Bispectrum from open inflation
aa r X i v : . [ a s t r o - ph . C O ] S e p YITP-13-85
Prepared for submission to JCAP
Bispectrum from open inflation
Kazuyuki Sugimura, a Eiichiro Komatsu b,c,d a Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan b Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild Str. 1, 85741 Garching, Germany c Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Ad-vanced Study, the University of Tokyo, Kashiwa, Japan 277-8583 (Kavli IPMU, WPI) d Texas Cosmology Center and the Department of Astronomy, The University of Texas atAustin, 1 University Station, C1400, Austin, TX 78712, USAE-mail: [email protected], [email protected]
Abstract.
We calculate the bispectrum of primordial curvature perturbations, ζ , generatedduring “open inflation.” Inflation occurs inside a bubble nucleated via quantum tunnelingfrom the background false vacuum state. Our universe lives inside the bubble, which can bedescribed as a Friedman-Lemaˆıtre-Robertson-Walker (FLRW) universe with negative spatialcurvature, undergoing slow-roll inflation. We pay special attention to the issue of an initialstate for quantum fluctuations. A “vacuum state” defined by a positive-frequency mode inde Sitter space charted by open coordinates is different from the Euclidean vacuum (whichis equivalent to the so-called “Bunch-Davies vacuum” defined by a positive-frequency modein de Sitter space charted by flat coordinates). Quantum tunneling (bubble nucleation)then modifies the initial state away from the original Euclidean vacuum. While most ofthe previous study on modifications of the initial quantum state introduces, by hand, aninitial time at which the quantum state is modified as well as the form of the modification,an effective initial time naturally emerges and the form is fixed by quantum tunneling inopen inflation models. Therefore, open inflation enables a self-consistent computation ofthe effect of a modified initial state on the bispectrum. We find a term which goes as h ζ k ζ k ζ k i ∝ /k k in the so-called squeezed configurations, k ≪ k ≈ k , in agreementwith the previous study on modifications of the initial state. The bispectrum in the exactfolded limit, e.g., k = k + k , is also enhanced and remains finite. However, these terms areexponentially suppressed when the wavelength of ζ is smaller than the curvature radius of theuniverse. The leading-order bispectrum is equal to the usual one from single-field slow-rollinflation; the terms specific for open inflation arise only in the sub-leading order when thewavelength of ζ is smaller than the curvature radius. ontents ϕ ζ A.1 Second-order action 21A.2 Third-order action 21
B Analytical structure of the universe in open inflation 23
B.1 Coordinate systems and Wightman functions 23B.2 Deformation of integration path using analyticity 25
C Order-of-magnitude estimate for sub-leading contributions 27
C.1 Contributions from outside the R -region 27C.2 Contributions from the sub-leading terms in Lagrangian 28 D Open harmonics in the sub-curvature approximation 29
D.1 Correspondence with flat harmonics 29D.2 Correspondence with Fourier modes in flat space 32D.3 Power spectrum and bispectrum 32 “Open inflation” models [1, 2] offer an attractive framework for understanding the origin ofour universe. According to this framework, our universe is contained within a single bubble,which was nucleated from a surrounding false vacuum state. This is attractive because there isno physical singularity at the beginning of our universe, which is the moment of the bubblenucleation in the de Sitter background. In the simplest scenario worked out by Colemanand De Luccia [3], the metric inside the bubble is a homogeneous and isotropic Friedman-Lemaˆıtre-Robertson-Walker (FLRW) metric with negative spatial curvature. Therefore, the– 1 –omogeneity and isotropy problems do not exist in this scenario. We still need inflation[4–8] to make geometry of the observable universe sufficiently flat, at the level compatiblewith observations [9–12]; hence the term, “open inflation.”How can we test open inflation models? While this is