Boundary Effects in Local Inflation and Spectrum of Density Perturbations
aa r X i v : . [ g r- q c ] N ov Boundary Effects in Local Inflationand Spectrum of Density Perturbations
Erol Ertan ∗ and Ali Kaya , † Bo˜gazi¸ci University, Department of Physics,34342, Bebek, ˙Istanbul, Turkey Feza G¨ursey Institute,Emek Mah. No:68, C¸ engelk¨oy, ˙Istanbul, Turkey (Dated: November 6, 2018)We observe that when a local patch in a radiation filled Robertson-Walker universe inflates bysome reason, outside perturbations can enter into the inflating region. Generally, the physicalwavelengths of these perturbations become larger than the Hubble radius as they cross into theinflating space and their amplitudes freeze out immediately. It turns out that the correspondingpower spectrum is not scale invariant. Although these perturbations cannot reach out to a distanceinner observer shielded by a de Sitter horizon, they still indicate a curious boundary effect in localinflationary scenarios.
I. INTRODUCTION
A remarkable property of the observed universe is itslarge scale homogeneity and isotropy. For instance, thetemperature of the cosmic microwave background (CMB)is isotropic one part in 10 across the sky. This smooth-ness appears to contradict with a generic big-bang sincein that case one would expect to see imprints of differentexotic objects on CMB. In particular, white holes shouldhave been created in big-bang because a common big-crunch necessarily contains black-holes and big-bang canbe viewed as a time reversed big-crunch. An importantopen problem in modern cosmology is to explain why theuniverse appeared in this very special state.It is generally claimed that the standard cosmologicalmodel cannot explain the isotropy of CMB, since in thismodel most of the cosmic photons we observe today orig-inates from causally disconnected regions in space andthus thermalization cannot take place to yield a uniformtemperature. A plausible way to solve this difficulty (andothers like flatness and monopole problems) is to modifythe standard model by assuming an early period of accel-erating expansion, called inflation. Although the detailsof how the expansion is achieved depend on the modelone considers, in general there is a scalar field whose po-tential energy acts like an effective cosmological constantyielding a de Sitter phase. By the huge exponential ex-pansion encountered in this period all inhomogeneitiesare smoothed out and the spatial sections are flattenedout. The temperature drops enormously at the end of in-flation. However, following a reheating process the tem-perature again raises and one ends up with a radiationdominated universe. ∗ [email protected] † [email protected] Since in a cosmological model without inflation causal-ity precludes thermalization, it seems impossible to jus-tify special properties of CMB. However, as pointed outby Penrose (see e.g. [1]), it is paradoxical to view thepresent specialness as a feature that should be explainedand use thermalization as the main mechanism to ac-count for it, since this already means that the universeshould have been more special in the past. Namely, oneshould try to understand possible cosmic origins of thesecond law of thermodynamics instead of using it in ex-plaining the current state of the universe. Furthermore,without understanding Planck scale physics it is impos-sible to determine the degree of anisotropy in the ab-sence of inflation. Therefore it is not possible to viewthe observed isotropy as a feature contradicting with thestandard model.Although inflation is proposed as a model explainingthe homogeneity and isotropy of the observed universe,one usually assumes a homogeneous and isotropic dis-tribution of a scalar field (having a suitable potential) torealize it. To resolve this conflict one supposes that infla-tion occurred in a homogeneous and isotropic local patchin a larger (radiation dominated) Freedman-Robertson-Walker-Lemaitre (FRWL) spacetime. One then wonderswhether inflation is ’natural’ in this set up. In some cases,it is possible to prove a cosmic no hair theorem whichasserts that a positive cosmological constant eventuallydominates the cosmic evolution irrespective of initial con-ditions and matter content [2, 3] (see also [4]). However,this theorem does not say too much about feasibility oflocal inflation which is not derived by a pure cosmologicalconstant. Although chaotic inflation appears to be an at-tractive point in simple scalar field models with suitablepotentials [5], the issue of initial conditions for inflationis not yet settled (for a review see e.g. [6]). As illustratedin [7], without having a natural measure in the space ofinitial conditions, it is impossible to assign a probabil-ity for inflation. Remarkably, a measure which yields anexponentially suppressed probability has been proposedrecently in [8]. Moreover, naive arguments indicate thatthe entropy of the pre-inflationary patch is much lowerthan a patch with a typical big-bang, which shows thatamong randomly chosen initial conditions the probabilityof seeing inflation should be negligible (see e.g. [9]). Onthe other hand, there are some obstructions in embeddinga ”small” inflating region into a FRWL universe imposedby the propagation of null geodesics [10]. A conceivableway to remedy this obstruction is proposed in [11].Even though the original motivations mainly arise fromthe ”shortcomings” of the standard model, one of themain successes of the inflationary paradigm turns out tobe its ability to produce a scale free spectrum of densityperturbations appropriate for the formation of structurein the universe. According to the standard picture, den-sity perturbations originate from quantum fluctuations,which are spontaneously created in de Sitter space andamplified over superhorizon scales. In this paper, wepoint out that in local inflationary models there also ex-ists (quantum) perturbations which inevitably enter intothe inflating region from outside, whose power spectrumis not scale invariant. These perturbations can spreadout a horizon distance from the boundary and they mayinfluence structure formation in that region.The plan of the paper is as follows. In the next sec-tion, we state our main argument. In section III, we ar-gue that due to the causal structure of local embeddingsome outside perturbations inevitably enter into the in-flating region and we illustrate how our main argumentis realized in the thin-shell approximation. In section IV,we make a further check of scale dependence by match-ing the interior and the exterior modes analytically. Webriefly conclude in section V.
