Bubble Universes With Different Gravitational Constants
aa r X i v : . [ g r- q c ] A p r Bubble Universes With Different Gravitational Constants
Yu-ichi Takamizu ∗ and Kei-ichi Maeda † Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Department of Physics, Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan (Dated: September 2, 2018)We argue a scenario motivated by the context of string landscape, where our universe is producedby a new vacuum bubble embedded in an old bubble and these bubble universes have not onlydifferent cosmological constants, but also their own different gravitational constants. We studythese effects on the primordial curvature perturbations. In order to construct a model of varyinggravitational constants, we use the Jordan-Brans-Dicke (JBD) theory where different expectationvalues of scalar fields produce difference of constants. In this system, we investigate the nucleationof bubble universe and dynamics of the wall separating two spacetimes. In particular, the primordialcurvature perturbation on superhorizon scales can be affected by the wall trajectory as the boundaryeffect. We show the effect of gravitational constant in the exterior bubble universe can provide apeak like a bump feature at a large scale in a modulation of power spectrum.
PACS numbers: 98.80.-k, 98.90.Cq
I. INTRODUCTION
In the context of the string theory landscape [1],there are various possibilities that multiple vacua canbe produced whose semiclassical tunnelling process frommetastable vacua to new vacua can occur. In this re-spect, it can allow us to argue that nucleation of spher-ical symmetric regions, namely Bubble in space whichare constructed by a new vacuum and expand into theold vacuum. One of any bubble universe created by suchfirst-order phase transition may be regarded as our uni-verse [2–14]. In particular, inspired by the bubble uni-verse scenario, an accelerated expansion of the universegenerated by its vacuum energy would give a natural ex-planation for inflation realization in the early universeas discussed old inflation [3] or open inflation [8, 9, 11]scenario in the past. If the bubble universe would beinside the old bubble universe, the so-called parent uni-verse, we are mainly interested in how much informationwe know or detect through any observational signatureof the parent universe.In the viewpoint of Anthropic principle [1, 15], it ispossible that several universes take their own choices ofnatural constants. It can allow us to argue a possibil-ity that landscape vacua have different values of gravi-tational constants as well as cosmological constants. Inthis paper, the bubble universes have their own differentgravitational constants and it can give us explore how adifferent gravitational constant in the exterior universeaffect and relates to physical (especially observational)quantities for the observer in the interior bubble universe(see also works [12] for the aim).In order to such model building that each bubble uni- ∗ Electronic address: [email protected] † Electronic address: [email protected] verse has its own different gravitational constant, it isuseful to employ the JBD theory [16] (see book [17] forreview). This theory can achieve a varying gravitationalconstant by identified a vacuum expectation value of thescalar field with gravitational constants in the original,the so-called Jordan frame. Therefore we will use thistheory to construct our model and investigate the conse-quence of the model to our (child) bubble universe, wherethe child bubble is inside the parent bubble with differentconstants.Inflationary scenario can give a remarkably successfulway of generating a consistent spectrum of curvature per-turbations with recent cosmological observations, suchas the cosmic microwave background (CMB) and galaxysurvey to explore a large-scale structure [3, 21, 22]. Thequantum fluctuations generated in a exponential expan-sion of the early stage of the universe were spread outa cosmological horizon scale and their wavelengths arethen pushed to superhorizon size, which can be reducedto classical perturbations seeding cosmic structure. Af-ter inflation ends, they reenter the horizon in a radiationdominated phase and can be detected by such as CMBobservation (see [22] for review). The recent data byPlanck satellite [23] gave a perfectly agreement with theprediction of inflationary theory.In order to discuss an observational consequence ofbubble universes with different gravitational constants,we analyze density perturbations during an inflationaryphase based on the JBD theory. Note that althoughseveral inflationary models with/without bubble forma-tions and the accompanying density perturbations arediscussed based on the scalar-tensor gravity theories in-cluding the JBD theory [18–20], the JBD scalar field isassumed to be an inflaton in those models. In the presentpaper, however, we assume that there exists an inflatonfield in addition to the JBD scalar field, which is used tofix only two different gravitational constants.If the superhorizon perturbations propagate eventhough their amplitudes freeze out and approach to theboundary of the bubble universe, they would be scatteredby the layer called bubble wall, separating a new and oldvacuum. Namely when one join the interior and exteriorperturbation mode over the bubble wall, a reflected wavecan be produced. Hence if there would be a possibilitythat one can get information of outside bubble, that isonly way to analyze perturbations coming back from su-perhorizon scales outside bubble as a trace of the outerregion (see [24] in a different setup, but in which a similareffect has been studied).The aim of this paper is to obtain physical signatureon a primordial curvature perturbation at superhorizonscales, which is affected by the boundary of bubble. Inparticular, we can provide how superhorizon mode ismodified by the effect of different gravitational constantin the exterior universe by employing the JBD theory.The paper is organized as follows. In section 2 weshow our model setup and discuss a nucleation of bubbleuniverse and a static bubble wall solution. In section3, we study dynamics of bubble wall after a nucleation.In section 4, we adopt linear perturbation theory to thesystem and match perturbation modes in both sides overthe bubble wall. In particular, our calculation is the casefor de Sitter background. Then we discuss the effect ofdifferent gravitational constant on the obtained results insection 5. Section 6 is devoted to conclusion.