an attractive framework, is ittestable/falsifiable? An obvious observable is the spatial curvature of the observable universe.Detection of negative curvature would greatly strengthen the case for open inflation models,while detection of positive curvature would rule them out [13]. In any case, in order forus to have any access to distinct signatures of open inflation, the total number of e -folds ofinflationary expansion must be close to the minimum that is required to make geometry of theobservable universe sufficiently flat. If so, we may find signatures of open inflation in scalar[14] and tensor perturbations [15, 16] in temperature anisotropy of the cosmic microwavebackground (CMB) [17–22]. Another possible observable is a signature of other bubblescolliding with ours [28–30].In this paper, we shall present the first computation of the bispectrum from single-fieldopen inflation. Our model is based on ref. [32], and the potential for the scalar (inflaton)field is shown in figure 1. In these models, a single scalar field is responsible for bothquantum tunneling and slow-roll inflation inside the bubble. For simplicity, we shall ignoremulti-field effects or the possibility of a rapid-roll era soon after quantum tunneling, althoughthese effects are potentially significant [33, 34]. The bispectrum is the three-point functionin Fourier space, and we define it as h ζ k ζ k ζ k i = (2 π ) δ D ( P i =1 k i ) B ( k , k , k ), where ζ k is the Fourier transform of a curvature perturbation computed on the uniform densityhypersurface.Our work is motivated in part by recent work on the effects of a modified initial state(non-Bunch-Davies initial state ) on the bispectrum [35–43]. In particular, it has been shownthat a modified initial state yields an enhanced bispectrum in the so-called squeezed limit, inwhich one of the wavenumbers is much less than the other two, i.e., k ≪ k ≈ k . Specifically, B ( k, k, k ) ∝ k − k − in the squeezed limit [38–40] instead of the usual B ( k, k, k ) ∝ k − k − for single-field inflation models with a Bunch-Davies initial state [44]. Such a distinct de-pendence on k has important implications for observations [45, 46]. However, most of theprevious work on a modified initial state puts an initial state at some arbitrary initial timeby hand, without specifying its physical origin. The only self-consistent computations thatwe are aware of use periodic features in the scalar-field potential or kinetic term [47, 48].In open inflation models, such a modified initial state naturally emerges as a result ofthe evolution from the Euclidean vacuum (which is equivalent to the Bunch-Davies vacuumin de Sitter space charted by flat coordinates) via quantum tunneling [18, 49, 50]. Therefore,open inflation enables a self-consistent computation of the effect of a modified initial stateon the bispectrum.This paper is organized as follows. In section 2, we review open inflation and our co-ordinate system. In section 3, we review quantum field theory for a free scalar field in openinflation. In section 4, we outline the in-in formalism on a Coleman-De Luccia instantonbackground, with which we calculate the bispectrum in section 5. Finally, we conclude in However, one may argue that this is merely a consequence of the assumptions made in the analysis ofColeman and De Luccia, who studied a bubble nucleation in a homogeneous and isotropic background. See refs. [23–27] for earlier calculations based upon a conformal vacuum state (rather than the Euclideanvacuum state we shall work with in this paper). See ref. [31] for related work on the bispectrum from inflation with positive spatial curvature. We shall define what we mean by the “Bunch-Davies initial state” in section 3. – 2 – unnelingslow-roll V( φ ) φ φ F φ N Figure 1 . Potential for single-field open inflation. Slow-roll inflation begins after a scalar fieldtunneling from the false vacuum state, φ F , to the nucleation point, φ N . The essential features of thispotential are: the potential heights before and after tunneling are approximately the same, V ( φ F ) ≈ V ( φ N ); the barrier height