II. THE BASIC ARGUMENT
Let us start by summarizing the well-known mecha-nism for the generation of scale free spectrum of densityperturbations in inflation. Consider a scalar field φ obey-ing the Klein-Gordon equation ∇ φ = 0 in a spatially flatRobertson-Walker spacetime ds = − dt + a ( t ) (cid:0) dx + dy + dz (cid:1) . (1)For a plane wave mode characterized by a comovingwavenumber k there are two important propagation lim-its. When the proper wavelength a/k is much smallerthan the Hubble radius R H = 1 /H , the mode will be-have like an ordinary harmonic oscillator with negligibledamping. In that case, the ground state of the oscillatoris a Gaussian wave function with spread given by (seeappendix A) ∆ φ k = 12 a k . (2)On the other hand, when the wavelength a/k is muchlarger than R H , the oscillator is overdamped and the fluctuation amplitude ∆ φ k freezes out.During inflation R H is constant but the physical wave-lengths increase exponentially. Therefore, a mode canevolve from an underdamped oscillator to an overdampedoscillator, but the opposite is not possible. One assumesthat throughout inflation quantum modes were born intheir ground states with wavelengths much smaller thanthe Hubble radius and evolve adiabatically with thespread given by (2) until their wavelengths become equalto R H . For a mode with wavenumber k this happenswhen a/k = R H and ∆ φ k = H / (2 k ). Later on, the os-cillator becomes overdamped, the wavelength leaves thehorizon and the amplitude freezes out. The correspond-ing power spectrum P ( k ) ≡ k ∆ φ k is independent of k ,i.e. P ( k ) is scale free. After inflation ends, the Hubbleradius grows more rapidly than the scale factor a , andthus the wavelengths created during inflation reenter thehorizon eventually.Let us point out that a superhorizon perturbation isstill propagating even though its amplitude freezes out.Consider, for instance, a mode in de Sitter space whosemetric in conformal time t c < ds = ( Ht c ) − (cid:0) − dt c + dx + dy + dz (cid:1) . (3)It is well known that the exact solution of ∇ φ = 0 fora plane wave perturbation can be written as (see e.g.chapter 7 of [12]) φ k = H √ k (cid:20) − t c + ik (cid:21) e i ( − kt c ± ~k.~x ) , (4)where the overall normalization is fixed by the harmonicoscillator commutation relation˙ φ ∗ k φ k − ˙ φ k φ ∗ k = i ( Ht c ) . (5)Note that φ k = σ k ( − Ht c ) where σ k is the canonical har-monic oscillator excitation. The subhorizon and super-horizon limits correspond to k | t c | ≫ k | t c | ≪ ∇ φ = 0. Therefore, a mode being frozen outdoes not mean it is non-propagating, this simply signifiesthat its amplitude becomes time independent.In visualizing local inflation, one usually imagines thatthe inflating region is exponentially expanding into theambient spacetime. Although it might be possible to pic-ture inflation using physical coordinates, this may createsome confusion in subtle considerations. It is a lot safer toconsider comoving coordinates in describing cosmologicalphenomena. In these coordinates the region itself doesnot necessarily expand. Inflation is realized by an expo-nentially growing metric in a (possibly fixed) comovingregion.Another crucial point is that in general the inflatingpatch cannot be considered as a closed system. Although t r Σ r S Inflation radiation t FIG. 1: A spacetime sketch of local inflation in comoving co-ordinates . The initial conditions at time t is suitable for aspherical region Σ of radius r to inflate in a radiation dom-inated FRWL space. Both regions are separated by the hy-persurface S . in some cases there may form ”isolated” inflating uni-verses [17], we will see in the next section that even inthis situation the inflating region is not causally discon-nected. Therefore, some exterior information can pene-trate inside. Indeed, one would expect a tendency of flow-ing from the surrounding region having positive pressureinto the inflating region with negative pressure (see e.g.[13]). It is even possible for this flow to stop inflation,however we will assume that this is not the case.Consider, for instance, a radiation dominated FRWLuniverse. Let us assume that at time t a local sphericalregion Σ of radius r starts inflating for some reason (seefigure 1), where r and t are comoving coordinates. Let S be the hypersurface that separates the inflating regionfrom radiation dominated space. In general S can be athick hypesurface, i.e. a four dimensional submanifold,but we assume that a thin wall approximation is valid.Denoting the scale factors as a I and a R , respectively, onecan normalize coordinates such that a I ( t ) = a R ( t ) = 1and thus a I = e H ( t − t ) , a R = (cid:18) tt (cid:19) / , (6)where H is the Hubble parameter in the inflating region.The corresponding Hubble radii are R I = 1 /H and R R =2 t . Causality requires that r should be smaller thanthe Hubble radius of the FRWL space at time t , i.e.2 t > r . Moreover, it is known that for stability vacuumenergy should dominate over a region larger than theinflationary horizon, r > /H (see e.g. [10]). In any casethere is no reason to expect a large hierarchy betweenthese three scales, i.e.2 t ∼ r ∼ H . (7)Let us think about a (quantum) scalar