II. MODEL
We consider the model of bubble universes which havetheir own gravitational constants. The wall of the bubblecan divide two bubble universes whose constants have dif-ferent values. We investigate the effect of different gravi-tational constant on the superhorizon scale perturbation.We will introduce the JBD theory [16, 17] with a po-tential V (Φ) in the Jordan frame as the model of varyingof gravitational constant, which is expressed as the action S = Z d x √− g (cid:20) Φ R − ∂ µ Φ ∂ µ Φ − V (Φ) (cid:21) , (2.1)where Φ is the scalar field whose potential is considered toprovide their semiclassical tunnelling, namely it has twominima called a true and false vacuum respectively. Weuse the units of 8 πG N = 1, where G N is the Newtoniangravitational constant.From such a potential, the bubble universe will be cre-ated as a standard instanton process. The gravitationalconstant G is related to the vacuum expectation value(VEV) of the scalar field as follows G = G N h Φ i . (2.2)The scalar field has a different value depending on eachpotential minima, and then it can read different valuesof gravitational constants. In order to analyze this system, it is useful to study dy-namics in the Einstein frame by using conformal trans-formation as g µν = Φ − ˜ g µν [25]. After transformation,the action in the Einstein frame is given by S = Z d x p − ˜ g " ˜ R − ∂ µ φ∂ µ φ − V ( φ ) . (2.3)Here we introduced a new scalar field: φ and its potential,which are related to the original ones in the Jordan frameas φ = φ + √ , V ( φ ) = V (Φ)Φ , (2.4)where φ corresponds to the VEV of the JBD scalar fieldin our universe (Φ = 1). Hereafter we will omit the tildewhich represents the variables in the Einstein frame forbrevity. In this paper, as expected to generate a bubble,we consider the potential which should give tunnelling,which is expressed by V ( φ ) = λ φ − µ ) + ǫ µ ( φ − µ ) + Λ . (2.5)Under the approximation that ǫ is small, two vacua areobtained by φ ± = ± µ − ǫ λµ + O ( ǫ ) where the suffixes ± describe two minima. In the case where ǫ >
0, thetunnelling occurs from φ + to φ − vacuum. Each potentialenergy is given by V ( φ + ) = Λ + O ( ǫ ) and V ( φ − ) =Λ − ǫ + O ( ǫ ), respectively.Since we assume that our universe was created by tun-nelling from de Sitter universe, our vacuum state is givenby φ = φ − . From the above potential (2.5), the poten-tial V (Φ) in the original Jordan frame is found by (2.4). In Fig.1, we plot two potentials V (Φ) and V ( φ ). Fromthe figure, it is clear that the tunnelling process can berealized from a false(+) vacuum to a true( − ) vacuum . A. Nucleation of Bubble Universe
We consider a spatially flat Freidmann-Lemaitre-Robertson-Walker (FLRW) spacetime in both sides ofthe wall. In this case, the instanton process is calculatedin the Euclidean spacetime with O (3) symmetry. In theLorentzian configuration, the line element takes the form ds = a ( η )( − dη + dr + r d Ω ) , (2.6)where η is a conformal time which is related to a cosmictime t by dt = adη . Note that the possibility for tunnelling is found by use of thepotential in the Einstein frame. We may misunderstand the pos-sibility if we look only at the potential in the Jordan frame. Seean example in the paper [26]
FIG. 1: (Left) We plot the potential V (Φ) in the Jordan frame. The tunnelling occurs from Φ + to Φ − . (Center) We plotthe potential V ( φ ) in the Einstein frame. The tunnelling occurs from φ + to φ − . (Right) We plot the symmetric potential(2.11) and asymmetric one (2.13). The positive vacuum φ + where the local minimum is located moves to the right directionby b -times with the same height of the potential barrier. Following [7, 10, 13], the calculation has been done byuse of the complex time path approach, in which dynam-ics is studied by the complex time η . By adopting thethin-wall approximation, in which a thickness of wall issmall enough compared to the size of a bubble, the bub-ble radius r ( η ) is solved as a function of η . The action isevaluated as S = Z dη h π ǫa ( η ) r ( η ) − πσa ( η ) r ( η ) q − ( ∂ η r ( η )) ) i . (2.7)This action contains both contributions from the volumeand surface area of three-dimensional sphere. Here wehave used ǫ and σ , which are the energy density differ-ence between two vacua and the surface energy densityof the wall, respectively. The equation of motion is givenby taking a variation of the action S with respect to r .Solving it, we find a classical trajectory of the wall r ( η ).In the backward of time η , we trace a shrinking bubbleradius and reach to a turning point where the canoni-cal momentum p = ∂ L /∂ ( ∂ η r ) vanishes. At this turningpoint ( η = η i ), a bubble nucleation occurs. By the an-alytic continuation, we find the wall trajectory in thecomplex η plane, which shrinks smoothly to zero size ofa bubble. In order to perform this procedure, it is usefulto rewrite the equation as one for η ( r ). The task to dois reduced to solve it with the boundary conditions: p = 4 πσ a r p ( ∂ r η ) − | η = η i = 0 , ∂ r η (0) = 0 . (2.8)where η = η i denotes a nucleation time.The tunnelling rate per unit four-volume is obtainedfrom the imaginary part of the action with the complex η , which becomesΓ( η i ) ≃ exp[ − S ( η i )] . (2.9)As a result [13], the tunnelling probability in a spatiallyflat de Sitter universe is obtained asIm S = π ǫ H sinh (cid:20)
14 ln(1 + (3
Hσ/ǫ ) ) (cid:21) . (2.10) Note that the tunnelling rate is independent of the choiceof η i in the de Sitter spacetime. This expression reducesto the result for Minkowski spacetime known as the Cole-man’s bubble solution [2]Im S = 27 π σ ǫ , in the limit of H → B. Wall solution
We discuss how the difference of the VEV’s φ ± , thatis the difference of gravitational constants in the Jordanframe, affects the observational consequence. So we an-alyze the dependence of the ratio G + /G − = Φ − / Φ =exp[ − √ ( φ + − φ − )] on the final result. For simplicity wediscuss only the potential in the limit of ǫ →
