Cosmic bubble collisions by matthew kleban

be found I will use natural units (8πG = c = h = 1) throughout Anotherreview of cosmic bubble collisions is [1]

11 Overview

First-order phase transitions are ubiquitous in physics During a first-ordertransition a meta-stable phase (the ldquofalserdquo vacuum) decays to a lower energyphase (the ldquotruerdquo vacuum which may itself be either metastable or trulystable) either by quantum tunneling or because the decay is stimulated bysome external influence The transition begins in a finite regionmdasha bubbleIf the bubble is sufficiently large when it forms it expands into the falsevacuum and collides with other bubbles Therefore in a static spacetimethe transition will eventually percolate and the false vacuum will disappearentirely

The situation is different when gravity is included The energy densityin a meta-stable phase does not change with the expansion of space it isa cosmological ldquoconstantrdquo Therefore if the false vacuum has positive en-ergy it will undergo cosmic inflationmdashthe volume of space containing it willexpand exponentially with time If the timescale for the exponential (thefalse-vacuum Hubble rate) is faster than the rate of bubble formation thefalse vacuum will continually reproduce itself and the transition will neverpercolate Bubbles of true vacuum will expand (but slower than the falsevacuum) and occasionally collide with each other (Fig 1) Contained insideeach bubble is an expanding Friedmann-Robertson-Walker (FRW) cosmol-ogy that is homogeneous and isotropic apart from random perturbations andthe aftereffects of collisions

In models consistent with current observations our observable universeis inside such a bubble embedded in and expanding into an eternally inflat-ing parent false vacuum Observational constraints require that our bubbleunderwent its own period of slow-roll inflation after its formation Colli-sions with other bubbles that nucleate in the (otherwise eternal) parent falsevacuum nearby occur with a non-zero probability per unit time and aretherefore guaranteed to happen eventually Their effects have already or willperturb the universe around us creating potentially observable signals Interms of the FRW coordinates describing the cosmology inside our bubblethese collisions occur before slow-roll inflationmdashin fact they occur before the(apparent) big bang of our FRW universe They can be regarded as creatinga special and predictable set of initial conditions at FRW time t = 0

2

Figure 1 A numerical simulation showing the spatial distribution of bubblesat a late time in an eternally inflating false vacuum Bubbles that appearedearlier expanded for longer and are larger but the physical volume of thefalse vacuum is larger at later times so there are more smaller bubblesEventually each bubble collides with an infinite number of others The late-time distribution is a scale-invariant fractal

3

12 Effective field theory coupled to gravity

Any model of spacetime fields coupled to gravity can give rise to bubblecollisions if there exist at least two meta-stable phases of the field theoryAfter forming bubbles expand and collide with each other The goal of thisreview is to describe the physics of the formation expansion and collision ofthese bubbles I will focus exclusively on models in which at least one of thephases (the false vacuum) has a positive vacuum energy

A region of spacetime filled with positive vacuum energy has a metric

ds2 = minusdt2 + a(t)2d~x2 (1)

and obeys Einsteinrsquos equations

(aa)2 = H2f = Vf3 (2)

where Vf is the energy density of the false vacuum and Hf is the associatedHubble constant The solution to (2) is de Sitter space a(t) = eHf t Regionsof space that are dominated by vacuum energy but contaminated by otherforms of matter or energy will exponentially rapidly inflate away the con-taminants and approach the metric (1) Regions not dominated by vacuumenergy will either expand more slowly or collapse into black holes which formany purposes is taken as justification for ignoring them after a few falsevacuum Hubble times (where the Hubble time is tH equiv 1H)

13 Decay

In metastable de Sitter space there is a dimensionless rate of bubble forma-tion γ When γ is small it can be defined as the expected number of bubblenucleations per unit Hubble time per unit Hubble volume that is the di-mensionful decay rate is Γ = H4

fγ Generally γ is the exponential of minusSwhere S is the action for an instanton Hence when S 1 γ is very smalland the rate of bubble nucleations is slow When γ gtsim 1 the semi-classicalmethods reviewed here are not adequate to describe the physics

When γ is small and the meta-stable phase has positive vacuum energythe exponential expansion means that the transition will never percolatemdashthere will always be some regions in which the unstable phase remains Theintuitive reason is simple in one Hubble time the de Sitter region increasesits volume by a factor of e3 In that same Hubble time one expects γ bubbles

4

of another vacuum each of volume less than the Hubble volume to appearTherefore if γ 1 a typical region will double in size many times beforeproducing a bubble The same applies to each ldquochildrdquo region and thereforethe meta-stable phase never ceases to exist This is known as ldquofalse vacuumeternal inflationrdquo (to distinguish it from slow-roll eternal inflation whichcan take place in an models with sufficiently flat positive potential energyfunctionals)

In the end the picture is an exponentially rapidly expanding spacetimein which bubbles of more slowly expanding phases occasionally appear andoccasionally collide It is important to note that this is a generic predictionof any model with multiple positive energy minimamdashwhile it seems to bea prediction of the string theory landscape it is certainly not unique to itNevertheless an observation that confirmed this model would be a confir-mation of a prediction of string theory and an observation that ruled it outwould be at least potentially a falsification

14 Motivation

In the last few years there has been a surge of interest in this problem Thereason is that string theory predicts the existence of many meta-stable min-ima the so-called ldquostring landscaperdquo [2 3] In string theory the geometryand topology of spacetime is dynamical String theories exist in 9 spatial di-mensions Since we observe only three spatial dimensions in string solutionsthat might describe our world six of the spatial dimensions are compactifiedthat is they form a geometry with finite volume while the other three spatialdimensions and time can form Minkowski or de Sitter space