perturbationthat appeared at time t in the FRWL space whose phys-ical wavelength λ p is smaller than the Hubble radius2 t > λ p . (8) FIG. 2: Left moving radial null rays of de Sitter (thick line)and radiation dominated FRWL space (thin line) drawn in thesame comoving coordinates in units where H = 1. Initiallythe Hubble constants are chosen to be equal to each other. This can be represented as an underdamped harmonicoscillator that was born in its ground state whose wave-function has the spread∆ φ k = 12 a R k , (9)where k is the corresponding comoving wavenumber k = a R /λ p . The mode can be assumed to evolve adiabaticallywith (9) in the geometric optics limit, i.e. on the nullrays of FRWL space. Moreover it cannot turn into anoverdamped oscillator since the Hubble radius is growingfaster than the physical wavelength.Assume now that (a portion of this) mode (containinga single pulse of one wavelength) enters into the inflatingspace at time t ∗ . Since we are using same comoving co-ordinates in both regions, the comoving wavenumber k does not change. Therefore, the physical wavelength ofthe mode as it enters into the inflating region is amplifiedby λ ∗ = a I ( t ∗ ) a R ( t ∗ ) λ p . (10)From (7) and (8), we see that there is no hierarchy be-tween λ p and 1 /H . Therefore, unless λ p is exponentiallysmall compared to 1 /H , (10) implies that λ ∗ ≫ H . (11)Consequently, this mode should freeze out in the inflatingregion. From (9), the fluctuation spectrum can be foundas ∆ φ k = 12 a R ( t ∗ ) k , (12)which acquire a scale dependent power spectrum with P ( k ) ∼ k .Although in the above argument we assume that themode enters into the inflating region at a single time t ∗ and amplified immediately, this actually happens in atime interval proportional to the wavelength of the mode(see next section). Therefore, it is important to notethat the amplification (10) does not occur immediately.Moreover, as it will be seen in the next section, it isnot always possible to use same comoving coordinates inboth regions. As a result, the comoving wavenumber mayalter during crossing. As we will show, however, thesecomplications do not change the main conclusion thatoutside perturbations acquire a scale dependent powerspectrum.As these perturbations propagate into the inflating re-gion, they can spread out a comoving radial distance 1 /H from the boundary, which is the inflationary horizon dis-tance (this is due to the behavior of the radial null linesin de Sitter space, see e.g. figure 2). Therefore, theycannot reach out an observer whose comoving distanceis larger than 1 /H from the boundary. In models where r ≫ /H , the probability of seeing a typical observer(us) near the boundary is small and thus these modeswould most likely be invisible to us to produce an obser-vational effect. III. A TOY MODEL AND THE CROSSING OFA SCALAR PERTURBATION ALONG THEBOUNDARY
As pointed out above, since the inflating region hasnegative pressure as oppose to the exterior with posi-tive pressure, there should be a tendency of flowing intothe inflating patch. This tendency can easily be seen bycomparing radial null lines of each space pictured in thesame comoving coordinates; it turns out that (when theHubble constants are comparable) the null curves in deSitter space is much more vertically aligned comparedto the ones in a radiation dominated spacetime (see fig-ure 2). Therefore, information spreads a lot faster in theradiation region.Whether outside perturbations can enter into the in-flating space or not depends on the causal structure ofthe hypersurface S separating two regions. To determine S one should study the coupled dynamics of the metricand the scalar field driving inflation. This problem canbe addressed in the thin-shell approximation [14] and in-deed the evolution of an inflating region (the false vacuumbubble) in a cosmological background has been studiedwell in this context (see e.g. [15–18]). In the thin-shellapproximation, two spaces are glued over the boundary S describing the history of the bubble, which is a time-like hypersurface in both regions. The induced metric istaken to be continuous over S and the difference of theextrinsic curvatures is equal to the energy momentumtensor on the shell.Consider, for instance, a well known example from [17],where the evolution equations describing a spherical falsevacuum bubble in a true vacuum region are explicitlysolved. It is argued in [17] that at least for a range ofinitial conditions there forms ”isolated” closed inflatinguniverses in this setup. A solution, as seen by an out- r=0 r=0IV FIG. 3: The sketch of a solution for S given in [17] as seenby an outside observer, which yields an ”isolated” inflatinguniverse on the left. side observer, is pictured in figure 3. The outside regionshould uniquely be described by the black hole metricas given by the Birkhoff’s theorem. Here, we see thatthe boundary is formed by a timelike curve starting fromthe past singularity at r = 0 and extending through-out future null boundary. As pointed out in [17], if theevolution is foliated on suitable constant time hypersur-faces, one sees a closed inflating universe that is detach-ing. However, it is clear that the inflating