0. That isthe case of the potential for which both vacua have thesame cosmological constant Λ .The potential (2.5) in this limit is V ( φ ) = λ (cid:0) φ − µ (cid:1) + Λ . (2.11)This gives G + G − = exp (cid:20) − µ √ (cid:21) . (2.12)So if we change the value of the parameter µ , we candiscuss the dependence of the different gravitational con-stants. However, it also changes the height of the poten-tial barrier. In order to keep the same potential height,we adopt the following toy potential V ( φ ) = 2 λ (1 + b ) ( φ − bµ ) ( φ + µ ) + Λ . (2.13) We give an analytic solution of a domain wall for small ǫ inAppendix. The values of the scalar field in the mother universe andour universe are given by φ + = bµ and φ − = − µ , respec-tively. The ratio of the gravitational constants is G + G − = exp (cid:20) − b + 1) µ √ (cid:21) . (2.14)Comparing (2.11) with the case b = 1 in (2.13) allowsus to discuss the change of the ratio of G + /G − with thesame height of the potential barrier (see Fig.1). Calculat-ing the wall solution and surface density of the wall σ forthis toy potential and varying the value of b ( > − σ ofa bubble, we consider a spherical symmetric and staticsolution φ ( r ). The basic equation is ∂ r φ + 2 r ∂ r φ − dV dφ = 0 . (2.15)Imposing the thin-wall approximation in which we canneglect the term of ∂ r φ , we can integrate it as Z φφ i dφ p V ( φ ) − V ( φ i )) = r − r i , (2.16)where φ i = φ ( r i ).For the potential (2.13), we can find the wall solutionas φ ( r ) = − µ + ( b + 1) µ (cid:20) (cid:18) r − r i d (cid:19)(cid:21) , (2.17)with d = (1 + b )2 µ √ λ . (2.18)Of course, in the limit of b = 1, we find that the solu-tion (2.17) is reduced to the usual wall solution with thepotential (2.11) φ ( r ) = µ tanh h µ √ λ r − r i ) i . (2.19)Then we obtain σ as σ = Z ∞ dr (cid:20)
12 ( ∂ r φ ( r )) + V ( φ ) (cid:21) ≃ Z φ + φ − dφ p V ( φ ) − V ( φ − )) , (2.20)where we have used the thin-wall approximation. Sub-stituting (2.13) into the above equation, we obtain theresult σ ( b ) = √ λµ b ) . (2.21)When b = 1, it gives the surface density of the originalpotential (2.11) as σ | b =1 = 2 √ λ µ . (2.22) We can argue how the difference of gravitational con-stants in the Jordan frame affects on the bubble nucle-ation rate Γ, the wall dynamics and the primordial per-turbation through this surface density σ ( b ), which is afunction of b .Note that we find the same surface density σ even whenwe take into account of small ǫ -modification of the po-tential (see Appendix). III. DYNAMICS OF BUBBLE WALL
In this section, we study dynamics of a bubble wallafter nucleation of a bubble universe. To obtain an ana-lytic solution of a wall trajectory, we assume that a wallis infinitely thin and impose a junction condition acrossthe wall. The bubble dynamics has been studied by sev-eral authors [5, 6, 13, 14, 24]. Following their works, webriefly summarize the results.Each spacetime is distinguished by the suffixes + and − , respectively. The bubble wall is represented as a time-like spherically symmetric hypersurface Σ dividing a false(+) vacuum and true ( − ) vacuum. We adopt a spa-tially flat slicing of de Sitter universe for both spacetimes,whose metrics are given by ds ± = − dt ± + exp(2 H ± t ± )( dr ± + r ± d Ω ) , (3.1)where H ± = p Λ ± /
3. Note that H − H − = ǫ/ . (3.2)When ǫ = 0, cosmological constants inside and outsideof the wall have different values. Up to the first order of ǫ , they can be approximated as Λ − = Λ − ǫ and Λ + = Λ,respectively. The metric of the bubble wall takes theform ds | Σ = − dτ + R d Ω . (3.3)We assume the stress-energy of this wall is given simply S µν = − σh µν , (3.4)where h µν = g µν − n µ n ν is the projection tensor on thehypersurface Σ, which is the metric tensor of (3.3). Theunit spacelike vector n µ normal to the hypersurface Σ isgiven by n µ = a (cid:18) − drdτ , dtdτ , , (cid:19) , n µ = (cid:18) a drdτ , a dtdτ , , (cid:19) . (3.5)From n µ n µ = 1, we obtain the relation equation (cid:18) dt ± dτ (cid:19) − a ± (cid:18) dr ± dτ (cid:19) = 1 . (3.6)By using drdτ = dtdτ × drdt , it can lead to dt ± dτ = 1 q − a ± ( drdt ± ) . (3.7)In order to find the dynamical equation for the wall,we use the Israel’s Junction conditions as[ g µν ] + − | Σ = 0 , (3.8)[ K µν ] + − = − ( S µν − h µν S ) , (3.9)where [ X ] + − = X | + − X | − for the variable X and S = S µµ . The extrinsic curvature K µν of the hypersurface Σis defined by K µν = h ρµ h σν ∇ σ n ρ , (3.10)where ∇ σ denotes a covariant derivative with respect to g µν . The continuity equation is given as S νµ || ν = − [ S ρσ n ρ h µσ ] + − , (3.11)where || denotes a covariant derivative with respect tothe projection tensor h µν . This equation is reduced tothe dynamical equation for σ as dσ ( τ ) dτ = (cid:20) a drdτ dtdτ ( ρ + P ) (cid:21) + − , (3.12)where ρ and P denote the energy density and pressurein the bulk universes, respectively. If ρ + P = 0, e.g. forthe radiation or matter dominant universe, σ changes intime. Throughout this paper, however, since we consideronly de Sitter background universes, σ turns out to beconstant.From the junction condition (3.8), we obtain theproper radius of the wall by R = a + r + = a − r − . (3.13)To find the dynamical equation for R , we use the ( θθ )component of the second junction equation (3.9), whichleads to K + θθ − K − θθ = − σ R , (3.14)and its square becomes( K + θθ ) = 1 σ R (cid:26) ( K − θθ ) − ( K + θθ ) − σ R (cid:27) . (3.15)While K θθ is calculated as K θθ = R ( (cid:18) dRdτ (cid:19) − H R ) . (3.16)Using (3.15), we can simplify the equation for R as (cid:26) dR ( τ ) dτ (cid:27) = B R ( τ ) − , (3.17)where B = H (1 + c ) = H − (1 + c − ) , (3.18) c ± = H − ± (cid:16) ǫ σ ∓ σ (cid:17) , (3.19) where the second equality in (3.18) is found from Eq.