Six dimensional manifolds have many parameters that describe theirshape and size These parameters are dynamical fields in string theory andtheir meta-stable solutions correspond to distinct possibilities for the shapeand size of the manifold Because they are meta-stable small fluctuationsaround these geometries behave like massive particles from the point of viewof the 4 large spacetime dimensions Therefore the low-energy physics asmeasured by a 4D observer is an effective field theory coupled to gravitywith the field content partially determined by which configuration the com-pact manifold is in

The compact manifold can make transitions from one meta-stable config-uration to another Among the parameters that can vary from configurationto configuration is the value of the vacuum energy In non-supersymmetric

5

solutions it will not be zero instead its typical value is set by the stringenergy scalemdashwith both positive and negative values possible [2 4 3] Theexact value in any given phase will depend on the details and if there areenough different phases it is plausible that a small but non-empty subsethave vacuum energies that are consistent with the tiny value we observe[5 6 7] Hence the theory is at least consistent with observation and theextraordinarily small but non-zero value was even predicted a decade in ad-vance in [8] under a set of assumptions that coincide with those just outlined

Because understanding the value of the cosmological constant is consid-ered one of the most important problems in theoretical physics and becausestring theory is our best candidate for a theory of quantum gravity it is worthtaking this scenario seriously One should look for observations beyond thecosmological constant which could test its validity hence this review

2 Earlier work

Alan Guthrsquos original model for inflationmdashnow known as ldquoold inflationrdquomdashwasan inflating false vacuum punctuated by bubbles [9] Inflation was the falsevacuum and the end of inflation was the nucleation of the bubble we inhabitThis model didnrsquot succeed because of difficulties with re-heating the universewhen a bubble first appears it is both strongly negatively curved and emptyReheating could potentially result from collisions with other bubbles butsuch collisions are either raremdashin which case reheating is anisotropicmdashorcommonmdashin which case the phase transition from the false vacuum completesafter order one Hubble time and there is little or no inflation Early studiesof this include [10 11]

21 Open inflaton

Old inflation was replaced by other models in which the inflaton slowly rollsand less attention was paid to models involving bubbles until the mid-90swhen the available data indicated that the universe was underdense and neg-atively curved At that time there was a burst of interest in ldquoopen inflationrdquomodels models in which our universe is inside a bubble that nucleated froma parent false vacuum (and hence is negatively curved) but where the bub-ble nucleation was followed by a period of standard slow-roll inflation Thatperiod of slow-roll produces density perturbations and reduces the curva-

6

ture thereby allowing Ωtotal sim 3 as the data seemed to indicate Therewere a series of papers by several groups of authors that worked out thepower spectrum of perturbations generated during inflation in such bubbles[12 13 14 15 16])

While not all authors agreed on the quantitative effects all agree thatthe power spectrum of inflationary perturbations is significantly modified onscales of order the radius of curvature and larger This can be understoodqualitatively in a simple way As I will review below when a bubble universefirst appears it is dominated by negative curvature As a result it expandswith a scale factor a(t) sim t After a time of orderHminus1i whereHi is the Hubbleconstant during inflation the curvature redshifts to the point it is belowthe inflationary vacuum energy and inflation begins Clearly perturbationsproduced before or during this transitional era will not have a scale-invariantspectrum identical to those produced after it

Mainly because of theoretical developments in string theory (see Sec 14)in the last few years there has been a resurgence of interest in these modelsfocussing both on the physics of individual bubble universes and on collisionsbetween bubbles [17 18 19 20 21 22 23 24 25 26 27 28 29 30 31]I will review much of that very recent work below A result from someyears back is [17 18] which demonstrate in several models (using argumentsparallel to [8]) that slow-roll inflation after the bubble nucleation is necessaryin order to avoid producing a universe devoid of structure With fixed δρρapproximately as many efolds of inflation N are needed as are necessary tosolve the flatness problem (approximately N sim 60 given typical values forthe reheating temperature etc)

22 Metrics and solutions

The single most important tool in understanding the physics of cosmic bub-bles is their symmetry The standard treatment of bubble formation in fieldtheory coupled to gravity was developed by [32] In that approach both therate of nucleation and the initial conditions just after the bubble forms areset by a solution to the equations of motion of the Euclidean version of thetheory Such solutions are known generally as ldquoinstantonsrdquo The justificationfor their application to this problem goes far beyond the scope of this reviewsee [32] and references therein

Another mechanism by which false vacua can decay is charged membranenucleation in a background 4-form flux [33] These instantons (and the ex-

7

panding bubbles they correspond to) possess the same symmetries as thoseof [32] and I expect most of the results reviewed here to hold for them aswell

In order to understand the symmetries of the bubble spacetime one canstart with the instanton solution of [32] In a model with a single scalar fieldφ coupled to Euclidean Einstein gravity the relevant solution is

ds2 = dψ2 + a(ψ)2dΩ23 φ = φ(ψ) (3)

where dΩ23 = dθ2 + sin2 θdΩ2

2 is the round metric on a 3-dimensional sphereThe functions a(ψ) and φ(ψ) are determined by solving Einsteinrsquos equationsand the equation of motion for the scalar field φ and the solutions depend onthe potential energy for the scalar V (φ) If V has at least two minima solu-tions φ(ψ) that oscillate once around the maximum1 of V (φ) that separatesthe two minima generally exist (but see [34] and [35])