region is notcausally disconnected from the black hole region. Indeedall left moving perturbations created in region IV in thefigure 3 enter into the de Sitter space and according tothe argument presented in the previous section they ac-quire a scale dependent power spectrum.Since in the thin-shell formulation S is always takento be a timelike hypersurface, it is inevitable that someexterior perturbations (propagating along null geodesics)enter into the inflating space. To explain how the mainmechanism discussed in the previous section works in thiscontext, we construct an explicit toy example in the ap-pendix B, where an inflating region and a flat Minkowskispace are glued over the worldline of a spherical bubble.The line element in each patch is taken as ds = − dt + a ( t ) (cid:0) dr + r d Ω (cid:1) , (13)where a = 1 in the flat space and a = e Ht in the inflatingregion. For a spherically symmetric configuration, thetwo spheres in each region can be identified by the bub-ble itself. However, r and t coordinates are in generaldiscontinuous. From appendix B (see (B20) and (B23))we quote the equations describing the boundary: in theflat space S is given by r = 1 α cosh( ατ ) , t = 1 α sinh( ατ ) , (14) A critical objection to this picture is raised in [19] by indicatingthat the solution involves the past of a bifurcating Killing horizonwhich is known to be unstable. The stability issue in the thinshell formulation of false vacuum bubbles is also considered in[20]. t r
FIG. 4: The worldline of the bubble boundary as seen by anobserver in the flat space. and in the inflating region it reads r = r cosh( ατ ) | sinh( ατ ) + H − σ Hσ | , (15) Ht = ln( | sinh( ατ ) + 4 H − σ Hσ | ) − ln( αr ) . Here r is an integration constant, σ > α is given by α = H σ + σ , (16)and τ is the proper time. To avoid any complication onecan assume that H , σ and thus α have the same order ofmagnitude.As pointed out in the appendix B, to solve the junctionconditions in this simplified set up one should identify flatspace as the interior and de Sitter space as the exteriorpatches. Therefore, these equations actually describe theevolution of a true vacuum bubble in an inflating space.The flat space trajectory (14) corresponds to the world-line of a particle moving with constant acceleration α equivalent to a hyperbola (see figure 4). To understandthe bubble solution in de Sitter space better, we notethat there is a singularity at τ = τ s , where the denomi-nator of (15) vanishes. One should restrict τ > τ s sincein that range t ′ >
0, where prime denotes differentiationwith respect to τ . Asymptotically as ατ → ∞ , r → r .The sign of r ′ is always negative when 4 H − σ ≤
0. If4 H − σ >
0, there is a turning point after which r ′ > λ in flat space φ k = 1 √ k e ik ( r − t ) r , (17) t rr FIG. 5: The worldline of the bubble boundary as seen by anobserver in de Sitter space when 4 H − σ ≤ t rr FIG. 6: The worldline of the bubble boundary as seen by anobserver in de Sitter space when 4 H − σ > where k = 2 π/λ . When a single pulse of one wavelengthextending in between the null interval r − t and r − t + λ reaches the boundary (see figure 7), it overlaps with aportion in the proper time interval ( τ, τ + ∆ τ ). We as-sume that ∆ τ ≪ τ , i.e. the crossing time is small com-pared to the whole history. Therefore, the approximatetime of passing can be defined as τ ∗ = τ +∆ τ /
2. To makesure that the inflating region already expanded enough,we focus on the perturbations entering at late times, i.e. e ατ ≫
1. Using (14), one finds that λ = 2 α e − ατ ∗ sinh( α ∆ τ / . (18)Given any τ ∗ this equation relates ∆ τ to λ .We would like to match this pulse with a sphericallysymmetric perturbation in de Sitter space. The modewhich has a comoving wavenumber ˜ k can be written as φ ˜ k = A H p k e i ˜ k ( r − t c ) r (cid:20) − t c + i ˜ k (cid:21) , (19)where t c is the conformal time defined by Ht c = − e − Ht . (20)Since this is not an oscillator mode spontaneously cre-ated in de Sitter space, there is a freedom in normaliza-tion which is indicated by the unknown constant A in(19). Using (15), the comoving wavelength ˜ λ of a pulseextending in between r − t c and r − t c + ˜ λ can be relatedto τ ∗ and ∆ τ as (see figure 8)˜ λ = 4 αr H e − ατ ∗ sinh( α ∆ τ / . (21)Comparing now (18) and (21), we obtain k = 2 α r H ˜ k. (22)From (22), it is easy to see that if the wavelength in theflat space obeys λ > πα e − ατ ∗ , (23)the corresponding mode in de Sitter space satisfy˜ k | t c ( τ ∗ ) | <
1. Therefore, these excitations become super-horizon perturbations in the inflating region. Matchingthe interior and exterior scalar modes up to phase on theboundary one further finds A = r r H e − ατ ∗ ˜ k. (24)Due to this ˜ k dependence, (19) is not canonically nor-malized and the overall amplitude gives ∆ φ k ∼ / ˜ k . Thecorresponding power spectrum becomes scale dependent P (˜ k ) ∼ ˜ k , which is consistent with the results obtainedin the previous section.At this point one may worry that the amplitude (24)is exponentially suppressed. However, this suppressionarises from 1 /r dependence of (17), i.e. this behavioris specific for a spherical wave created at the origin anddoes not indicate a generic feature. It is also interestingto note that (18) and (23) impliessinh( α ∆ τ / > π. (25)This shows there is a minimum time width of boundarycrossing for perturbations which become superhorizon inthe inflating region.In general, one has to include a scattered left movingwave in flat space corresponding to reflection from theboundary (see next section). The reflection can be ig-nored if the radius of curvature of the boundary (in ourcase 1 /α ) is much larger than the wavelength λ . From(23) we see that for e ατ ∗ ≫ λ ≪ /α . Thus for thesemodes the reflection can be neglected and the above con-clusions must hold. IV. ANALYTICAL MATCHING