is small compared to V ( φ F ); and the barrier is narrow, V ≪ | d V /dφ | atthe barrier. section 6. In appendix A, we derive the second- and third-order actions for scalar perturba-tions. In appendix B, we show how we choose the paths of integration when computing thecorrelation function. In appendix C, we show that all but one term in the third-order actionafter field redefinition is important in the sub-curvature approximation. In appendix D, weshow the correspondence between the open harmonics and the Fourier modes in flat space inthe sub-curvature approximation.We adopt the units c = ~ = 8 πG = 1. We describe quantum tunneling using a Coleman-De Luccia (CDL) instanton, ¯ φ ( x ) and¯ g µν ( x ) [3]. A CDL instanton, which is an Euclidean O(4)-symmetric solution, can be writtenas a function of only one variable, ¯ φ ( τ ) and ¯ a ( τ ), where τ is the imaginary (Euclidean) time.We study quantum tunneling using the Euclidean metric given by ds = dτ + ¯ a ( τ )( dχ + sin χd Ω ) , (2.1)with d Ω being the metric of a 2-sphere ( S ). We then analytically continue the solutionbeyond tunneling (i.e., bubble nucleation) in order to describe the post-tunneling world.The equations of motion (EOMs) for the instanton are given by d ¯ φ ( τ ) dτ + 3 d ln ¯ a ( τ ) dτ d ¯ φ ( τ ) dτ − dV [ ¯ φ ( τ )] d ¯ φ = 0 , (2.2) (cid:20) d ln ¯ a ( τ ) dτ (cid:21) = − V [ ¯ φ ( τ )]3 + 1¯ a ( τ ) , (2.3)where the potential, V ( φ ), has a false vacuum at φ F . We have a slow-roll inflation phasefollowing the quantum tunneling to a nucleation point, φ N (see figure 1).– 3 –or simplicity, we shall assume that the potential barrier is small compared with thevacuum energy of the false vacuum, and the potential heights before and after tunneling aswell as during the subsequent slow-roll phase are approximately equal, i.e., V ( φ F ) ≈ V ( φ N ) ≈ V ( φ I ) ≡ V I , where φ I denotes the scalar field values during slow-roll inflation. Then, we canapproximate V ( φ ) as V ( φ ) ≈ V I in eq. (2.3), and ¯ a is approximately given by the scale factorof Euclidean de Sitter spacetime, ¯ a ( τ ) = 1 H I sin( H I τ ) , (2.4)with H I = V I /
3. With this choice of coordinates τ , the two ends of the Euclidean 4-sphere in τ correspond to H I τ = ± π/
2. The boundary conditions for the instanton are d ¯ φ ( τ ) /dτ = 0 at τ = ± π/ (2 H I ). We shall also assume that the potential barrier is narrow, H I ≪ | d V /dφ | .With these assumptions, ¯ φ is approximately given by a thin-wall instanton solution:¯ φ ( τ ) ≈ φ F (cid:18) − π H I ≤ τ < π H I − R W (cid:19) φ N (cid:18) π H I − R W < τ ≤ π H I (cid:19) , (2.5)where R W is the radius of the bubble wall.The world after bubble nucleation can be described by analytical continuation of a CDLinstanton to Lorentzian regions. For the Euclidean de Sitter spacetime defined by eq. (2.4),analytical continuation is made with the following coordinate system [51], E - and E -regions: ( τ ( − π/ (2 H I ) ≤ τ ≤ π/ (2 H I )) ,χ (0 ≤ χ ≤ π ) , (2.6) R -region: ( t R = i ( τ − π/ (2 H I )) (0 ≤ t R < ∞ ) ,r R = iχ (0 ≤ r R < ∞ ) , (2.7) L -region: ( t L = i ( − τ − π/ (2 H I )) (0 ≤ t L < ∞ ) ,r L = iχ (0 ≤ r L < ∞ ) , (2.8) C -region: ( t C = τ ( − π/ (2 H I ) ≤ t C ≤ π/ (2 H I )) ,r C = i (cid:0) χ − π (cid:1) (0 ≤ r C < ∞ ) . (2.9)The E - and E -regions are the original Euclidean regions we have used to describe quantumtunneling, which consist of two 4-hemispheres E and E defined by 0 ≤ χ < π/ π/ < χ ≤ π (“south”), respectively. The others are the Lorentzian regions, whichdescribe the world after quantum tunneling. Each region and its physical meaning are il-lustrated in figure 2. The coordinates of a 2-sphere, Ω i , are commonly used for all regions.The boundaries between the C - and R -regions and the C - and L -regions are coordinatesingularities.With the coordinate system given in eq. (2.6) to (2.9), the metrics in the E -, R -, L -,and C -regions are given, respectively, by ds = dτ + H − I cos H I τ (cid:0) dχ + sin χd Ω (cid:1) = − dt R + H − I sinh H I t R (cid:0) dr R + sinh r R d Ω (cid:1) = − dt L + H − I sinh H I t L (cid:0) dr L + sinh r L d Ω (cid:1) = dt C + H − I cos H I t C (cid:0) − dr C + cosh r C d Ω (cid:1) , (2.10)– 4 – wallwallreheatingnucleationnucleation E E E E L C