(3.2). For c + , although ǫ is assumed to be small, c + canbe positive because4 ǫ σ = 12(1 + b ) (cid:18) ǫλµ (cid:19) (cid:18) µm PL (cid:19) − , (3.20)can be smaller than unity, if we assume µ ≪ m PL , where m PL (= 1) is the reduced Planck mass.For de Sitter bulk universes, which we have assumedhere, B is a constant and then a solution of a bubbleradius R ( τ ) is obtained by R ( τ ) = 1 B cosh Bτ . (3.21)We can also solve the wall motion in the interior orexterior coordinates ( t + , r + ) or ( t − , r − ). In what follows,although we give the solution for the exterior coordinates,the same form of the solution is obtained for the interiorcoordinates just by replacement of + with − .From (3.13), we haveln R = H + t + + ln r + , (3.22)and differentiate both sides of the equation with respectto τ . Taking its square and using (3.6) and (3.21), wefind the equation for d (ln r + ) /dτ as d (ln r + ) dτ = ∂ τ (ln R ) ± H R | c + | − H R , (3.23)which is integrated by using the solution of (3.21) as r + = r ∞ + cosh Bτ | sinh Bτ ± c + | , (3.24)where r ∞ + is an integration constant to be determinedlater and ± take +( − ), if the sign of c + takes a positive(negative) value. Inserting (3.24) into (3.22) leads to thesolution of t + as H + t + = ln( | sinh Bτ ± c + | ) − ln Br ∞ + . (3.25)Finally we solve the wall motion as r + ( t + ), which de-notes a dynamics of the comoving bubble radius r for theobserver in the exterior universe.Combining dRdτ = dtdτ × ddt ( ar ), (3.7) and (3.21), we havea differential equation − B R H + ∂ t r ( t + ) r = 1 − | c + | p B R − . (3.26)It is integrated when rewritten as equation for B R as r + ( t + ) = q ( a − − + c − H − . (3.27)Here we take a nucleation time of a bubble as t + = 0,when we impose ∂ t r + (0) = 0. The solution takes thesame form for both interior coordinates too.We can summarize dynamics of bubble wall in the in-side and outside observer as follows. At the initial time t ± = 0, the bubble is created at the comoving radii r ± (0) = r i ± = (cid:12)(cid:12)(cid:12) ǫ σ ∓ σ (cid:12)(cid:12)(cid:12) − , (3.28)and then expands. The comoving radii eventually con-verge to r ± ( t → ∞ ) = q c ± r i ± ≡ r ∞± . (3.29)The integration constant r ∞ + given in (3.24) should bethe same as the above value. The asymptotic comovingradii are rewritten by r ∞± = q H − ± + ( r i ± ) . (3.30)If we ignore small initial radii r i ± , this equation basi-cally yields r ∞± ∼ H − ± , which means the final conver-gent radii are approximately described by the Hubbleradii H − ± in each coordinates. It is obviously noted thatphysical scale of the bubble radius continues to expandas R = a ± r ± ∝ e H ± t ± , although the comoving radii con-verge to about the Hubble radii. IV. PERTURBATIONS
In this section, in order to discuss observational effectsof the different gravitational constant in the mother uni-verse, we analyze metric perturbations in the Newtonian(longitudinal) gauge. The perturbed metric is given by ds = a [ − (1 + 2 ψ ) dη + (1 − ψ )( dr + r d Ω )] . (4.1)In the interior and exterior spacetimes, the tunnellingJBD scalar field takes the values of φ − and φ + , respec-tively, which provide non-zero cosmological constants. Asa result, each bulk universes expand exponentially as deSitter spacetime. However, such a de Sitter expansionwill not end. In order to discuss more realistic cosmo-logical scenario, instead of a cosmological constant, weintroduce another scalar field ϕ , which is confined at ourvacuum state φ = φ − . This scalar field ϕ is responsiblefor inflation and will finish the exponential expansion.The quantum fluctuation of this scalar field provides thedensity perturbations, which we will discuss here . In the outside mother universe, we may also find an inflaton,which will finish the exponential expansion. In our analysis ofperturbations, however, we assume de Sitter expansion in bothbulk universes. Hence our result does not change unless theoutside inflation will end earlier than our inflation. There maybe some effects on the perturbations from the inflaton field, whichwe ignore in our analysis.
The new potential U ( ϕ ) is assumed to be independentof V ( φ ). In this case, we find the background equationsas H = 13 (cid:18) ϕ ′ + a U (cid:19) , (4.2) H ′ − H = − ϕ ′ , (4.3) ϕ ′′ + 2 H ϕ ′ + a U ,ϕ = 0 , (4.4)where H = Ha and a prime denotes a derivative withrespect to a conformal time η . We assume a slow-rollinflation for ϕ field for simplicity, i.e. a ( t ) = exp( Ht ) , H = − η , − ˙ ϕ H , (4.5)where a dot denotes a derivative with respect to the cos-mic time t . We introduce slow-roll parameters as ε = − ˙ HH , ε = ˙ ε Hε . (4.6)Adding the perturbation of scalar field δϕ and expand-ing the basic equations up to linear order, we find onemaster equation by use of the Mukhanov-Sasaki variable v , which is a linear combination of δϕ and ψ . Defining u = aψϕ ′ , v = a (cid:18) δϕ + ϕ ′ H ψ (cid:19) , (4.7)we find the basic equations for these variables as∆ u = z ( v/z ) ′ , v = θ ( u/θ ) ′ , (4.8)where we have used θ = H aϕ ′ = 1 a √ ε , z = 1 θ . (4.9)Then we obtain the closed master equation for u : u ′′ − (cid:18) ∆ + θ ′′ θ (cid:19) u = 0 . (4.10)Under a slow-roll condition, the term θ ′′ /θ is evaluatedas θ ′′ θ = 1 η (cid:18) ν − (cid:19) , ν = 14 + ε + ε . (4.11)For spherically symmetric perturbations, a solution u ( r, η ) = e ± ikr /r satisfies the equation∆ u = ( ∂ r + 2 ∂ r /r ) u = − k u . The mode function with a comoving wave number k isobtained by u = e ± ikr r √− η h C H (1) ν ( − kη ) + C H (2) ν ( − kη ) i , (4.12)where C , are arbitrary constants. In the interiorcoordinate( − ) system, v is reduced to v k → e − ikη / √ k in the high-frequency limit, which corresponds to an adi-abatic vacuum. This condition fixes the above constantsas C , . Then the solution u is determined up to a phasefactor as u in = e ik − r − r − √− πη − k − H (1) ν ( − k − η − ) . (4.13)This mode solution is proportional to e ik − ( r − − η − ) inthe large | η − | limit, so interpreted as an incoming wavefrom the past. It propagates in the outward (large r − )direction from inside the bubble and then approaches tothe boundary wall.This incoming wave is scattered at the wall and thendivided into reflected and transmitted waves, which arewritten by u rf = e − ik − r − r − √− πη − k − H (1) ν ( − k − η − ) , (4.14) u tr = e ik + r + r + √− πη + k + H (1) ν ( − k + η + ) , (4.15)respectively. Here the outside wavenumber: k + is re-lated to k − as k + = ( a + /a − ) k − because the phase factor e ik ( r ± η ) and the proper time dτ = a ± ( dη ± − dr ± ) haveto be invariant across the wall.The scattered mode u rf is moving in the inward direc-tion and then is going back to the interior bubble. Whilethe mode u tr is transmitted to the outside bubble (theexterior universe).Expressing each perturbation as