Solutions of this form have a large degree of rotational symmetry they areinvariant under the group of rotations in 4D Euclidean space (in which onecan embed the 3-sphere) namely SO(4) This is a group with 4 times 32 = 6symmetry generators (each corresponding to a rotation in one of the sixplanes of 4D space) The action S(a φ) for a solution is determined by anintegral involving a(ψ) and φ(ψ) and is related to the false vacuum decayrate by γ sim eminusS

Under certain conditions on the potential V (φ) the solutions a φ areldquothin wallrdquo meaning that they vary sharply with ψ In particular in thethin wall limit φ(ψ) sim φ0 + microΘ(ψ minus ψ0) where Θ(ψ) is a step functionand φ0 micro and ψ0 are constants The thin wall limit is convenient it makesit possible to calculate many quantities in closed form However it is notrequired for the validity of most of the physics discussed in this review nordo the symmetries of the solutions discussed here depend on it

To determine the spacetime after the bubble nucleation occurs one cansimply analytically continue (3) back to Lorentzian signature The resultis a spacetime that contains a bubble expanding in a bath of false vacuumThere is more than one choice of analytic continuation different choices givemetrics that cover some part of the full Lorentzian spacetime These metricscan be patched together to determine the global structure of the spacetimeOf most relevance for the cosmology seen by observers inside the bubble isthe continuation that produces a metric that covers (most of) the interior of

1In Euclidean signature the equations of motion have the sign of the potential reversed

8

the bubble Details can be found in eg [18] the result is

ds2 = minusdt2+a(t)2dH23 = minusdt2+a(t)2

(dρ2 + sinh2 ρdΩ2

2

)and φ = φ(t) (4)

where H3 is 3D hyperbolic space (the homogeneous and isotropic 3D spacewith constant negative curvature) As can be seen at a glance this is anFRW metric describing a homogeneous and isotropic cosmology scale factora(t) and scalar field ldquomatterrdquo φ(t) The evolution of a will be determined byV (φ(t)) in the usual way as well as by any other sources of stress-energy

After the continuation the SO(4) invariance of the Euclidean solutionhas been replaced by the SO(3 1) invariance associated with H3 In otherwords the symmetries of the instanton are what give rise to the homogeneityand isotropy of the constant-time surfaces of this FRW cosmology Observersliving inside the bubble live in a negatively curved universe that satisfies thecosmological principle

One of the most intriguing aspects of this result is that the ldquobig bangrdquot = 0 where the scale factor a(0) = 0 vanishes is not a curvature singularityIt is merely a coordinate singularitymdashthe spacetime just outside the surfacet = 0 (which is null) is regular and described by another analytic continuationof the metric Of course there is a quantum nucleation event (a finite distanceoutside the surface t = 0) that may make the description there in terms ofa classical spacetime plus field configuration problematic but so long as theenergy densities involved are sub-Planckian even the region outside the ldquobigbangrdquo surface t = 0 is no more singular than the nucleation of a bubble in afirst-order phase transition in field theory

23 Bubblology

The cosmology inside the bubble in the region described by (4) is fairly simpleto describe The scale factor obeys the Friedmann equation

H2 = (aa)2 = ρ3 + 1a2 (5)

where ρ sim V (φ) + (φ)22 + is the energy density in the scalar plus thatof any other matter or radiation At t = 0 the initial conditions are suchthat φ = 0 As a result the dominant term on the right hand side is thecurvature term 1a2 and one obtains a(t) = t+O(t3) [18]

To produce a cosmology consistent with observations there must be aperiod of slow-roll inflation that begins at some time t gt 0 [18] The most

9

00 02 04 06 08 10Φ

02

04

06

08

10VHΦLM4

Figure 2 A scalar field potential for an open inflation model where a singlefield φ tunnels through a barrier and then drives slow-roll inflation inside theresulting bubble

economical way to accomplish this is to assume that the field that tunneledto produce the bubble in the first place is also the inflaton and that itspotential V (φ) has a slow-roll ldquoplateaurdquo that the field evolves into after thetunneling (see Fig 2)

In such a model inflation begins at a time ti such that 1a(ti)2 sim 1t2i sim

V (φ(ti)) sim H2i Therefore inflation begins at t = ti sim 1Hi From that

time on the scale factor will begin to grow exponentially rapidly diluting thecurvature and (given enough efolds) solving the curvature problem

24 Curvature and fine-tuning

During inflation the radius of spatial curvature of the universe grows expo-nentially In terms of the total energy density Ω the curvature contributionto the Friedmann equation is defined by Ωk equiv 1 minus Ω = minusk(Ha) where

10

k = minus1 for negative curvature As described above Ωk ltsim 1 at the start ofslow-roll inflation Since during inflation H is approximately constant and agrows exponentially by the end of inflation after N efolds Ωk sim eminus2N

After inflation Ωk grows by a very large factor e2Nlowast so that its valuetoday is Ωk sim e2(NlowastminusN) (Nlowast sim 63 depends logarithmically on the reheatingtemperature and various other factors see eg [36])

Therefore in open inflation models the value of Ω today is exponentiallysensitive to the length of inflation As with nearly all relics of the state ofthe universe prior to the start of inflation inflation ldquoinflates awayrdquo curvaturewith exponential efficiency

