N R R ( N ( r G t 1 2 − GN L ) can be written in terms of the variables N are denoted by primes and dots, respectively. = N ˙ t , − + R R − ˙ 3.3 + L R R – 34 – ˙ 0 = π R π and ( ˙ R 0 L 0 L . r + ) ) and π ] R , π 0 ˙ L L R ˙ , ˙ R G t R L LR R 0 47 [ r Lπ π N t − R − ( r N − 0 17 ( 2 2 L L 2 t N R R GN r 2 R π + N 0 ) as + 2 drdt N GLπ R L 3.2 π Z = = = t R L = drdt π GN Z − Gravity Gravity r t G 1 = Gravity H H 2 S ˙ R = Gravity S The dynamical terms in the action ( Again, partial derivatives with respect to 17 It remains to write thewe matter consider contributions in a terms matter of canonical energy variables. momentum For tensor simplicity that originates from a static scalar field We can also solve for the velocities where and the Hamiltonian density is given by which can be written in terms of canonical variables as the details omitted in the main text. We willappearing mostly in follow the the metric references ( [ The results in this workgravitational configuration. are derived Even from though the thethe relevant first-order literature, action action a has of number appeared of a severalcrucially signs spherically times relies are in symmetric inconsistent on in the some accuracy of of the references. the relative Since our signs work in the action, we now fill in some of ical Physics. Research atthrough Perimeter the Institute Department is supported ofProvince by of Innovation, the Ontario Science through Government and of the Economic Canada Ministry of Development Research, and Innovation by andA Science. the First-order action of domain walls I would likeMatthew to Kleban, thank Liam Adam McAllister,der Brown, Ruben Vilenkin, Kate Monten, Erick Henry Eckerle, Weinberg,This Tye, Frederik work Timm Eve was Denef, Vavagiakis, Wrase supported Alexan- and Matthew byDE-SC0011941. the Claire Johnson, U.S. This Zukowski Department research of for Energy was useful (DOE) supported under discussions. in under part grant no. by Perimeter Institute for Theoret- Acknowledgments JHEP12(2016)155 , 2 α , (A.8) (A.9) (A.6) (A.7) ) . m (A.10) (A.11) ) + r ( Shell r,α 2 2 α ρ ˆ 2 p H L r r N , ) ν πLR α ]. In the present + ˆ r 2 r ) dx 53 r − H = 4 µ α , . r ) ˆ r Shell t,α ρ t N α ( dx 50 N ˙ [ ˆ r δ H H + t µν 1 Shell r,α − + ˆ g α , − N =1 t ˙ ˆ r H r ( α X α − N α ( 1 ADM H ˆ ! p 2 α t N ) − )ˆ p =1 ( ˆ L α 2 X α α N N 2 α ) + ˆ r ˆ r ˆ dtdr r dtM R ˆ L − . ( − = α − α − 2 R ρ Z t ˙ σ α 2 r R r m 1 ˆ − ( ( N − R δ − δ =1 π = 0, the second derivative terms are Matter t,r α α X α N q ˙ − Matter r ˆ r πLR σ H + r α H Z = = ˆ ˙ p =1 L N + N + 4 π α X α , L ˆ 4 p dt π L Shell r,α →∞ − ) + Z r − Shell t,α H – 35 – ) r 1 Gravity t,r r ( 2 0 H − ( =1 ρ L H 1 R . ρ X α N dtdr