Figure 2 . Penrose-like diagram of open inflation. The top half (above the line labeled “nucleation”)corresponds to the world after bubble nucleation, and the time is real. The bottom half correspondsto either one of the two hemispheres of the Euclidean 4-sphere, E (“north”) or E (“south”), and thetime is imaginary. (Also see figure 3.) The solid and dashed lines show r = constant and t = constantlines, respectively, in the R -, L -, and C -regions, while they show χ = constant and τ = constant,respectively, in the E - and E -regions. The R -region describes the interior of the bubble, i.e., ouruniverse. The C -region contains a worldline of the bubble wall, which separates the interior (locatedto the right of the line labeled “wall”) and exterior of the bubble. The exterior of the bubble is in thefalse vacuum state with ¯ φ = φ F . The big filled circle at the right edge of the line labeled “nucleation”shows the location of the bubble center at the time of nucleation. Note that the bubble has a finite sizeupon nucleation. The L -region is in the false vacuum state with ¯ φ = φ F . The coordinate values arechosen such that the right vertical edge of the R -region has r R = 0, and the thick solid line betweenthe R - and C -regions have t R = 0 and r R = ∞ as well as t C = π/ (2 H I ) and r C = ∞ . The line labeled“nucleation” has r C = 0 as well as χ = π/
2. The left vertical edge of the L -region has r L = 0, and thethick solid line between the L - and C -regions have t L = 0 and r L = ∞ as well as t C = − π/ (2 H I ) and r C = ∞ . Finally, while the top edge of the L -region has t L = ∞ , that of the R -region is somewhatfuzzy because slow-roll inflation in the R -region must end in a finite time and our universe must entera radiation-dominated era after reheating (which is not shown in this diagram). While t and r inthe R - and L -regions coincide with the familiar time and radial coordinates in a FLRW universe,those in the C -region do not. For example, a t C = constant line gives a worldline of a body withconstant acceleration (which is analogous to the Rindler coordinates). Therefore, the bubble wall inthe C -region is accelerated and its worldline approaches a null line as r C → ∞ . In this paper, thewall worldline is along t C = R W where R W is the radius of the bubble, and φ ( t C ) = φ F and φ N for − π/ (2 H I ) ≤ t C < π/ (2 H I ) − R W (exterior of the bubble) and π/ (2 H I ) − R W < t C ≤ π/ (2 H I )(interior), respectively (eq. (2.5)). – 5 –nd φ ’s in the corresponding regions are given, respectively, by φ ( τ ) = ¯ φ ( τ ) , (2.11) φ ( t R ) = ¯ φ [ − it R + π/ (2 H I )] , (2.12) φ ( t L ) = ¯ φ [ it L − π/ (2 H I )] , (2.13) φ ( t C ) = ¯ φ ( t C ) , (2.14)where ¯ φ ( τ ) is a solution to eqs. (2.2) and (2.3).We live in the future of the R -region, in which slow-roll inflation occurs after quantumtunneling, and reheating follows after inflation (see figure 2). In the R -region, a t R = constsurface is both a 3-hyperboloid and a φ = const surface; thus, we observe the R -region as aFLRW universe with negative spatial curvature.If we could solve eqs. (2.2) and (2.3) exactly for a given potential written by an analyticalfunction, analytical continuation given by eq. (2.7) to (2.9) would give a complete descriptionof the world after quantum tunneling. However, since the solutions given in eqs. (2.4) and(2.5) are approximation, we need to re-solve the EOM for φ and the Friedmann equation ina FLRW universe with negative spatial curvature, in order to describe the evolution of theuniverse in the region where the solutions given in eqs. (2.4) and (2.5) are not valid. Weshall therefore use eqs. (2.4) and (2.5) as the initial conditions for the subsequent evolutionof φ ( t ) and a ( t ).For brevity, in the following we omit the subscript R from t R and r R unless otherwisestated. The metric in the R -region is given by ds = − dt + a ( t ) γ ij dx i dx j , (2.15)where γ ij is the metric for the