In the previous section, in matching the interior andthe exterior modes over the boundary, we have simplyconsidered a pulse of one wavelength in both regions. Al-though this has been sufficient to illustrate our argument, t r τ τ +∆ ττ * λ FIG. 7: A right moving spherical perturbation of wavelength λ crossing the hypersurface S . ! rr ττ * τ +∆ τ λ ~ FIG. 8: The perturbation in figure 7 exits as a spherical waveof comoving wavelength ˜ λ in de Sitter space. to be more precise, one should actually glue the whole so-lutions. To address this problem, we first note that theKlein-Gordon equation can be expressed in terms of thederivative operator D µ of the induced metric on S andthe normal vector n µ as ∇ φ = D φ + Kn µ ∂ µ φ + n µ ∂ µ ( n ν ∂ ν φ ) − ( n µ ∇ µ n ν ) D ν φ. (26)Since K is discontinues but does not involve a delta func-tion, this equation implies that two fields φ − and φ + ,which are obeying ∇ φ ± = 0 in respective regions, canbe glued on the boundary if they satisfy φ − = φ + , (27) n µ ∂ µ φ − = n µ ∂ µ φ + . (28)We now use these junction conditions to study the cross-ing of a perturbation.In our case, there is an incoming wave which is theproperly normalized, right moving, spherical perturba-tion φ in = 1 √ k e ik ( r − t ) r . (29)The reflected and the transmitted modes can be taken asthe general, spherically symmetric, left and right movingwaves in flat and de Sitter spaces, respectively. Thesecan be written as φ ref = g (1 / ( r + t )) r , φ tr = f ′ ( r − t c ) t c r + f ( r − t c ) r , (30)where f, g are arbitrary functions and prime denotes dif-ferentiation with respect to the argument. The generalsolution in de Sitter space can be obtained from (19) bysuperposition. Two conditions (27) and (28) can be usedto determine two unknown functions f and g . Note that φ − = φ in + φ ref and φ + = φ tr .We consider the late time matching of the solutions, i.e.we assume e ατ ≫
1. From (14) and (15), the componentsof the normal vector n µ can be found in flat space as n t = n r = e ατ , (31)while in de Sitter space they read n t = 4 H − σ Hσ , n r = 2 r α H e − ατ , (32)where we only keep the leading order contribution in anexpansion in e − ατ . We define a (dimensionfull) variable x ≡ αe − ατ . In units where α = 1 we have x ≪
1. Tofirst order in x , the continuity of the modes (27) implies r k x e ikx/α + 2 x g = 1 r f − H xf ′ , (33)where prime denotes differentiation with respect to theargument and g = g ( x ) , f = f ( r + 2 r H x ) . (34)On the other hand, the continuity of normal derivatives(28) gives i √ kα x − α √ k ! e ikx/α − αxg ′ − αg = − r σH xf ′′ + 2 αH f ′ − αr H f. (35)Solving g from (33) and using it in (35) we obtain thefollowing second order equation for f :2 r ( σ + 2 α ) H xf ′′ − αH f ′ + 2 αr H f = α √ k − i √ kα x ! e ikx/α . (36)The right hand side of this equation is the source termrelated to the incoming wave.It is possible to solve (36) exactly. The homogenoussolution involves two Bessel functions and the particularsolution can be obtained by variation of parameters. Weneed to keep the unique particular solution supported by the source incoming wave. However, (36) has correctionsin powers of x , i.e. both the functions multiplying f andthe source in the right hand side are just the leadingorder contributions. Therefore, only the first term in theexpansion of the exact solution in x can be trusted.Since the source term in (36) is oscillating, the solutioncan be written as f = e ikx/α F ( x ), i.e. f will oscillateexactly the same way as the source. From (34) we thenfind f ( y ) = e ikHy/ (2 α r ) ... , (37)thus the comoving wavenumber ˜ k in de Sitter space canbe identified as ˜ k = kH/ (2 α r ), which is precisely therelation (22) derived in the previous section.Let us now determine the solution of (36) in the shortand long wavelenght limits. To avoid any complicationwe assume that H , σ (and thus α ) and 1 /r have thesame order of magnitude.We start with the short wavelength limit kx/α ≫ f = A e ikx/α (1 + O ( x )) . (38)In (36), f ′′ term becomes much larger than the two otherterms in the left hand side and the constant A can befixed as A = i α r ( σ + 2 α ) 1 k √ k . (39)Here k dependence is crucial. Using f in (30), we find φ tr = " αe − iHk/ α ( σ + 2 α ) H √ k e i ˜ k ( r − t c ) r (cid:18) − t c + i ˜ k (cid:19) . (40)The constant in the square brackets is an order unity con-stant. Comparing with (4) and nothing that ˜ k and k arenearly equal to each other (recall that we assume H , α and 1 /r have the same order of magnitude), we see