This raises the question of how fine-tuned an open inflation model withobservable curvature would be The answer depends on the expected numberof efolds of slow-roll inflation As mentioned above [18] demonstrated thattoo little inflation (N lt Nlowast) prevents structure formation at least given afixed (N -independent) value of δρρ The argument parallels that of Wein-berg [8] Negative curvature can be thought of as a velocity for the expansionof the universe that exceeds ldquoescape velocityrdquo As such overdensities col-lapse only if the overdensity exceeds a certain boundmdashin effect in order tocollapse an overdense region must be dense enough that it is locally a closeduniverse Therefore for a given amplitude of δρρ insufficient N makescollapsed structures (ie stars and galaxies) exponentially rare

Following Weinbergrsquos logic [8] one may therefore expect that N is notmuch greater than Nlowast How much greater we expect it to be is a stronglymodel-dependent question in [18] a toy model gave a measure of for theprobability of N efolds P (N) sim Nminus4

3 Collisions

In any given Lorentz frame every bubble must eventually undergo a firstcollision In this section I will discuss the effects on cosmology of a singlecollisionmdashstrictly speaking the analysis does not apply once there are regionsaffected by multiple collisions However if the effects are sufficiently weak inthose regions one can use linear cosmological perturbation theory in whichcase the effects of collisions simply superpose

When two bubbles collide the spacetime region to the future of the col-lision is affected (see Fig 3 for a spacetime diagram of a bubble collision)Precisely what occurs inside that region depends on the model However

11

Figure 3 A spacetime diagram showing the causal structure of a cosmicbubble collision Coordinates are chosen so that light propagating in theplane of the diagram moves along 45 lines

for applications to observational cosmology one is generally interested in re-gions in which the effects are a small perturbation on the spacetime prior tothe collision As I will discuss below the assumption that the perturbationis small plus the symmetries of the collision suffice to extract the genericleading-order effects on cosmology

The most important features of a collision of this type are related to itssymmetries As reviewed above a single bubble has an SO(3 1) isometrya group generated by 6 symmetry generators It turns out that a bubblecollision breaks the SO(3 1) to SO(2 1) a group with 3 generators [11] Thisis enough symmetry to solve Einsteinrsquos equations in a situation in which thestress tensor is locally dominated by vacuum energymdashthe situation is verysimilar to that of a black hole (with three rotation isometries) in a vacuumdominated spacetime (de Sitter Minkowski or anti-de Sitter) As in thoseexamples there is a one parameter family of solutions Details can be foundin eg [11 20 22 23]

We will see that as a result of these symmetries the effects of the collision

12

break isotropy and homogeneity but are azimuthally symmetric (and non-chiral) around a special directionmdashthe axis pointing toward the center of thecolliding bubble

One important feature of these symmetries is that there exists a timeslicing of the collision spacetime in which the entire collision occurs at oneinstant everywhere along a 2 dimensional hyperboloid At later times in thesecoordinates the effects of the collision spread along a light ldquoconerdquo which isreally a region of space bounded by two hyperboloids To visualize this itmay help the reader to visualize the intersection of two lightcones cones orldquohourglass-typerdquo timelike hyperboloids in 2+1 dimensions The intersectionis a (spacelike) hyperbola which can be chosen to correspond to an instantof time and a point in one transverse coordinate (eg t and x in (6))

31 Thin-wall collision metric

In the approximation that the spacetime everywhere away from thin surfacesis dominated by vacuum energy one can write down exact solutions Thesemetrics are of the form

ds2 = minusdt2g(t) + g(t)dx2 + t2dH22 (6)

where dH22 = dρ2 + sinh2 ρdφ2 is the metric on a 2D hyperboloid and

g(t) = 1 +H2t2 minusmt (7)

Here H is the Hubble constant related to the local vacuum energy densityρV by H2 = ρV 3 and m is a constant [34 20 22] With m = 0 this issimply de Sitter space But with m non-zero it represents a solution that isessentially an analytically continued Schwarzschild-de Sitter black hole (see[22] for a discussion of the causal structure of these solutions)

Across a wall that separates two vacua (either the wall that separates twocollided bubbles or the walls of the bubbles themselves in the parent falsevacuum) one expects the vacuum energy density ρV to jump along withmany other quantities In the limit the wall is thin regions described by (6)can be ldquogluedrdquo together using Israel matching conditions [37 38 11 20] Thisallows one to determine the trajectory of the domain walls and therefore theconditions under which the domain walls that form in a collision acceleratetowards or away from inertial observers inside the bubbles When the wallaccelerates it rapidly becomes relativistic Anything it collides with will

13

almost certainly not survive as the ultra-relativistic wall typically has avery high energy density even in its rest frame In such cases there are nocosmological observables since the wall arrives almost immediately after itsfirst light signal

The results of this analysis which hold at least when the spacetime oneither side of the domain wall is dominated by vacuum energy are as followsthe domain wall always accelerates away from an inertial observer in thebubble with lower vacuum energy andmdashdepending on the tension in thewall and the difference in the two vacuum energiesmdashit can either acceleratetowards away from or not accelerate with respect to an observer in thebubble with larger vacuum energy [22 23] One interesting implication ofthis result is that inertial observers in bubbles with small positive vacuumenergy are shielded from the walls separating them from regions with highervacuum energy and might be shielded from walls with negative vacuumenergy beyond them if the potential satisfies certain conditions

Generalizing the solution (6) to the case of non-vacuum energy dominatedspacetimes is difficultmdashroughly as difficult as embedding a black hole into anon-trivial FRW cosmology However the basic symmetry structure will notbe affected the metric must still posses SO(2 1) symmetry and therefore canbe written in the form ds2 = ds22 +f(t x)dH2