unit 3-hyperboloid given by γ ij dx i dx j ≡ dr + sinh rd Ω , (2.16)and a ( t ) is the FLRW scale factor. The EOM for φ and the Friedmann equation are given,respectively, by ¨ φ ( t ) + 3 H ( t ) ˙ φ ( t ) + dV [ φ ( t )] dφ = 0 , (2.17) H ( t ) = V [ φ ( t )]3 + 1 a ( t ) , (2.18)where the over-dots denote derivatives with respect to t and H ( t ) ≡ ˙ a ( t ) /a ( t ) is in generaldifferent from H I . The initial conditions are given by a (0) = 0, ˙ a (0) = 1, φ (0) = φ N , and˙ φ (0) = 0, according to the CDL instanton solution.At the onset of slow-roll inflation in the R -region, the universe is still in a curvature-dominated era, i.e., the spatial curvature term is still dominant in the Friedmann equation.However, spatial curvature decays away as the universe expands, and an inflationary erabegins when the potential of φ becomes dominant in the Friedmann equation. After reachingthe inflationary era, the universe evolves in the same way as the usual slow-roll inflationscenario. We thus impose slow-roll conditions: ǫ ≡ ( dV /dφ ) V ≪ , ǫ η ≡ ( d V /d φ ) V ≪ . (2.19)– 6 –lthough the slow-roll conditions may be broken during the curvature-dominated era, weshall assume that the slow-roll conditions are satisfied in the whole R -region.Let us define the conformal time, η ≡ R t ∞ dt ′ /a ( t ′ ). To the leading order of the slow-roll parameters, we have η ≈ − (1 /
2) log ((cosh H I t + 1) / (cosh H I t − −∞ < η . − − . η <
0, respectively. The scale factor and the Hubble parameter are approximatelygiven by a ( η ) ≈ e η H I ( −∞ < η . − − H I η ( − . η < , H ( η ) ≈ − H I η ( −∞ < η . − H I ( − . η < . (2.20)The first and second derivatives of the scalar field with respect to time are approximatelygiven by˙ φ ( η ) ≈ − √ ǫ H I e η ( −∞ < η . − −√ ǫH I ( − . η < , ¨ φ ( η ) ≈ − √ ǫ H I ( −∞ < η . − √ ǫ ( ǫ + ǫ η ) H I ( − . η < . (2.21)We assume that the potential is a monotonically increasing function of φ (except for thebarrier), and thus ˙ φ and ¨ φ have definite signs. In this section, we review quantum field theory (QFT) for a free scalar field in open inflation[18, 49, 50]. Hereafter, we shall consider only scalar-type perturbations. We write theperturbed scalar field around the uniform background, φ ( t ), as φ ( t, x ) = φ ( t ) + ϕ ( t, x ) , (3.1)where ϕ is the scalar field perturbation. We write the spatial coordinates as x ≡ ( r, Ω) forbrevity. From now on, we shall use φ to denote the background scalar field, φ ( t ), unlessotherwise stated.The perturbed metric in the R -region around the metric given by eq. (2.15) is writtenin the Arnowitt-Deser-Misner (ADM) form as [52] ds = − N dt + h ij (cid:0) dx i + N i dt (cid:1) (cid:0) dx j + N j dt (cid:1) , (3.2)where h ij = a ( t ) e ζ γ ij is the spatial metric with the curvature perturbation ζ , N the lapsefunction, and N i the shift vector. We fix the gauge degrees of freedom by taking the uniformcurvature (flat) gauge, ζ ≡
0, and obtain the quadratic action for scalar perturbations by sub-stituting the constraint equations into the original action. As we show in appendix A.1, thequadratic action for scalar perturbations up to the leading order in the slow-roll parametersis given by S (cid:18) ≡ Z d x L (cid:19) = Z dt d x a √ γ (cid:20)
12 ˙ ϕ − a ∂ i ϕ∂ i ϕ (cid:21) , (3.3)– 7 –here L is the Lagrangian density for free theory, ∂ i denotes 3-dimensional covariant deriva-tives with respect to γ ij , and ∂ i is defined by ∂ i ≡ γ ij ∂ j . eq. (3.3), derived in the R -region,can also be used outside the R -region by analytical continuation, except near the bubblewall, for which a non-zero mass term should be inserted into the action.Next, we obtain the 2-point function of ϕ . We promote ϕ to an operator, ˆ ϕ , and writeˆ ϕ ( x ) = X k h u k ( x )ˆ a k + v k ( x )ˆ a † k i , (3.4)where ˆ a k and ˆ a † k are the annihilation and creation operators, respectively, that satisfy thecommutation relation, [ˆ a k , ˆ a † k ′ ] = δ kk ′ ; and u k ( x ) and v k ( x ) are mode functions, which form acomplete set. The mode functions are solutions of the linearized EOMs. Whilst v k ( x ) = u ∗ k ( x )in