2 − =1 α G = 2 − X + α R p N Z 2 t 0 )ˆ , ρ t α t,r + gR + 2 α 0 ˆ r H ρ t t H √ α m L ˙ ˆ − r RR H = 1 and lim 2 α r + ˆ p ( t = dtdrLN dtdr 2 δ 2 α ) contains second derivatives. Restricting to asymptotically flat ˆ 1 p N L + dt dtdr N Z Z − =1 R r X α π π N Z A.9 , ) Z π 4 4 Matter →∞ t 1 α 2 α r L − ˆ − − − − r =1 H ˆ π R 0 X α L N α − = = = R G r = Lπ m πσ ( − δ S L − 2 L g 2 = = 4 R √ R π α 0 2 4 GLπ R m Shell t,α dx = = H t r Z H H Note that the actionsolutions, ( where lim precisely canceled by including acase non-dynamical term we only considerterm is solutions irrelevant. with constant asymptotic mass and therefore the ADM where the total Hamiltonian density is given by The Hamiltonian densities in full are given by Combing the above expressions, we can write the action as Let’s also define a matter Hamiltonianenergy density that density includes and the contribution the from domain a vacuum wall tension, and we defined energy density. We then have the matter action theory with thin domain walls, such that the domain wall tension equals the domain wall JHEP12(2016)155 , (A.17) (A.14) (A.15) (A.16) (A.12) (A.13)   ], . +1 α 47 ˆ α A ˆ A α s A 2 +1 α L away from the shells ˆ α A α ˆ − A 2 α L . A 2 2 α ˆ 0 L ˆ . √ L ! R − L ) − √ + − α + 2 δR r − 0 α, 2 R 0 0 α, =ˆ ˆ R ! r π ˆ . This gives

R R . 0 L q + + π q α , α r ! ∂S − is implicit. Collecting all the terms = 0 and vary ∂R =ˆ α − = at the location of the shell to ensure log δL r − α 0 .

0 L A Shell − + ˆ +1 0 0 α, L R π S R α 0 α, δ ˆ ( − R 1 − ˆ ˆ ∂S A − R = 0 ∂R α 2 − =1 = 2 ) 0 r α δS/δL r X α dr N R   L =ˆ α R − A r ˆ H r

+ 2 Z δ 0 r − – 36 – log L = and δRπ α π + = r π t ∂S ) in the main text, ∂R − R G η α =ˆ + Space H Rη r ˆ π 2 r G α