2 (just as a black hole embeddedin an expanding FRW could be written in a form ds2 = ds22 + f(t r)dΩ2

2 Inparticular the region affected by the collisionmdashwhich takes place at somedefinite values of t and xmdashis still bounded by the future light ldquoconerdquo of thehyperboloid H2 at t x

32 A new solution

The metric (6) is a specific analytic continuation of an anti-de Sitter Schwarzschildblack hole metric There exist generalizations known as de Sitter-Vaidya oranti-de Sitter-Vaidya metrics that represent the formation of a black holein de Sitter or anti-de Sitter spacetime from the collapse of a shell of nullradiation [39 40] The shell does not have to be thinmdashin fact these metricscontain a free function that describes the profile of the ingoing radiationThe only criterion is that the radiation be purely radially ingoing (or purelyoutgoing)

Performing the appropriate analytic continuation one obtains the follow-

14

ing metric2

ds2 = (1minusm(w)t+H2t2)dw2 minus 2dwdt+ t2dH22 (8)

If m(w) = m is constant (8) is equivalent to (6) by a coordinate transfor-mation In general it is easy to check that it represents a spacetime withSO(2 1) isometries that contains both vacuum energy (with energy densitydetermined by H as above) and radiation (with energy flux proportional tomprime(w))mdashbut with all the radiation moving coherently in the same directionAs such it may be a fairly good approximation to the spacetime inside abubble after a collision at least during the curvature domination and slow-roll inflation phases To my knowledge this is the first time this solution hasappeared in print

4 Cosmological effects of collisions

To determine the effects of the collision on cosmological observables I willmake the following assumptions

-1 We are inside a bubble that has been or will be struck by at least oneother bubble

0 The collision preserves SO(2 1) that is one can choose coordinateslike (6) or (8) in which it occurs at one instant everywhere along ahyperboloid

1 The effects of the collision are confined entirely to the interior of thefuture ldquolightconerdquo of the collision event that is inside a region x gtxc(t) where xc(t) is a null geodesic in the (x t) plane transverse to thecollision hyperboloid

2 There was sufficient inflation to make the spatial curvature of the uni-verse and of the hyperbolic collision ldquolightconerdquo negligible

3 The effects on the observable part of our universe are small enough tobe treated using linear perturbation theory

4 The collision affects the inflaton by creating an O(1) perturbation init near the beginning of inflation

2I thank T Levi and S Chang for discussions on this metric

15

5 The inflaton perturbation is generic apart from constraints imposedby the symmetries and assumption 1

Using these assumptions the effects on the cosmic microwave background(CMB) temperature were first worked out in [24] and on CMB polarizationin [30]

41 Inflaton perturbation

Given these assumptions one can simply expand the inflaton perturbationin a power series at a time near the beginning of inflation at which time ththe metric (6) will be a good description of the spacetime Dropping the H2

factor (on which the perturbation is constant) for clarity

ds2 = minusdt2g(t)+g(t)dx2 asymp minusdt2(Hit)2+(Hit)

2dx2 = (Hiτ)minus2(minusdτ 2 + dx2

)

(9)The approximation is valid after the beginning of inflation Hit gt 1 and thecoordinate τ = minus1(H2t) is the conformal time The reader should recallthat τ increases towards zero during inflation and is exponentially small byits end

The constraint from assumption 1 means that the collision perturbationis non-zero only inside the affected region [24 41]

δφ(x τi) = M

( infinsumn=0

anHni (x+ τi)

n

)Θ(x+ τi) (10)

˙δφ(x τi) = M

( infinsumn=0

bnHn+1i (x+ τi)

n

)Θ(x+ τi) (11)

Here δφ is the perturbation in the inflaton field an and bn are dimensionlesscoefficients τi is a time near the start of inflation where the slow-roll approx-imation is valid M is a mass scale associated with the range over which thefield varies and the coordinates are chosen for convenience so that the edgeof the region affected by the collision is located at x = minusτi at that time Notethat this expansion is completely generalmdashit does not assume that the fieldthat tunneled is the inflaton and depends only on the assumptions outlinedjust above It does however ignore the effects of the collision on fields otherthan the inflaton at least in single-field models one expects that primordialperturbations of the inflaton field will have the largest imprint on cosmologytoday

16

Without a model for the inflaton and the field(s) that participated in thetunneling we cannot compute the coefficients an and bn Generically one ex-pects that the perturbation will be at least O(1) on scales corresponding tothe radius of curvature of the universe That is one expects the coefficientsan sim 1 Moreover given enough inflation to solve the flatness problem (andproduce a universe consistent with current constraints on curvature) the ob-

servable universe today corresponds to a region of size |x|Hi simradic

Ωk(t0) 1at the beginning of inflation Therefore we are interested in the perturbation(10) in the limit Hx 1 and so the first term will dominate the observablesignatures (unless for some reason a0 an for some n gt 0) Notice thata0 6= 0 is a spatial discontinuity in δφ

To determine the effects of this initial inflaton perturbation on the CMBand other cosmological observables one can employ the standard machineryof inflationary perturbation theory The first step is to determine the evolu-tion of the perturbation during inflation During slow-roll inflation at lowestorder in the slow-roll parameters inflaton perturbations satisfy the equationof a free massless scalar in de Sitter space (see eg [42]) One can solve thatequation in full generality the solution is [24]