the Lorentzian region, the same relation does not necessarily hold in the Euclidean region,as the argument x may contain an imaginary part.The choice of u k selects a “vacuum state,” | i , which is annihilated by the annihilationoperator in eq. (3.4), ˆ a k | i = 0. The most natural choice of a vacuum state in de Sitterspace is the Euclidean vacuum, | E i . The Euclidean vacuum is defined by a purely positive-frequency function which is analytical in E . The two-point function (Wightman function)computed with respect to this vacuum state, h E | ϕ ( x ) ϕ ( x ′ ) | E i , is invariant under de Sittergroup SO(4,1) (see section 5.4 of ref. [53]). However, a positive-frequency function of de Sitter space charted by open coordinatesdoes not give a purely positive-frequency function of the Euclidean vacuum state. This isnot a surprise: a positive-frequency function defined in a given coordination is typically amixture of positive- and negative-frequency functions defined in another coordination, and therelation between them is given by the Bogoliubov transformation. For example, a positive-frequency function of Minkowski space charted by Rindler coordinates is a mixture of positive-and negative-frequency functions of Minkowski space charted by ds = − dt + d x (whosepositive-frequency function defines the Minkowski vacuum), and thus a comoving Rindlerobserver (who experiences a constant acceleration in Minkowski space) detects particles (seesection 4.5 of ref. [53]).Sasaki, Tanaka, and Yamamoto derive an appropriate mixture of positive- and negative-frequency functions of de Sitter space charted by open coordinates, whose annihilation oper-ator annihilates the Euclidean vacuum [51]. They expand ˆ ϕ asˆ ϕ ( x ) = X plm X σ = ± h u ( σ ) plm ( x )ˆ a ( σ ) plm + v ( σ ) plm ( x )ˆ a ( σ ) plm † i , (3.5)with ˆ a ( σ ) plm | E i = 0 for σ = ± . The mode functions, u ( σ ) plm ( v ( σ ) plm ), are chosen to be analyticalin E ( E ); hence the Euclidean vacuum. They are linear combinations of positive- andnegative-frequency functions defined in the R and L regions. The Euclidean vacuum state is equivalent to the Bunch-Davies vacuum state [54], which is perhaps morefamiliar to cosmologists. The Bunch-Davies state is defined by a positive-frequency function of de Sitter spacecharted by flat coordinates. Specifically, for flat coordinates of ds = a ( η )( − dη + d x ) with a ( η ) = − / ( Hη ),a positive-frequency function for a minimally-coupled scalar field with mass m is given by the Hankel functionof the first kind as u k = √− πη π ) / a H (1) ν ( − kη ) e i k · x with ν = − m H for η <
0. This function is analytical in theentire lower half complex η plane [Im ( η ) < u k is finite in the limit of η → − (1 + iǫ ) ∞ . This procedureyields a purely positive-frequency function of the Euclidean vacuum state. The annihilation operator definedwith respect to this u k annihilates the Euclidean vacuum, ˆ a k | E i = 0, and a comoving observer in de Sitterspace charted by flat coordinates does not detect particles (see section 5.4 of ref. [53]; also see ref. [55]). – 8 –hey then write the mode functions as u ( σ ) plm ( x ) = u ( σ ) p ( η ) Y plm ( x ), where Y plm ( x ) areharmonics on a 3-hyperboloid, and the indices p , l , and m take on 0 < p < ∞ , l = 0 , , · · · ,and m = − l, − l + 1 , · · · , l − , l , respectively. The explicit form is given by Y plm ( x ) = f pl ( r ) Y lm (Ω), where Y lm (Ω) is the spherical harmonics on a 2-sphere, and [51] f pl ( r ) = (cid:12)(cid:12)(cid:12)(cid:12) Γ( ip + l + 1)Γ( ip + 1) (cid:12)(cid:12)(cid:12)(cid:12) p √ sinh r P − l − / ip − / (cosh r ) , (3.6)with P νµ ( z ) being the associated Legendre functions. This function goes as f pl ( r ) ∝ e − r for p >