u − = u in + βu rf , u + = αu tr , (4.16)we match those wave solutions at the hypersurface by useof the junction condition.The matching has to be done for each proper time ofthe wall τ . We then impose that the wave amplitudesand their normal derivatives are continuous at the wallhypersurface as u − ( τ ) = u + ( τ ) , (4.17) n µ − ∂ µ u − ( τ ) = n µ + ∂ µ u + ( τ ) . (4.18)The similar argument was given in [24] as the matchingperturbation across a hypersurface.From the wall motion (3.24) and (3.25) with (3.27),the normal vector n µ (3.5) is found as n t ± = | c ± | sinh Bτ − Bτ + | c ± | , n r ± = B r ∞± cosh BτH ± (sinh Bτ + | c ± | ) . (4.19) Rewriting a ± and η ± as a function of τ and using thematching conditions (4.17) and (4.18), we obtain the am-plitudes of the reflected and transmitted waves. We plotsome wave trajectory in Fig.2.In the comoving radial coordinate r , the bubble radius(see the red line in Fig.2) increases from an initial value r i to a convergent value r ∞± ≃ /H ± , while a comovinghorizon 1 / ( H ± a ± ) (the green line) decreases monotoni-cally from an initial value 1 /H ± where we have set a i = 1at an initial time t = 0 and approaches asymptoticallyto zero during de Sitter expansion. However if de Sitterexpansion continues forever, a typical perturbation mode(as written in a blue line in Fig.2) keeps to be located insuperhorizon scale and cannot enter in the horizon again,which is not observable. Such a model does not describea realistic scenario for cosmological perturbations.Hence as discussed above, we have added anotherscalar field ϕ , which is responsible for inflation. In arealistic inflationary scenario, inflation will end and leadto reheating of the Universe, connecting to a usual radi-ation dominated FLRW universe. Taking such a realisticscenario into account, we find the horizon scale increasesagain from the convergent value, and then such super-horizon perturbation will reenter into the horizon. Theinitial incoming wave of perturbations approaches to thewall, which is located on over the horizon (see the solidlight-blue line with an arrow), and then is scattered intothe reflected and transmitted waves. Now we shall ana-lyze how this reflection mode affects the observed quan-tities, i.e. the CMB power spectrum of the perturbationsby the observer lived in the interior bubble universe.Generally speaking, if one of the bulk spacetimes is notde Sitter, it is difficult to find an analytic solution for theperturbations as well as the background spacetime. Weneed a complicated numerical analysis. However, sincethe universe expands exponentially during inflation, weassume the simplest situation for the calculation of thewave propagation, that is, the case of a de Sitter expan-sion in both sides of spacetime.In a de Sitter background, we can calculate the reflec-tion amplitude β and discuss how this mode affects onperturbations. We match perturbations in both sides ateach proper time τ . Eliminating α in the junction con-ditions (4.17) and (4.18), we find β as FIG. 2: We plot the trajectories representing the bubble wall (red) and horizon scale (green) as seen by a observer outsidethe wall in the units of H − = 1. The blue line denotes a typical perturbation scale, which is in superhorizon scale during deSitter phase. It is shown by the light-blue lines that an incoming wave propagates from inside the bubble to the boundarywall, scattered at the wall and then divided into a reflected β and transmitted α waves. The pink line denotes incoming wavefrom the exterior bubble and scattered into a reflected ˜ β and transmitted ˜ α waves similar as light-blue lines. (Left) We plotthem in the coordinate t and comoving radius r . (Right) We plot them in the coordinate conformal time η = − / ( aH ) andcomoving radius r . It is shown that the slope of incoming wave (light-blue) is an angle of 45 ◦ . So at late time, the reflection isnot generated since the slope of incoming asymptotically becomes a same slope of the wall trajectory, i.e., β ≃ α ≃ β = e ik − r − × h − (cid:16) f + ( τ ) − f − ( τ ) (cid:17) H (1) ν ( − k + η + ) H (1) ν ( − k − η − )+ g + ( τ ) H (1) ν +1 ( − k + η + ) H (1) ν ( − k − η − ) − g − ( τ ) H (1) ν +1 ( − k − η − ) H (1) ν ( − k + η + ) i × h(cid:16) f + ( τ ) − f ∗− ( τ ) (cid:17) H (1) ν ( − k + η + ) H (1) ν ( − k − η − ) − g + ( τ ) H (1) ν +1 ( − k + η + ) H (1) ν ( − k − η − ) + g − ( τ ) H (1) ν +1 ( − k − η − ) H (1) ν ( − k + η + ) i − , (4.20)where ∗ denotes a complex conjugate and f ± ( τ ) = h ( | c ± | sinh Bτ − H ± (1 + 2 ν )2 B + (1 − ik ± r ∞± y ± ( τ ) − cosh Bτ ) BH ± i y ± ( τ ) − , (4.21) g ± ( τ ) = k ± r ∞± ( | c ± | sinh Bτ − y − ± ( τ ) , (4.22) − k ± η ± ( τ ) = k ± Br ∞± H ± y ± ( τ ) , (4.23)with y ± ( τ ) = sinh Bτ + | c ± | . (4.24)We note that in the above calculation we have used the same value of ν = 1 / ν ± , which depend on the slow-roll parameters ε , ε . In our calculation, however, both sides of spacetime are approximated by de Sitter, that is, slow-roll parametersvanish and then we approximate the wave functions with ν ± = ν = 1 /
2. We have also used the formula of the Hankelfunctions: ∂ z H ν ( z ) = νz − H ν ( z ) − H ν +1 ( z ).Using the definition of H / ( z ) and H / ( z ), we find β = e ik − r − × f + ( τ ) − f − ( τ ) + g + ( τ ) k + η + − g − ( τ ) k − η − + i [ g + ( τ ) − g − ( τ )] − ( f + ( τ ) − f ∗− ( τ )) − g + ( τ ) k + η + + g − ( τ ) k − η − − i [ g + ( τ ) − g − ( τ )] . (4.25)If ǫ vanishes exactly, i.e., there is no energy difference between two vacuum states, since H + = H − , f + = f − , g + = g − , and η + = η − , then we find β = 0. However, we assume ǫ/ H − H − = 0, which results in β ∝ ǫ = 0 if ǫ is small. In fact, assuming ǫ ≪ λµ × ( µ/m PL ) as well as ǫ ≪ Λ, we find β = e ik − r × c p c cosh( B τ ) ǫℓ n ik − ℓ (cid:0) sinh( B τ ) − c − (cid:1) + p c c cosh( B τ ) (cid:16) η ℓ (cid:17) − sinh( B τ ) (cid:18) c + η ℓ (cid:19) + η c ℓ o , (4.26)where we define ℓ = r