thatthe transmitted wave is a properly normalized mode inde Sitter space. Indeed, in the limit that we consider 1 / ˜ k term can be neglected compared to t c in the parentheses,which shows that this is a subhorizon mode. Therefore,perturbations which have short wavelengths in flat spacecan be matched by subhorizon modes in the inflating re-gion which acquire a scale free power spectrum. This isconsistent with the results of the section II since the am-plification (10) is not enough to push the physical wave-lengths of these modes out of the inflationary horizon.Consider now the long wavelength limit kx/α ≪ f = A (1 + O ( x )) . (41)The amplitude can be fixed from (36) as A = r H/ √ k which yields a spherical, superhorizon mode in de Sitterspace φ tr = r H √ kr . (42)However the normalization is different than the normal-ization of the canonical superhorizon perturbation, whichis 1 /k √ k . This gives a scale dependent power spectrumwith P ( k ) ∼ k , which is consistent with the results pre-sented in the previous sections. V. CONCLUSIONS
Inflation offers a successful way of generating a scalefree spectrum of density perturbations consistent withcosmological observations. According to the generalview, all classical inhomogeneities are smoothed out bythe exponential expansion, leaving only room for quan-tum fluctuations. Naturally, quantum modes are as-sumed to be born as harmonic oscillators in their groundstates, which have physical wavelengths smaller than theinflationary horizon. The wavelengths are then pushedto superhorizon sizes by expansion and their amplitudesfreeze out. After inflation ends, they reenter the horizonin a radiation dominated phase as classical perturbationsseeding cosmic structure.The success of the above mechanism profoundly de-pends on the very special properties of the generationand the propagation of quantum fluctuations in de Sit-ter space. Indeed, even in de Sitter space the well knownambiguity in the choice of an invariant vacuum may altersome of the predictions of the theory (see e.g. [22]). Inthis paper, we point out that in local inflationary modelsthe causal embedding of an inflating patch into an ambi-ent spacetime generally allows outside perturbations toenter into the inflating region. It is clear that these per-turbations may not have a scale-free power spectrum.In this paper we show that quantum fluctuations en-tering into a local inflating patch have a scale dependentpower spectrum. In section II, we first give a generalargument and in sections III and IV we illustrate it byan explicit toy example in the thin-shell approximation.Although the set up studied in this context is artificialand cannot be considered as a part of a realistic scenario,it nicely touches to the salient points raised in section II.It would be interesting to construct a numerical or anasymptotic false vacuum bubble solution in a radiationdominated FRWL space and verify that scale freeness isspoiled for the quantum modes crossing into the inflatingregion, as suggested by the basic argument of section II.One may think that the above conclusions are avoidedin models where the inflating patch detaches from theambient spacetime. We consider one such example from[17] and indicate that the patch is still causally connectedto the surrounding region. Indeed, in the thin-shell ap-proximation causal detachment is impossible since theboundary in between the two regions is taken to be atimelike hypersurface in both sides. The key issue is then to determine whether causal dis-connection is possible in local inflationary models at all.One of the nicest features of inflationary paradigm is thatit uses classical general relativity in determining the evo-lution of the metric and thus its predictions are not sensi-tive to the quantum gravitational corrections. In generalrelativity, on the other hand, causal detachment impliesthe existence of a real event horizon. To our knowledgeblack holes are only examples where observer indepen-dent causal separations take place. In some models usingchaotic inflation, it is argued that after big bang (or afteruniverse emerges from a spacetime foam) regions suitablefor inflation expands and others recollapse. In such localmodels it seems that the above conclusions are avoided.However, it is not clear to us how this picture can beverified without understanding the end point of a col-lapse, which requires a detailed knowledge of quantumgravity. Indeed, even in semiclassical gravity black holesradiate energy and thus it appears that a complete causaldetachment is not possible.Finally, these findings do not imply a disagreementwith observations in local inflationary models. To claima discrepancy, one should first make sure that the con-tribution of outside quantum perturbations to structureformation is not negligible. Moreover, as pointed outpreviously, depending on the size of the initial inflatingpatch and the location of the observer inside, the out-side perturbations may not be able to reach the observersince in de Sitter space there are particle horizons and anobserver is shielded by a horizon. In any case, our resultsindicate that the boundary effects should be consideredas an issue in local inflation.