δφ = g(τ + x)minus τgprime(τ + x) + f(τ minus x)minus τf prime(τ minus x) (12)

where g f are arbitrary functions of one variable and gprime f prime their derivativesThis solution may be familiar from the more standard treatment of in-

flationary fluctuations in momentum space the terms proportional to τ arethe usual decaying modes in the expression δφk sim kminus32 (1plusmn ikτ) Evidentlythe τ -dependent terms arise due to the effects of inflation since this solutionis otherwise identical to that of a massless scalar field in 1+1-dimensionalMinkowski space

Given the initial conditions in (10) one can use (12) to determine thesolution for all future times during slow-roll inflation The result can bewritten in closed form [41] but is not particularly illuminating Ratherthan reproduce it here it is easier to instead expand the functions f andg of (12) (which contain the same information as and are determined byδφ and ˙δφ) Since g is purely left-moving and f is purely right-moving forτ gt τi the perturbation near the edge of the lightcone Hi(xminus xc(τ)) 1 isdetermined only by g Therefore the effects on the CMB are determined by

17

the coefficients in the expansion of g

g(x+ τ) =

( infinsumn=1

cn(x+ τ)n)

Θ(x+ τ) (13)

Note that the sum begins with the linear term n = 1 This is because ann = 0 term in this expansion would lead to a δ-function singularity in δφwhich could not be expanded as in (10) and in any case would invalidateperturbation theory Put another way a non-zero c1 corresponds to a per-turbation δφ with a non-zero a0 in (10)

By the end of inflation τ = τe is exponentially small and one can dropthe terms proportional to τ in (12) leaving simply

δφ(x τe) asymp g(x) =

( infinsumn=1

cn(x)n)

Θ(x) (14)

Note the leading term is now a discontinuity in the first derivative not adiscontinuity in the perturbation itself Inflation smooths sub-horizon pri-mordial gradients by effectively integrating them once

42 Post-inflationary cosmology and Sachs-Wolfe

To precisely determine the effect of this perturbation on cosmological observ-ables at later times requires numerically integrating the coupled equationsthat control the perturbations in the various components of the energy den-sity in the early universe There is a standard set of software tools availablefor this purpose [43 44] The results of such a numerical analysis for a bub-ble collision can be found in [45] and analytically (and approximately) in[24 30]

One can understand the qualitative results with a simple analytic ap-proximation A primary component of the collision signal is its effect on theCMB temperature The most important contribution to CMB temperaturefluctuations comes from the Sachs-Wolfe effect [46 47] To first approxima-tion all CMB photons last scattered at redshift zdc sim 1100 when electronsrecombined with protons and photons decoupled from the plasma and thenpropagated freely (without scattering again) until now Their temperatureis determined by the intrinsic temperature T (tdc x y z) at the point in thelast scattering volume at the time tdc corresponding to z = zdc Given that aphoton free-streamed it originated at some point satisfying x2+y2+z2 = D2

dc

18

on a sphere of radius Ddc the radius of the current earthrsquos past lightconeat t = tdc The radius Ddc asymp 14Gpc can be determined from the expansionhistory a(t) by integrating back along a null geodesic in the FRW metric

Because a locally higher temperature at last scattering corresponds to alocally higher density there is a corresponding negative gravitational poten-tial energy fluctation at that location In the Sachs-Wolfe approximation thecurrent temperature of a photon originating from such a point is a combi-nation of the intrinsic temperature of the plasma it last scattered from andthe local gravitational potential at that point as well as an overall redshift(1 + zdc)

minus1 (and some additional contributions from time variations of thepotential along the way the ldquointegrated Sachs-Wolfe effectrdquo) A simple cal-culation shows that the negative potential energy contribution is larger thanthe increased intrinsic temperature perturbation Φ(tdc) sim minus(32)δTT (tdc[47] Therefore locally hot spots at last scattering make cold spots on theCMB

43 CMB temperature

Given a perturbation to the Newtonian potential at reheating Φ = λxΘ(x)one can easily estimate the effect on the CMB temperature today Theessential points are the following

bull Perturbations that are linear in the spatial coordinates (such as Φ =λx) remain so regardless of the cosmological evolution (to first order inperturbation theory) because the equations that govern the evolutionof cosmological perturbations are all second-order in spatial derivatives

bull The evolution equations are causalmdashthe effects of an initial perturba-tions are confined to within its light- or sound-cone

bull By the time of last scattering the radius of the sound-cone rsh of a per-turbation present at reheating corresponds to approximately 6 degreeson the CMB sky (rshDdc asymp 01)

Given this one can easily determine the temperature perturbation in theCMB The ldquokinkrdquo at x = 0 will evolve but its effects must be confinedwithin a slab of thickness 2rsh Outside that slab the perturbationmdashwhichwas initially either zero or linear in xmdashcannot evolve in shape although

19

Figure 4 The universe at decoupling showing the earthrsquos last scatteringsphere (our past lightcone at the time of decoupling) the regions affectedand unaffected by the collision and the angular size of the affected disk inthe CMB temperature map θc

20

its amplitude can (and does) change Therefore the perturbation to theNewtonian potential at t = tdc must be of the following form

Φ(tdc x y z) =

0 x lt minusrshf(x) minus rsh lt x lt rshλx x gt rsh

(15)

where f(x) is some function that must be determined by a more carefulanalysis but which one may expect to be a smooth deformation of the ldquokinkrdquoxΘ(x) (see Fig 4)

To determine the effects of such a perturbation on the CMB in the Sachs-Wolfe approximation one needs to project Φ(tdc x y z) onto the earthrsquos lastscattering sphere (of radius Ddc) This is trivial if the earthrsquos comovinglocation is xminusxe = y = z = 0 and choosing spherical coordinates centered onthe earth with the north pole θ = 0 on the x-axis one has xminusxe = Ddc cos θTherefore the effects of the perturbation on the CMB will be confined to theinterior of a disk of angular radius