L 3.5 δR S δ 0 ˆ 0 R R = α ˆ R ˆ = δL p α ∂S δ p ∂R δLπ 1 ˆ R S ( − =0 =1

= d L L X α 0 N π =0 π ˆ L L , α R Z δS/δR Z G = π Rη Z − + dr Shell α α dS ) and the argument of ˙ ˙ dr dr ˆ ˆ r r + α δS α α 1 − r ˆ ˆ + + p p (ˆ α − 1 1 − − ˆ r α ˆ α α r − − dt dt +1 ˆ ˆ r r α α Z α ˆ ˆ r r ˆ Z Z Z Z A is discontinuous at the location of the shells. Considering an arbitrary varia- = = =1 =1 =1 N N N ≡ 0 X X X α α α , α R +1 = = = , the action receives an additional contribution at the location of shells, α ˆ R Shell A S Space Finally, we turn to the question relevant for quantum tunneling, where we need to S A.1 The junction conditions The action does nothave contain the a Hamiltonian kinetic constraints term for the shift and lapse functions, we therefore where then gives the full action quoted in ( We need to subtractthat derivatives the with full respect action to satisfies Note that tion of To integrate this expression weto keep satisfy the constraints. This gives the following contribution to the action [ evaluate the action for anthe arbitrary tunneling shell event. trajectory. The This variation shell of trajectory the will action parametrize is given by JHEP12(2016)155 (A.22) (A.23) (A.24) (A.18) (A.19) (A.20) (A.21) and find 0 ]. M 57 , , , ) )[ , − Shell r,α α − H 3.13 0 α, A L,α, α ˆ . ˆ π R − ˆ ) L L,α r − , ˆ − α 2 π 2 0 ( ˆ + ρ R G 3 and L + R 2 0 / L,α 1 0 α, α π − ˆ =1 L,α, 0 R 2 2 X α π N L R (ˆ R G = . dR R α + πGρ = ˆ α L = Z ··· − = 8 π R Shell t,α + R,α 4 GM 2 α α = 0 H 2 H − 0 α α R 0 ··· − ˆ that do not contribute to the integral and ˆ , π L GM R 2 L 2 α 2 r α 1 + 2 0 RR , we can evaluate the derivative − H =1 L A 0 L R + 2 X α α N , 2 α − – 37 – R − α + r 2 H 2 dr 0 1 0 − 2 L dr Lπ R R + M − α G + r ˆ − r = 1 α R − 2 α ˆ r ˆ r 1 + α R G α Z = ˆ r π + 1) constraint equations that fix the classical dynamics of A r Z − α α η 2 ˆ L ˆ L − 2 R H − 2 0 R = L 2 L R Gπ 1) matching conditions by integrating the Hamiltonian con- N = = π can be interpreted as the asymptotic mass parameter. For con- RL α 2 α L,α − = ˆ p 2 2 ˆ π L + ], consider the following constraint M 2 α t R G N M H m 53 , 0 = L + R 47 2 α 2 L,α ˆ p π 1, we defined the Hubble constants ) over the shell located at ˆ 0 = q α G ˆ = R A.11 π − η , respectively. These are precisely the Israel junction conditions ( − / In this appendixHawking we provide and some Kruskal-Szekeres supplementaldynamics coordinates material and of on the bubble thevacuum constraint bubble. definition with equations of governing a Gibbons- the Schwarzschild interior and a de Sitter exterior, a true where the dots represent termsquantities continuous evaluated in on the inside/outside+ of the shell are denoted by a hat and the index B Classical domain wall dynamics We now findstraints the ( 2( or where stant energy densities withinthe the momentum domains where In a static spacetime, the system by considering ashell. linear Following combination [ of these constraints at the location at each We obtain the required 2( JHEP12(2016)155 . , , , (B.5) (B.2) (B.3) (B.4) (B.1) t t GM GM t t 4 4 GM GM 4 4 . , ) ) , cosh sinh , ) ) sinh cosh Ht Ht , . GM GM Ht Ht . | | 4 4 GM GM , 6 V V r/ r/ 4 4 | | 2 2 e e sinh ( cosh ( r/ r/ , 1 Ω e e < < 1 sinh ( cosh ( 2 2 d 1 | | − r 2 1 Ω Hr Hr Hr − GM r U U d − r 2 | | Hr Hr Hr . 