3Λ = H − , c = ℓσ , r = ℓ p c c ,B = p c ℓ , η = − (1 + c ) ℓc (sinh( B τ ) + c ) . (4.27)Similarly, we also have to estimate the contributionfrom the perturbation in the exterior universe (see thepink line in Fig.2). Now the incoming wave u is given as u in = e − ik + r + r + √− πη + k + H (1) ν ( − k + η + ) . (4.28)It is scattered at the wall and then divided into reflectedand transmitted waves, which are written by u rf = e ik + r + r + √− πη + k + H (1) ν ( − k + η + ) , (4.29) u tr = e − ik − r − r − √− πη − k − H (1) ν ( − k − η − ) , (4.30) respectively. Expressing each perturbation as u + = u in + ˜ βu rf , u − = ˜ αu tr , (4.31)we match those wave solutions at the hypersurface by useof the junction condition.Eliminating ˜ β in the junction conditions (4.17) and(4.18), we find the transmitted waves of the perturbationsin the exterior universe, that is moving inward directionto the interior universe. The transmitted rate ˜ α is givenby˜ α = r η + η − k − r − k + r + e i ( k − r − − k + r + ) × h(cid:16) f + ( τ ) − f ∗ + ( τ ) (cid:17) ( H (1) ν ( − k + η + )) i × h(cid:16) f + ( τ ) − f ∗− ( τ ) (cid:17) H (1) ν ( − k + η + ) H (1) ν ( − k − η − ) − g + ( τ ) H (1) ν +1 ( − k + η + ) H (1) ν ( − k − η − )+ g − ( τ ) H (1) ν +1 ( − k − η − ) H (1) ν ( − k + η + ) i − . (4.32)Using the definition of H / ( z ) and H / ( z ), we find˜ α = s k − k + H − H + e i ( k − ( r − + η − ) − k + ( r + + η + )) × f + ( τ ) − f ∗ + ( τ )( f + ( τ ) − f ∗− ( τ )) + g + ( τ ) k + η + − g − ( τ ) k − η − + i [ g + ( τ ) − g − ( τ )] . (4.33)When ǫ is small, we find˜ α = 1 + ǫℓ " ik − η + c p c cosh( B τ ) ( ik − ℓ (cid:0) sinh( B τ ) − c − (cid:1) − p c c cosh( B τ ) (cid:18) − c + 2 η ℓ (cid:19) − sinh( B τ ) (cid:18) c + η ℓ (cid:19) + η c ℓ ) . (4.34)Once we would obtain the solution of u , we have to convert it to the curvature perturbation ζ and es-0timate its power spectrum. In Fourier space, we find ζ k = v k /z = θ ( u k /θ ) ′ . Hence the curvature perturbation is given by ζ k = √− πη − k − e − ik − r − ( H (1)1 / ( − k − η − ) h H − √ ε − ε − β − dβdτdη − dτ η − ! + H + √ ε ε − η + η − dη − dτdη + dτ ! ˜ α − d ˜ αdτdη + dτ η + !i − k − η − H (1)3 / ( − k − η − ) h H − √ ε − β + H + √ ε η + η − dη − dτdη + dτ ˜ α i) . (4.35)Using the definition of Hankel function, a dimensionless power spectrum can be obtained by∆ ζ ( k − ) ≡ k − π | ζ k | = 18 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 + ik − η − ) H − √ ε − β + ik − η − η + η − dη − dτdη + dτ ! H + √ ε ˜ α − η − dη − dτ H − √ ε − dβdτ − η + dη + dτ H + √ ε d ˜ αdτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.36)where we neglected a slow-roll correction, i.e., ε ± = 0.We can divide the deviation from the perturbations in a standard inflationary model into three parts s , s and s ,which are of order O ( ǫ ), as 8 π ℓ ε ∆ ζ ( k − ) ≡ s + s + s + O ( ǫ ) , (4.37)with s = 2Re (cid:18)r ε ε − β + (˜ α − (cid:19) ,s = 2Re − η − dη − dτ r ε ε − dβdτ − η + dη + dτ d ˜ αdτ ! ,s = 2 ik − η − × Im r ε ε − β + η + η − dη − dτdη + dτ (˜ α − ! . (4.38)where Re and Im denote a real and imaginary part, respectively.At a superhorizon scale ( k − η − ≪ s and s are important, which we can calculate s = ǫℓ c √ ε p c cosh( B τ ) " − sin(2 k − r ) k − ℓ √ ǫ − (cid:0) sinh( B τ ) − c − (cid:1) + p c c cosh( B τ ) (cid:26) cos(2 k − r ) √ ǫ − (cid:16) η ℓ (cid:17) − √ ǫ (cid:18) − c + 2 η ℓ (cid:19)(cid:27) − sinh( B τ ) (cid:18) cos(2 k − r ) √ ǫ − + 1 √ ǫ (cid:19) (cid:18) c + η ℓ (cid:19) + η c ℓ (cid:18) cos(2 k − r ) √ ǫ − + 1 √ ǫ (cid:19) , (4.39)and s = ǫℓ c y √ ε p c cosh( B τ ) ( − sin(2 k − r ) k − ℓ √ ε − y c cosh ( B τ ) + (cid:16) cos(2 k − r ) √ ε − + 1 √ ε (cid:17) × (cid:20) − ( B τ ) (cid:16) c + η ℓ (cid:17) − η c y ℓ (cid:16) cosh( B τ ) q c − c sinh( B τ ) + 1 + tanh( B τ ) y cosh( B τ ) (cid:17)(cid:21) + 3 η c y ℓ cosh( B τ ) p c √ ε ) , (4.40)1where y = sinh( B τ ) + c . (4.41)The correction s is derived from the time derivative terms of β and ˜ α , which plays as the order O (1 / cosh( B τ )) andthen it can be ignored at the late time, although it can become a leading correction to (4.39) at the initial time.At the subhorizon scale ( k − η − > ∼ s , which is evaluated by s = − ǫℓ k − η √ ε ((cid:16) cos(2 k − r ) √ ε − + 1 √ ε (cid:17) c sinh( B τ ) − k − ℓ p c cosh( B τ ) − k − η √ ε + c sin(2 k − r ) √ ε − p c cosh( B τ ) " p c c cosh( B τ ) (cid:16) η ℓ (cid:17) − sinh( B τ ) (cid:16) c + η ℓ (cid:17) + η c ℓ . (4.42)In order to discuss how those correction terms can beobserved, we shall depict them numerically with the spe-cific values of parameters, which we choose ǫ = 10 − ℓ − , and σ ≥ . ℓ − . (4.43)This inequality guarantees to satisfy the condition 4 ǫ < σ given as (3.20). In Fig.3, we show a modulationfactor of power spectrum (1 + s + s + s ) for differentvalues of σ . It shows that the modulation factor canbe larger in a subhorizon regime: k − ℓ > ∼
1. For a largevalue of c , the wall radius has still stayed to be constant.Namely c > τ = 1. Smaller value of c shows that the modulationfactor is increasing with oscillations in terms of k (e.g.see the plot for c = 0 . c . FIG. 3: We plot a modulation factor of power spectrum (1 + s + s + s ) given in (4.37). We set ǫ = 0 . ℓ − and τ = ℓ for each value of c = ℓσ/ √ ε ± = 0 . In Fig.4, we plot how each correction contributes tothe total modulation of power spectrum. We find thatthe term s can be ignored, while s plays an important FIG. 4: All parameters except for c are same as used in Fig.3.(Top) We plot three contributions of s , s and s for c = 0 . s with total one. Theterm s can give us a good approximation for describing thetotal modulation function. The increasing oscillation appearsaround k − ℓ > role in the superhorizon scale k − ℓ ≤
1. On the otherhand, s plays an important role in the subhorizon scale,which gives an increasing function of k with oscillationsfor k − ℓ >
1. This subhorizon effect comes from the oneproportional to ( k − η ) in (4.42). That is transmittedwave propagating from the exterior universe. Note thatthe contribution of s becomes larger than unity, which2means a breakdown of perturbative approach for ǫ . Themost effective term comes mainly from the one propor-tional to ( k − η ) in (4.42). It gives the condition (4.44).We can then evaluate a breakdown scale k max of our per-turbative treatment for ǫ as13 ǫℓ ( k − η ) < O (1) −→ k max = r ǫℓ c y ( τ )(1 + c ) ℓ . (4.44)For c = 0 .