0, and r = 1 corresponds to the curvature radius of a 3-hyperboloid. Therefore,the modes with p > p > p < ∂ Y plm ( x ) = − ( p + 1) Y plm ( x ) ,Y ∗ plm ( x ) = ( − m Y pl − m ( x ) , Z d x √ γ Y ∗ p l m ( x ) Y p l m ( x ) = δ ( p − p ) δ l l δ m m , Z ∞ dp X lm Y ∗ plm ( x ) Y plm ( x ′ ) = δ (3) ( x − x ′ ) . (3.7)The conformal-time dependence of the mode function for a massless minimally-coupled scalarfield, u ( σ ) p ( η ), is given by [51] u ( σ ) p ( η ) = 1 p − e − πp ) ˜ u p ( η ) − σ e − πp p − e − πp ) ˜ v p ( η ) (3.8)with ˜ u p ( η ) = H I cosh η + ip sinh η p p (1 + p ) e − ipη , ˜ v p ( η ) = H I cosh η − ip sinh η p p (1 + p ) e ipη . (3.9)As ˜ u p and ˜ v p are positive- and negative-frequency functions naturally defined in the R -region(recall that eq. (3.9) is written in coordinates in the R -region), a comoving observer in the R -region detects particles with respect to the Euclidean vacuum. To see this, let us expand ˆ ϕ in the R -region as ˆ ϕ ( x ) = P plm [˜ u p ( η ) Y plm ( x )˜ a plm + ˜ v p ( η ) Y ∗ plm ( x )˜ a † plm ]. Then, the occupationnumber of particles is given by N plm = h E | ˜ a † plm ˜ a plm | E i = 1 e πp − , (3.10)which is a thermal spectrum. The argument and result given here basically parallel those forparticle creation in Rindler space, i.e., a comoving observer in Rindler space sees particleswith a thermal spectrum (see eq. (4.97) of ref. [53]).The Euclidean vacuum state given by eq. (3.8) is a natural choice for quantum fluc-tuations in homogeneous de Sitter space before bubble nucleation. The initial state then– 9 –volves away from the Euclidean vacuum state via bubble nucleation; thus, eq. (3.8) must bemodified. In the R -region, we find u k ( x ) by solving the following linearized EOM, δLδϕ ≡ − ¨ ϕ ( x ) − H ( t ) ˙ ϕ ( x ) + 1 a ( t ) ∂ ϕ ( x ) = 0 , (3.11)where ∂ ≡ γ ij ∂ i ∂ j , and L is given by S = R d xa √ γ L , and is also related to the Lagrangiandensity, L , by L = a √ γL . This EOM is obtained by the variation of the second order actiongiven in eq. (3.3). For our model, we can choose the mode functions such that one vanishesin the R -region and another vanishes in the L -region [18]. Specifically, we expand ˆ ϕ asˆ ϕ ( x ) = X plm h u Rplm ( x )ˆ a Rplm + v Rplm ( x )ˆ a Rplm † + u Lplm ( x )ˆ a Lplm + v Lplm ( x )ˆ a Lplm † i , (3.12)where u Rplm and u Lplm vanish in the L - and R -regions, respectively. (Similarly for v Rplm and v Lplm .) In this paper, we shall consider u Rplm (and v Rplm ) only, as we have no access to the L -region observationally; henceforth we shall drop the superscript R , i.e., u Rplm → u plm and v Rplm → v plm , unless stated otherwise. We then write u plm ( x ) = u p ( η ) Y plm ( x ) . (3.13)Yamamoto, Sasaki, and Tanaka find [18] u p ( η ) = 1 √ − e − πp ˜ u p ( η ) + e − πp − iδ p √ − e − πp ˜ v p ( η ) , (3.14)with ˜ u p ( η ) and ˜ v p ( η ) given by eq. (3.9). Unlike the previous mode function given in eq. (3.8),the annihilation operator defined with respect to eq. (3.14) does not annihilate the Euclideanvacuum. Again, ˜ u p and ˜ v p are positive- and negative-frequency functions naturally definedin the R -region; thus, writing ˆ ϕ ( x ) = P plm [˜ u p ( η ) Y plm ( x )˜ a plm + ˜ v p ( η ) Y ∗ plm ( x )˜ a † plm ] in the R -region, we find that the occupation number of particles detected in the R -region with respectto the new vacuum state, | i , is still given by a thermal spectrum, N plm = h | ˜ a † plm ˜ a plm | i =( e πp − − .As a comoving observer in the R -region, which is inside the nucleated bubble, detectsparticles (also see ref. [58] which discusses particle creation due to bubble nucleation inMinkowski space), the state for quantum fluctuations set at the time of bubble nucleation inopen inflation is a non -Bunch-Davies vacuum state.With the definition of the field operator given in eq. (3.12), the Wightman functions, G ± ( x, x ′ ), are given by G + ( x, x ′ ) ≡ (cid:10) (cid:12)(cid:12) ϕ ( x ) ϕ ( x ′ ) (cid:12)(cid:12) (cid:11) = X plm (cid:2) u Rplm ( x ) v Rplm ( x ′ ) + u Lplm ( x ) v Lplm ( x ′ ) (cid:3) ,G − ( x, x ′ ) ≡ (cid:10) (cid:12)(cid:12) ϕ ( x ′ ) ϕ ( x ) (cid:12)(cid:12) (cid:11) = X plm (cid:2) u Rplm ( x ′ ) v Rplm ( x ) + u Lplm ( x ′ ) v Lplm ( x ) (cid:3) . (3.15) Here, we do not give the explicit expression for the complex phase, δ p , which does not affect the conclusionof this paper (see ref. [18] for the way to obtain δ p ). While eq. (3.14) is written with the coordinates