2 − GM − r 2 r , , 1 1 + Hr 1 + − − GM ) + 0 0 r 2 2 1 GM 1 1 + Hr 1 + − GM r r 2 ) + 2 1 2 r r . r/ r r − − dU e V > V < − r r − + = = = = dU Hr Hr = = = = 1 + I II III IV ], we introduce Kruskal-Szekeres (KS) co- II: dV GH GH GH GH − − 2 I II IV VI : III KS KS KS KS 1 + 1 − 56 ( dV , = r ,V ,V − | GM ,V 2 GM ,V ( – 38 – ,V 2 , ,V 2 ) U ) 2 ,V | | ,V V r/ ) ) U ) in domains with de Sitter metric. In the following Ht − | − t < Ht t = e GM | GM 2 Ht t Ht 2 t 3 GH 4 < Hr 4 GM ) V GM U | | H V 4 4 , φ cosh ( sinh ( V | − 1 + r 1 cosh ( sinh ( GM GH 2 , , cosh sinh 1 Hr Hr Hr ) in domains with Schwarzschild metric and Gibbons-Hawking 0 0 U − , θ 32( = sinh cosh Hr Hr Hr − − KS 2 GH GM GM = 1 + 1 1 + Hr − 4 4 , φ U > U < GM GM ds 2 1 1 + Hr 1 + ,U r/ r/ 4 4 r r e e KS ds r/ r/ r r − − 1 I: GH e e , θ V 1 III: − r = = = = GM KS − r 2 GM I II III IV GH GH GH GH ,U 2 r − GM U U U U r 2 KS 1 − GM V 2 1 r r is given by − − r r r is defined by = = = = r III IV I II KS KS KS KS U U U U where The coordinate systems for both spacetimes are illustrated in figure The de Sitter metric becomes and Similarly, we have for the Gibbons-Hawking coordinates The Schwarzschild metric in Kruskal-Szekeres coordinates is given by The coordinates for Schwarzschild regions are defined as The KS and GH coordinates areall defined of piecewise in de four Sitter regions and that, taken Schwarzschild together, space, cover respectively. These regions are defined as follows, In order to illustratenates the that dynamics are of smooth domain everywhere.ordinates walls ( Following that [ cross horizons(GH) we coordinates require ( coordi- we drop the subscripts GH and KS since the distinction will be obvious from the context. B.1 Gibbons-Hawking and Kruskal-Szekeres coordinates JHEP12(2016)155 z − → (B.8) (B.9) (B.6) (B.7) (B.10) . , 4 / 3 . 2 | / 8 γ + 3 E / | 4 H − + 2 3 / 1 2 2 4 2 4 γ γ κ ) s 4 S + + 2 = 2 GM 4 γ 4 γ =0 S (2 A − 2 + 2 + ˆ ˆ r , r . = , z q q γ | γ r < r > E 3 2 ,E | ) with the replacement of indices 2 + 2 2 − for for H H ρ . = 3.32 3 4 ) = , , − + =0 2 2 2 dS κ κ max R ]. However, their result for the tunneling rate, 2 R A z – 39 – 2 GM pl ( 2 29 2 H σ = 2 M , z − − 2 ,V 3 (by exchanging the definition of inside and outside), z 2 | , γ ) with . The full dynamics can be read off from the effective γz = 1 = 1 E r γ | 4 γ 3.3 3.1 B A H − − p 4 . All possible bound and unbound solutions are summarized A A 2 + 2 2 17 − 2 + = = r 2 const. + 0 + 2 when the domain wall tension in Planck units is small compared + ), ˆ → R . The effective potential has a maximum at 4 γ ,H , and the position of the horizons in terms of the new radial coordinate 0 r 3 , ≈ − 3.31 ˆ = 16 R ˆ ] only applies for domain wall extrinsic curvatures that do not change sign R 2 γ 1 z A | 29 3 max 2 − + E z | 3 H GM z p 2 . = 2 = 0 z − ˆ We can write For the case of a Schwarzschild/de Sitter domain wall the rescaled radial coordinate R as C Alternate expression forColeman the and tunneling rate deapproximation Luccia and computed including gravity the in(3.12, vacuum 3.13) [ in decay [ rate in the leading semiclassical An example of an unboundordinates, trajectory, is as shown in seen figure bothin in table the Schwarzschild and de Sitter co- + and coordinates potential in figure z As expected, these expressions are identical to ( Remember that to the Hubble scale, but now the metric is given by ( is again given by ( In this subsection weSchwarzschild give space the that equations isdynamics governing separated are the identical by to dynamics the a dynamics ofbackground, of thin a discussed a wall bubble in of spherical from de section patch Sitter awe phase of de provide within a the Sitter Schwarzschild dynamics phase. for the While reverse the situation for quick reference. B.2 Schwarzschild/de Sitter domain wall dynamcis JHEP12(2016)155 3 7 4 (left)