θc = cosminus1(minusxe minus rsh

Ddc

) (16)

The effects can be very easily determined everywhere except within the an-nulus cosminus1

(minusxe+rsh

Ddc

)lt θ lt cosminus1

(minusxeminusrsh

Ddc

)near the rim of the disk where

it is determined by the function f(x) The angular width of this annulus isclose to 2times 6 = 12 for a large radius disk (|xe| Ddc) and increases forsmaller disks

Inside the inner edge of the annulus where θ lt cosminus1(minusxe+rsh

Ddc

) the CMB

temperature perturbation in the Sachs-Wolfe approximation is simply linearin cos θ as first predicted in [24] (see also [30]) The numerical evolution of[45] confirms that this is an accurate approximation and that the tempera-ture profile within the annulus is smooth and relatively featureless

44 CMB polarization

The free-streaming approximation discussed in the previous subsection is notentirely valid Approximately 10 of CMB photons Thomson re-scatter atz lt zdc This re-scattering leads to a net linear polarization of the CMBradiation if the radiation incident on the scatterer has a non-zero quadrupolemoment [48 49] CMB polarization can be decomposed into two types

21

E-mode and B-mode B-mode polarization originates from tensor perturba-tions Because the bubble perturbation is invariant under rotations aroundthe x-axis and does not break chirality it will not generate B-modes An-other way to understand the same fact is to note that the symmetries of atwo bubble collision suffice to prevent the production of gravity waves see[11 50]

However the perturbation will generate E-mode polarization In factthe pattern of polarization is completely determined by the collision pertur-bation at decoupling Φ(tdc x y z) It can be determined accurately usingnumerical evolution and as for the temperature this is necessary to deter-mine the detailed structure near the edge of the disk In contrast to thetemperature perturbation the polarization signal contains a distinct and in-teresting substructure within the annulus at the diskrsquos edge see [30 45] fordetails

Analytic approximation for polarization One can estimate the polar-ization signal using an analytic approximation [30] In this approximationone computes the quadrupole moment of the temperature distribution seenby scattering electrons along the earthrsquos line of sight making use of the per-turbation at decoupling (15) but with f(x) = λxΘ(x) The result is that thepolarization is non-zero in a wide annulus surrounding the edge of the diskand has an additional feature confined to a much narrower annulus [30]

To understand this result the key point is to realize that a linear per-turbation at decoupling Φ(tdc) = λx produces precisely zero quadrupolemoment in the radiation incident on a scatterermdashit is pure dipole There-fore any scatterer with a last scattering sphere (the scattererrsquos lightcone atdecoupling) that is entirely in the linear region x gt rsh or x lt rsh will notproduce linear polarization in the CMB Only scatterers with lightcones thatintersect x = 0 at decoupling will see a bath of incident radiation with anon-zero quadrupole component

Most re-scattering of CMB photons occurs at two times in the history ofthe universe around the time of decoupling zdc sim 1000 and much later atreionization at zri sim 10 The lightcone of a scattering electron at z sim 1000 ismuch smaller than the lightcone of the earth todaymdashit is roughly 1 acrossThe lighcone of an electron at z sim 10 is by contrast much larger Thereforescattering at reionization produces an effect within a broad annulus whilescattering near decoupling produces a sharp feature that must be resolved

22

numerically

45 Other cosmological observables

A number of cosmological observables other than the CMB will be affected bya bubble collision A linear gradient or xΘ(x) kink in the Newtonian potentialΦ leads to a coherent flow for large-scale structures a problem that wasstudied in a radiation dominated toy universe in [27] The kink correspondsto a wall of over- or under-density that could potentially be detected in large-scale structure or lensing surveys Observations of 21cm radiation could serveas a sensitive probe of large-scale structures with unusual symmetries suchas this [51] If the collision affects light fields other than the inflaton it couldlead to large-scale variations in other observables (for example if the finestructure constant can vary as in [52]) Generally speaking any measure ofthe large-scale structure of density perturbations would be sensitive to theeffects of a bubble collision

5 Probabilities and measures

The late-time statistics of the clusters of bubbles that form from collisionsturns out to be a very rich and interesting topic [10 19 53 25 54 55 56 5758] but one that goes beyond the scope of this review Here I will discussonly the question of the probability for observing these signals given whatwe know about cosmology and the underlying microphysics and what themost probable ranges are for the parameters describing their effects

To begin to address this consider an observer inside a bubble expand-ing into an inflating false vacuum To the extent that we can regard thespacetime outside the bubble as unperturbed by its presence the spacetimeoutside is pure de Sitter space de Sitter spacetime has an event horizonmdashevents separated by more than one Hubble length cannot influence each otherTherefore the only bubbles that can collide with the observerrsquos bubble arethose that nucleate within one false-vacuum Hubble length 1Hf of the wallof the bubble

At any given time one can imagine a spherical shell of false vacuumwith area equal to the surface area of the observerrsquos bubble at that timeand thickness one false vacuum Hubble length 1Hf If a new bubble formswithin that shell it will collide with the observerrsquos Therefore we should

23

expect the rate of bubble collisions to be

〈dNdT 〉 sim A(T )Hminus1f Γ (17)

where T is a time coordinate in the false vacuum and T = A(T ) = 0 is thetime of nucleation of the observerrsquos bubble