1, we find a limited maximum value ofwavenumber as k max ≃ . ℓ − , beyond which we cannotuse this formula.Of course, these plots also depend on the parameter ofthe proper time τ when the waves are scattered. Howeverthe plot shows a similar behavior when τ < ℓ , althoughwe set τ = ℓ in Fig.3. For c = 0 .
1, the parameter τ > ∼ ℓ gives no modulation of power spectrum since the wallradius converges to a constant value.In the observational point of view, the observed powerspectrum would be generated by a wave scattered at thewall around which an e -folding number takes about 50 −
60. If an inflationary period is much longer than 60 e -foldings, it maybe difficult to observe such effect on theCMB power spectrum because the observed modulationis generated at τ ≫ ℓ when the wall radius is almostconstant. So we have to fine-tune the inflationary modelwhose period ends around 60 e -foldings. In this case, thevalue of the modulation is determined by the scatteredwaves at the wall position when inflation has just startedand the radius of the wall is expanding. It will show thedeviation from a standard inflationary model.Note that if the initial radius of the wall is much largerthan the horizon scale H − = ℓ , i.e., c = σℓ/ ≪
1, wemay observe the original perturbation before the scatter-ing at large scale. So there may not appear the modula-tion for such a larger scale perturbation.
V. THE EFFECT OF DIFFERENTGRAVITATIONAL CONSTANTS
In the original Jordan frame, the gravitational con-stants are different in both vacua, which is given by theVEV’s of the scalar field Φ by Eq. (2.2). Since the scalarfield φ is given by Eq. (2.17), the value changes from φ − to φ + suddenly near the bubble wall. Hence the grav-itational constants are almost constant both inside andoutside of the bubble except near the wall region. TheVEV’s in a false(+) and true( − ) vacua are written by φ + = bµ, φ − (= φ ) = − µ . (5.1)Then we findΦ + = e (1+ b ) µ/ √ , Φ − = 1 , (5.2)which gives G + = e − b ) µ/ √ G N , G − = G N . (5.3) Hence when we analyze the effect of the different grav-itational constant in a false vacuum, we should study thedependence of b on our results discussed in the previoussection. If one wishes to set almost the same values forboth gravitational constants, one needs to choose a neg-ative value of b such that b ∼ −
1, which implies thatthe value of φ in a false vacuum is close to that in thetrue vacuum ( φ + ). In Fig. 5, we show the gravitationalconstants for two values of b ( b = 1 and 2). FIG. 5: We plot the gravitational constants G for b = 1 and b = 2, by setting µ = √ λ = 1. We have used the radialcoordinate r in the Jordan frame. r i denotes the position ofbubble wall. Since the parameter b is related to the surface densityof the wall σ ( b ) as σ ( b ) = √ λµ b ) , (5.4)which is one of the key parameters in our results, we candiscuss the effect of the different gravitational constantby changing σ . In fact, the ratio of two gravitationalconstants is given by G + G − = exp h − σ √ λµ i . (5.5)Now we study how change of the gravitational con-stant outside of our universe affects on the final resultsobtained in the Einstein frame in the previous section.The effects are found mainly in two points; one is thebubble nucleation rate and the other is the power spec-trum of curvature perturbations, which we shall discussin due order. A. Nucleation rate of the bubble withdifferent gravitational constant
The bubble nucleation rate is given by (2.9) and (2.10),which is described asΓ ≃ exp h − π σ H i , (5.6)3in the limit qof ǫ →
0. This result shows that the largervalue of σ makes tunnelling process suppressed highlyby a huge exponential factor. If σ ≃
0, i.e., b ≃ − µ and λ are fixed, when the value of b changes from − ∞ , the surface tension σ takes the value from 0 to ∞ .It is because the wall barrier height is constant but thewidth becomes narrower as b decreases. When we changethe gravitational constant in a false vacuum, however, ifwe fix σ , i.e., the potential is given by the relation suchthat λµ ( φ + − φ − ) , (5.7)is constant, the nucleation probability does not change.Hence whether the universe with a different gravitationalconstant is plausible or not depends on the choice of thepotential form. B. Curvature perturbations
The power spectrum of curvature perturbations whichreenter inside the horizon of our bubble universe is eval-uated by considering both reflected β and transmitted˜ α waves. The modulation factor of a normalised powerspectrum of curvature perturbations (4.37)-(4.42) is af-fected by changing σ ( b ). In what follows, we assume µ = √ b = 1 with that of b = 2. If the surface density takes thevalue σ = 0 . ℓ − for b = 1, we find σ = 0 . ℓ − for b = 2.The gravitational constant G + for b = 2 of the exteriorof the bubble takes smaller value than the case of b = 1by the factor e − ≃ . b ) becomes threehalves (see Fig.5).For small value of ǫ , which we assume through thispaper, which have been obtained as (4.39)-(4.42), weplot the effect of difference of gravitational constants onthe power spectrum in Fig.6. The modulation factor(1+ s + s + s ), which describes a correction from a stan-dard scale-invariant spectrum, shows oscillatory bumpsat a small scale k − > ∼ /ℓ and its amplitude increases forsmaller k − . It also increases for a smaller value of the pa-rameter σ . By using (4.42), the term such as cos(2 k − r )gives the oscillation period ∆ k ≃ π/r ∝ σ for small σ .Therefore when b becomes a half, the period ∆ k becomesmore sharp by twice.We conclude the effect of different gravitational con-stant on the final power spectrum as follows. The typicalsignature appears on a small scale and it gives an in-creasing oscillation. Especially, if the gravitational con-stants for both sides of the wall is close to each other, i.e., σ ≈ b ≈ − α wave of perturbation in the exterior bubble universe (see (4.34)), and can be constrained by the observationof CMB. FIG. 6: We plot the final power spectrums for σ = 0 . ℓ − and σ = 0 . ℓ − . VI. CONCLUDING REMARKS