=0 =0 2

S A =0 κ < H 3 0 dS A >

) 1 (z ′ − V , for R z κ < H 0 , < 0 2 z ′ + max max max max R 1 < V < V − =0 =0 1 0 0 − + < E < V < E < V

ˆ ˆ

R R 0 =0

2 < E R < =0 − =0 0 0 − + ′ − ˆ ˆ R max R R E < V V V V E < V 0 3 − 0 2 4 6 8 2 2 1 0 1 − − − − − − V 3 4 – 40 – =0 2

=0 S

=0 3 A 0 ) as a function of rescaled radial coordinate 0 dS z > z)

A (

1 > ( =0

′

+ =0 V R V ′ − R R R 2 z 0 1 − 1 0 0 < 2 < ′ + − (right) for a Schwarzschild/de Sitter domain wall. The situation is equivalent to ﬁgure Unbound IV - III - II I Unbound IV - I - II I Unbound I - II IV - I Bound I IV - I - II Bound I IV - III - II Type de Sitter Schwarzschild Conditions ′ − R . Eﬀective potential . Illustration of the unbound trajectory of a Schwarzschild/de Sitter domain wall that tra- R . Summary of all possible trajectories of a Schwarzschild/de Sitter domain wall. We show 0 3 0 2 4 6 8 − κ > H − − − − 2 1 0 2 1 − − V Table 2 the regions of the corresponding conformal diagrams that are traversed by the trajectories. Figure 17 verses the regions IV-III-II indiagram. the de The Sitter spacetime diagram diagram and is is contained not in applicable region in I of the the faded Schwarzschild region. and after a coordinate change that reverses the inside and outside regions. Figure 16 JHEP12(2016)155 ) ] Ω ) is 3.41 (C.3) (C.1) (C.2) ! (1998) C.3 1 − 116 2 ]. / ]. Curiously, 3 ) for arbitrary i 2 29 ˆ R , 3.41 Measurements of SPIRE A astro-ph/9812133 ! IN [ ]. Astron. J. ) . The result ( ][ , 0 + 2 A πGρ / , ˆ ρ R 8 3 1 ) vanishes. For positive 100 2 , − ] − 2 − 3) C.1 61 Observational evidence from . 1 Θ( / ) set the sign of the extrinsic ) h (1999) 565 2Λ) A 2 ) − / 0 0 + 0 A F B 3.15 ˆ ˆ , ρ R 517 R πGρ πGσ A 6 N sgn( ∓ − arXiv:1502.01589 [1 + ( B [ − I ρ ] is identical to our general results ( , = 1 A − 61 2 − N ) Planck 2015 results. XIII. Cosmological A ], 2 2 Astrophys. J. ρ / + – 41 – , 3 29 , and Λ = (8 i F A 4 collaboration, S. Perlmutter et al., 2 , πGσ (2016) A13 σ ˆ ]. B R 2 B σ/ρ N π + 6 B ) = sgn( 594 − A ρ 27 0 B ρ = 3 SPIRE collaboration, A.G. Riess et al., I πGρ ˆ ), which permits any use, distribution and reproduction in ( R ρ , 8 3 IN 0 B A ˆ ρ ][ R − ) agrees with the result obtained by Coleman and de Luccia. N 2 sgn( 1

), h ) and the junction conditions ( ). Our result is valid for arbitrary domain wall curvatures, but = 2 C.1 3 A ) 3 ρ G 0 − B 8 ˆ (2 R 3.39 3.41 CC-BY 4.0 / + 4 This article is distributed under the terms of the Creative Commons σ σ sgn( Astron. Astrophys. 2 3 , π ˆ

R collaboration, P.A.R. Ade et al., 2 ]. 2 π from 42 high redshift supernovae G 3 astro-ph/9805201 = 27 [ Λ 16 is given in ( + 2 ) even though their calculation, again, only applies in the small tension regime. 0 SPIRE ˆ = B R IN Supernova Search Team supernovae for an accelerating universe1009 and a cosmological constant Planck parameters Supernova Cosmology Project and [ C.1 B We derived the transition probability using a Hamiltonian framework and obtained the [2] [3] [1] References a sign “error”. Similarly,and the ( result given in [ Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. where identical to (3.16) —Coleman and but de at Luccia largeand found small tensions with tension regimes, their disagrees although (3.16) they with the considered (3.13) only correct the — answer latter scenario for of and the then [ both made the large domain wall curvatures ( In the limit of vanishingdomain energy wall density curvatures inside the bubble, we find with ( where curvature, In the absence of disconnected spacetimes the last term in ( Their result was rewritten by Parke in a moretunneling compact exponent form ( [ we wrote the expression inin a a more way compact that form. closely We resembles now (3.13) express of the [ tunneling exponent across the wall and transitions that do not contain any causally disconnected spacetimes. JHEP12(2016)155 ]. , 49 , D , ]. , Phys. Phys. Phys. , , SPIRE , (1987) IN B 108 (1985) 299 SPIRE ][ ]. IN 59 ][ Phys. Rev. ]. , Astrophys. J. 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