APPENDIX A: SCALAR FIELD QUANTIZATIONIN A FRWL SPACE
In this appendix we review the quantization of a real,massless scalar field in a FRWL spacetime with the met-ric ds = − dt + a ( t ) (cid:0) dx + dy + dz (cid:1) . (A1)The canonical action can be taken as S = − Z √− g g µν ∂ µ φ ∂ ν φ, (A2)which gives ∇ φ = 0. Since in a free field theory differentFourier modes decouple, we consider a single excitationwith a fixed comoving wave-vector ~k : φ = φ k ( t ) e i~k.~x + φ † k ( t ) e − i~k.~x , (A3)where the field equations imply¨ φ k + 3 H ˙ φ k + k a φ k = 0 . (A4)From (A1) and (A2) the momentum conjugate to φ be-comes P φ = a ˙ φ . The quantization can be achieved byimposing [ φ, P φ ] = i, φ k | > = 0 , (A5)where | > is the vacuum as seen by a comoving observer.For a subhorizon mode k/a ≫ H the friction term in(A4) can be ignored and an approximate solution for φ k can be found as φ k = a k e − ikt/a . (A6)The commutator in (A5) then implies[ a k , a † k ] = 12 ka . (A7)Therefore, a k and a † k are creation and annihilation oper-ators and the system is equivalent to the harmonic oscil-lator. One can easily calculate the two-point function orthe spread of the Gaussian ground state wave-functionas ∆ φ k ≡ < | φ | > = 12 a k . (A8) APPENDIX B: DERIVATION OF THE BUBBLESOLUTION
The aim of this appendix is to obtain an explicit equa-tion for the hypersurface S which is separating a flatMinkowski space from an inflating region. We start byconsidering the actual problem, i.e. a false vacuum bub-ble in a FRWL space. The metrics in both regions canbe written as ds ± = − dt ± + a ± ( t ± ) (cid:0) dr ± + r ± d Ω (cid:1) , (B1)where + and − denote the exterior and interior patches,respectively. For a spherical bubble one can identify theinterior and the exterior spheres with the bubble itself.However, t and r coordinates are in general discontinuousover S and they should be differentiated in both regions.The third coordinate on S can be chosen as the propertime τ . The hypersurface will be completely specifiedwhen the trajectories t ± ( τ ) and r ± ( τ ) are solved. Byspherical symmetry, the induced metric h ij on S , where i, j indices refer to the coordinates ( τ, θ, φ ), can be writ-ten as dh = − dτ + R ( τ ) d Ω . (B2)In the thin-shell approximation h ij is assumed to be con-tinuous over S . Note that R ( τ ) is the proper radius ofthe bubble at time τ . If one is only interested in the in-trinsic properties, it would be enough to determine R ( τ ).However, we are also concerned with the embedding ofthe bubble into the ambient spacetime. By Einstein’s field equations the discontinuity of theextrinsic curvature can be related to the energy momen-tum tensor on the hypersurface S ij . Moreover, the fourdimensional energy-momentum conservation also impliesa conservation equation for S ij . One can show that theserespectively yield the following equations for the bubble(known as junction conditions),[ − K ij + Kh ij ] + − = S ij , (B3) D i S ij = (cid:2) − T jµ n µ (cid:3) + − , (B4)where D i is the covariant derivative of h ij .In either region, the tangent vector v µ correspondingto the derivative operator ∂ τ can be calculated as v = t ′ ∂ t + r ′ ∂ r , (B5)where prime denotes differentiation with respect to theproper time τ . By definition v µ v µ = −
1. The two tan-gent vectors on the sphere together with v µ form a basesin the tangent space of S . This information is enough todetermine the unit normal vector n µ as n = (cid:0) a r ′ ∂ t + a − t ′ ∂ r (cid:1) sign( t ′ ) , (B6)where the sign function is inserted to make sure that n µ ispointing from inside out, i.e. in the growing r direction.Since S is now uniquely specified, it is straightforward tocalculate the extrinsic curvature which has components K ττ = (cid:18) ar ′ t ′′ − at ′ r ′′ + a dadt r ′ − dadt t ′ r ′ (cid:19) sign( t ′ ) ,K θθ = (cid:18) a r dadt r ′ + art ′ (cid:19) sign( t ′ ) (B7) K φφ = K θθ / sin θ, where the last relation follows from spherical symmetry.For a spherical false vacuum bubble associated witha minimally coupled scalar field, the energy momentumtensor on the hypersurface can be assumed to have theform [17] S ij = − σh ij . (B8)Thus in our problem there are five dynamical fields ofinterest which