To proceed we need some information about A(T ) The first observationis that since the bubble is expanding it is a monotonically increasing functionof T Therefore the total number of bubble collisions after infinite time willbe infinite This is to be expectedmdashthe bubble expands eternally and therate for collisions does not go to zero (it increases) As a result bubbles forminfinite clusters [10]

51 Observable collisions

We are primarily interested in collisions that could be visible to us today Tosee what this implies recall that bubbles when they form are dominated bynegative spatial curvature This persists for a physical time interval t sim 1Hiwhere Hi is the Hubble scale during slow roll inflation inside the bubble be-cause the energy density in curvature redshifts like a(t)minus2 sim tminus2 during cur-vature domination Inflation begins when the energy density in the inflatonpotential ρV sim H2

i is of order the energy density in curvature tminus2If a given bubble collision is visible to us today we must be inside the

lightcone of the nucleation event of the colliding bubble Therefore we shouldtrace back our own past lightcone and determine the area A of our bubblersquoswall at the moment our past lightcone intersects it Collisions which occurlater when A is larger are not yet visible

The details were first worked out correctly in [25] but the result is intu-itive Null rays propagate a comoving distance equal to the conformal timeConformal time τ inside the bubble universe is related to physical time t byτ =

intdta(t) Inflation makes a(t) exponentially large which means that

during and after inflation very little conformal time passes In fact if there isenough inflation to solve the curvature problem most of the conformal timeinside the bubble is before the beginning of inflation when the universe iscurvature dominated

During curvature domination a(t) sim t and therefore τ sim ln t Hence weexpect τ sim ln 1Hi To complete the estimate recall that negative spatialcurvature means that the areas of spheres grow exponentially with comov-ing distance a sphere of comoving radius ρ has an area of order cosh2 ρ

24

Therefore we can expect the area of our bubble wall at the moment out pastlightcone intersects it to be of order A sim cosh2 ln(1Hi) sim 1H2

i Puttingthis together gives

〈dNdT 〉 sim Hminus2i Hminus1f Γ = γH3fH

2i (18)

We are not quite finished To complete the estimate we should integratedNdT back to T = 0 the nucleation time of the observerrsquos bubble Thisintegral will introduce an additional factor of Hf on the right-hand sidePerhaps the simplest way to see this is to apply the argument about null raypropagation to the false vacuummdashone immediately sees that the lightconesof bubbles travel a comoving distance 1Hf in a time 1Hf and then slowdown and stop (in comoving distance) exponentially rapidly

Finally we might be interested in bubble collisions with lightcones thatdivide the part of the last scattering surface we can see today into two piecesrather than completely encompass it The details can be found in [25] but

taking this into account introduces one more factorradic

Ωk(t0) sim Ddca(t0)mdashthe ratio of the comoving size of the earthrsquos decoupling sphere to the radiusof spatial curvature today

In the end the result is [25]

〈N〉 simradic

ΩkHminus2i Hminus2f Γ = γ

radicΩk (HfHi)

2 (19)

The significance of this result is that even though γ is small the ratio(HfHi)

2 is expected to be very large Indeed lack of B-mode polarizationin the CMB indicates that (MPlHi)

2 gtsim 1010 If Hf is a fundamental scaleone therefore expects this ratio to be very large The decay rate γ sim eminusS

must be small in order for this semi-classical analysis to be valid However inthe string theory landscape there are an enormous number of decay channelsand the relevant decay rate is the fastest one available for our parent falsevacuum Finally observational constraints are consistent with

radicΩk lt 1

Given the current state of understanding of the landscape and the scale ofinflation it is impossible to make a definite statementmdashbut it doesnrsquot appearthat any true fine-tuning is necessary to achieve 〈N〉 gt 1

52 Spot sizes

Given the symmetries of the parent false vacuum one would expect thedistribution of spot centers to be uniform on the sky (cf [19] but also [25])What about the distribution of sizes

25

This can be estimated very easily A glance at (17) shows that the rate ofbubble collisions increases with time (since A gt 0) Earlier collisions makelarger spots on the skymdashin fact very early collisions completely encompassthe part of last scattering surface visible to us Therefore we might expectmore smaller spots

However this effect is quite small so long as Ωk 1 today The reasonis that at least if we focus only on collisions with lightcones that divide ourlast scattering sphere we are looking at collisions that occurred over a timeinterval where A(T ) didnrsquot vary significantly As a result the distribution ofcollision times over that interval is close to uniform

The implications for spot sizes are easy to see Each collision producesa light ldquoconerdquo which at any given time is really a hyperbolic sheet In theapproximation in which we ignore the curvature of that sheet it is simplya plane situated at some transverse coordinate x(t) Uniform distributionof time T corresponds to a uniform distribution of x over the small range(|x| ltsim

radicΩk) that is visible today Therefore we should expect the edges of

the regions affected by bubble collisions to be well approximated by randomlysituated randomly oriented planes

Since x = Ddc cos θ a uniform distribution dx implies a distributiond(cos θc) = sin θdθc for the angular radius of the affected spots in the CMBmap Spots with small angular size are quite rare as are those that nearlycover the entire sky

One interesting implication of this result is that super-horizon ldquospotsrdquomdashcollisions with lightcones that encompass our entire last scattering surfacemdashare significantly more common than those that divide it and produce a visiblespot Such large collisions are difficult to detect because they are expectedto produce a pattern that is primarily dipole and therefore is masked by theearthrsquos peculiar motion However it is important to note that not all effectsof such superhorizon perturbations are masked and some are in principleobservable