We have argued a possible scenario that each universetakes its own different value of gravitational constant inthe context of the emergence of the bubble universes. Inorder to construct a model that makes different gravita-tional constants corresponding to several vacua, we adoptthe JBD theory, with which taking different values of theVEV of the JBD scalar field, the universes with differentgravitational constants are achieved. In particular, weinvestigate the effect of different gravitational constantin the exterior universe on primordial curvature pertur-bations observed in the interior bubble universe.Via a conformal transformation, we can simply analyzethe physics in the Einstein frame, that is the nucleationof bubbles, the evolution of bubble wall and matchingcurvature perturbations of the interior and exterior re-gion. In the Jordan frame, it becomes clear that the fi-nal power spectrum of curvature perturbation is affectedby the bubble boundary through the surface energy den-sity σ of the wall. The key parameter is b , which is b = φ + / | φ − | , where φ ± is the VEV’s of the JBD scalarfield in the interior and exterior universes. The param-eter b is related to the ratio of gravitational constantsas − b + 1) µ/ √ G + /G − ). In Fig. 3, for differ-ent values of σ , we show the modulation of the powerspectrum, which has several peaks with growing oscilla-tion in Fourier space. If the gravitational constant in theexterior universe gets smaller, b or σ increases and thenthe modulation becomes smaller. Choosing a smaller pa-rameter σ will heighten the amplitude of peak shown inthe modulation of power spectrum of curvature pertur-bations.In our result, we assume that the e -foldings of inflation-ary period is about 60, because the initial modulation of4perturbations will not be observed for inflationary modelswith the e -foldings more than 60. In this paper, we havenot discussed after inflation. The result obtained fromperturbations matching at later time such as a radiationdominated era will be more interesting and important,although it is difficult to describe a trajectory of the bub-ble wall in an analytic form. In order to solve dynamicsof wall in this phase, we may need numerical treatment.Along this direction, we hope to make a progress in ourfuture work. Acknowledgments
The work of YT is supported by a Grant-in-Aidthrough JSPS Fellow for Research Abroad H26-No.27.KM would like to thank DAMTP, the Centre for The-oretical Cosmology, and Clare Hall in the University ofCambridge, where this work was started. This work wassupported in part by Grants-in-Aid from the ScientificResearch Fund of the Japan Society for the Promotionof Science (No. 25400276).
Appendix A: domain wall solution and surfaceenergy for small ǫ We consider the following potential V = V + ǫV = 2 λ ( b + 1) ( φ − bµ ) ( φ + µ ) + ǫ ( b + 1) µ ( φ − bµ ) + Λ . (A1)There are three extrema φ + = bµ − ǫ ( b + 1)4 λµ , (A2) φ B = b − µ + ǫ ( b + 1)2 λµ , (A3) φ − = − µ − ǫ ( b + 1)4 λµ . (A4) φ + and φ − correspond to the local minima, while φ B isthe local maximum, at which the potential values are bygiven V ( φ + ) = Λ + O ( ǫ ) , (A5) V ( φ B ) = Λ + λ µ − ǫ O ( ǫ ) , (A6) V ( φ − ) = Λ − ǫ + O ( ǫ ) . (A7)Although φ + can be shifted by changing a free parameter b , the potential barrier and the minimum values do notdepend on b .Now we find a domain wall solution for this potential V . Plugging φ = φ ( r ) + ǫφ ( r ) into the basic equation and expanding it up to the first order of small parameter ǫ , we find the following equations: φ ′′ + 1 r φ ′ − dV dφ ( φ ) = 0 , (A8) φ ′′ + 1 r φ ′ − d V dφ ( φ ) φ − b + 1) µ = 0 . (A9)By using thin wall approximation, Eq. (A8) gives the0-th order domain wall solution as φ = ( b − µ + ( b + 1)2 µ tanh (cid:18) r − r i d (cid:19) . (A10)Using this solution, we find d V dφ ( φ ) = 4 λµ ( b + 1) " −
32 cosh (cid:0) r − r i d (cid:1) . (A11)If we ignore the deviation form unity near r = r i thisterm can be treated as a constant. Then we find anapproximate solution for φ as φ = − ( b + 1)4 λµ " − Dd cosh (cid:0) r − r i d (cid:1) , (A12)where D is an arbitrary constant.This solution φ = φ + ǫφ can be rewritten as φ = ( b − µ − ǫ ( b + 1)4 λµ + ( b + 1)2 µ tanh (cid:18) r − r i + δr ǫ d (cid:19) , (A13)where δr ǫ = ǫλµ D . This domain wall solution is justshifted from one with ǫ = 0 by − ǫ ( b +1)4 λµ and − δr ǫ in φ -and r -directions, respectively. The domain wall structureitself does not change.Next we shall calculate the surface energy density,which is defined by σ = Z ∞ dr (cid:20)
12 ( ∂ r φ ) + V DW ( φ ) (cid:21) , (A14)where V DW is the part of the potential which contributesto the structure of a domain wall.Plugging the above solution (A13) into the kinetic termand potential, we find up to the first order of ǫ as12 ( ∂ r φ ) = λµ (cid:0) r − r i + δr ǫ d (cid:1) , (A15) V ( φ ) ≈ λµ (cid:0) r − r i + δr ǫ d (cid:1) + ǫ tanh (cid:0) r − r i d (cid:1) (cid:0) r − r i d (cid:1) + ǫ (cid:20) tanh (cid:18) r − r i d (cid:19) − (cid:21) + Λ . (A16)The potential V contains a back ground bulk energy, i.e.,cosmological constants; V bulk = Λ − ǫ − sgn( r − r i )] . (A17)5where the sign function sgn( x ) is defined bysgn( x ) = (cid:26) x > − x < . (A18)Since the bulk energy should not be included in the sur-face energy of the wall, we define V DW := V ( φ ) − V bulk = λµ (cid:0) r − r i + δr ǫ d (cid:1) + ǫ tanh (cid:0) r − r i d (cid:1) (cid:0) r − r i d (cid:1) + ǫ (cid:20) tanh (cid:18) r − r i d (cid:19) − sgn( r − r i ) (cid:21) . (A19)In the thin-wall approximation ( d ≪ r i ), the surface en- ergy σ is evaluated approximately as σ ≈ Z ∞−∞ dr (cid:20)
12 ( ∂ r φ ) + V DW ( φ ) (cid:21) = λµ Z ∞−∞ dr cosh (cid:0) r − r i + δr ǫ d (cid:1) , (A20)which gives the same result as the case with ǫ = 0, i.e., σ ≈ √ λµ ( b + 1)3 . (A21)It is plausible because the domain wall structure does notchange by the small ǫ -modification of the potential. [1] L. Susskind, In Universe or multiverse? ed by B. Carr,pp247-266, hep-th/0302219.[2] S. R. Coleman and F. De Luccia, Phys. Rev. D , 3305(1980).[3] K. Sato, Mon. Not. Roy. Astron. Soc.
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