are t ± ( τ ), r ± ( τ ) and σ ( τ ). The fact that τ is the proper time gives two equations t ′ ± − a ± r ′ ± = 1 . (B9)The continuity of the induced metric yields R = a r = a − r − , (B10)where R is defined in (B2). The conservation of stressenergy tensor (B4) can be satisfied provided σ ′ = [ a t ′ r ′ ( ρ + P )] + − (B11)0Finally ( θθ ) component of (B3) implies (cid:20) ( a r dadt r ′ + a r t ′ ) (cid:21) + − = − σ a r , (B12)where we assume sign( t ′ ) > τ τ ) component of (B3) gives a second orderequation (cid:20) ar ′ t ′′ − at ′ r ′′ + a dadt r ′ − dadt t ′ r ′ (cid:21) + − = σ , (B13)which is satisfied identically provided the above five equa-tions and the Einstein’s equations in the exterior andinterior regions hold.Following [21] it is possible to obtain a useful equationfor R . After some simple algebraic manipulations one canshow that ( θθ ) component of the discontinuity equation(B3) implies K +2 θθ = 1 σ R (cid:18) K − θθ − K +2 θθ + R σ (cid:19) . (B14)From (B7) it follows that K θθ can be expressed in termsof R as K θθ = R (cid:2) R ′ − H R (cid:3) . (B15)Using (B15) in (B14) yields R ′ + 1 = R σ (cid:0) H − H − (cid:1) + R (cid:0) H + H − (cid:1) + R σ , (B16)where H ± = d ln a ± /dt ± are the corresponding Hubbleparameters. Although this looks like a single equation for R , let us note that there is a dependence on the variable σ and a hidden dependence on t + through H + . Note that H − ≡ H is constant in the interior de Sitter space.It turns out that it is difficult to solve the above fieldequations to obtain an explicit solution for the boundary.Since our aim in this paper is to examine how perturba-tions propagate from one region into the other, ratherthan studying the bubble dynamics, we make some sim-plified assumptions.First note that for local inflation it is desirable to have H − ≫ H + . Therefore, in (B16) one can ignore the termsinvolving H + . This is nearly equivalent to taking outsideregion to be the flat space, except σ is still time depen-dent. In (B11), the interior contribution to the righthand side vanishes since the cosmological constant obeys ρ − + P − = 0. By field equations, the exterior contribu-tion ρ + + P + is proportional to dH + /dt + = − /t . Weassume that inflation occurs at a sufficiently later timeafter big bang (i.e. t + ≫
1) such that this term is verysmall and thus σ ′ ∼
0. This effectively means that theoutside region can be taken as the flat space. Summariz-ing we take a − = e Ht − , a + = 1 , σ ′ = 0 . (B17)These assumptions may not be suitable for a realisticscenario using inflating bubbles in a cosmological back-ground, but they are not harmful for our purposes. After these simplifications, (B16) can be solved for R as R = 1 α cosh( ατ ) (B18)where α = | H σ + σ | (B19)Since a + = 1, (B10) and (B9) can be used to determine r + and t + as r + = 1 α cosh( ατ ) , t + = 1 α sinh( ατ ) . (B20)To solve the interior fields we first note that (B10) canbe used to fix t − in terms of r − as Ht − = ln R − ln r − . (B21)Using (B21) in the proper time equation (B9), one getsa quadratic equation for ln r ′− which can be solved alge-braically as ln r ′− = ln R ′ ± HR | H σ − σ | − H R . (B22)Although it looks complicated, this equation can be inte-grated to obtain r − ( τ ). Further using (B21) the interiorembedding coordinates can be fixed as r − = r cosh( ατ ) | sinh( ατ ) ± H − σ Hσ | , (B23) Ht − = ln( | sinh( ατ ) ± H − σ Hσ | ) − ln( αr ) . The correlation between ± signs in (B23) and (B22) de-pends on the sign of 4 H − σ . It may first be seen sur-prising that there appears two solutions since the initialsystem of differential equations are first order. However,it is easy to see that ± solutions are related by time re-versal τ → − τ . Without loss of any generality we choose+ sign in (15).At this point we recall that although (B16) was ob-tained from the original junction condition, the two equa-tions are not equivalent. Therefore, a final check is re-quired to see whether (B20) and (B23) obey (B12). Itturns out that to satisfy (B12) one should either choose σ < σ <
0, which amounts toassume a negative cosmological constant on the shell, wesimply switch the roles played by the interior and exteriorspaces.
ACKNOWLEDGMENTS