in the R -region, analytical continuation lets us use it also in parts of the C - and E -regions which are inside thebubble. – 10 –hile G + ( x, x ′ ) = G − ( x, x ′ ) ( ≡ G ( x, x ′ )) for space-like separated x and x ′ , G + ( x, x ′ ) and G − ( x, x ′ ) are different for time-like separated x and x ′ .Let us now calculate the power spectrum. The multipole expansion coefficients of ϕ aredefined by ϕ plm ( η ) = Z d x √ γ Y ∗ plm ( x ) ϕ ( η, x ) , (3.16)and the power spectrum, P ( p ), is defined by (cid:10) ϕ ∗ plm ϕ p ′ l ′ m ′ (cid:11) = δ ( p − p ′ ) δ ll ′ δ mm ′ P ( p ). Fromeqs. (3.4), (3.7), (3.14), and (3.16), P ( p ) at late time (i.e., η ≈
0) is given by [18] P ( p ) = H I cosh πp + cos 2 δ p p (1 + p ) sinh πp . (3.17)We recover a scale-invariant spectrum in the large p limit, P ( p ) → H I / (2 p ). This is anexpected result, as large- p modes do not feel curvature of space, and the occupation numberof particles falls off exponentially at large p . Here, we outline the in-in formalism on a CDL instanton background, which can be used tocalculate the bispectrum in open inflation models. The formalism is a natural extension ofthe in-in formalism in de Sitter space charted by flat coordinates [44] to the case of a universewith quantum tunneling. The free QFT in open inflation, as reviewed in section 3, is basedon a WKB analysis of a tunneling wave function without non-linear interaction terms [59–61].By performing a similar analysis with interaction terms, the in-in formalism can be extendedto the case of open inflation [62–64].The Lagrangian density is given by L full = L + L int , (4.1)where L and L int are the free and interaction parts of the Lagrangian density, respectively.Non-trivial time evolution in L causes the state to evolve away from the Bunch-Daviesvacuum state, and L int is a source of non-Gaussianity. The in-in formalism on a CDLinstanton background tells us that the N -point function of ϕ for x i s on a spacial hypersurfaceΣ is given by D ϕ ( x ) ϕ ( x ) · · · ϕ ( x N ) E = D (cid:12)(cid:12)(cid:12) P ϕ ( x ) ϕ ( x ) · · · ϕ ( x N ) e i R C × Σ λ dλd x L int ( x ) (cid:12)(cid:12)(cid:12) ED (cid:12)(cid:12)(cid:12) P e i R C × Σ λ dλd x L int ( x ) (cid:12)(cid:12)(cid:12) E , (4.2)where the λ integral is performed along the path C = C + C , and the x integral is over thehypersurface Σ λ for a given λ (see figure 3). The path ordering operator, P , orders operatorsin the expression according to the order along C . Covariance of eq. (4.2) guarantees that the N -point function is not affected by the choice of C or Σ λ .The path of integration is shown in figure 3. In the first integration domain, V ≡ C × Σ λ (the green arrows), the path C starts in E and runs through the nucleation surface Σ N toΣ . In the second domain, V ≡ C × Σ λ (the red arrows), the path C starts from Σ and runsthrough Σ N into E . The contributions to the integral from the Euclidean and Lorentzianregions correspond to contributions from during and after bubble nucleation, respectively.– 11 – igure 3 . A 4-dimensional CDL instanton embedded in 5-dimensional Minkowski spacetime. Theintegral that gives an N -point correlation function on a spatial hypersurface Σ (eq. (4.2)) is performedalong a path, λ , and over space, x . The λ integration is done along C = C + C , and the x integrationis done over a hypersurface Σ λ for a given λ . The upper cylinder (light blue) is the Lorentzian regionafter bubble nucleation. The upper and lower hemispheres (dark blue) are the Euclidean regions E and E , respectively. A bubble is assumed to be nucleated on the hypersurface Σ N . (The bubblewall is not shown.) The hypersurfaces Σ λ are chosen to be Cauchy surfaces in the Lorentzian region,and are chosen to cover E and E in the Euclidean regions of the integration domain V (the greenarrows ending on Σ ) and V (the red arrows ending in E ), respectively. To evaluate eq. (4.2), we use Wick’s theorem after Taylor expanding the exponential.Due to the path-ordering operator, P , each pair of field operators becomes G P ( x, x ′ ), whichis given by G P ( x, x ′ ) ≡ (cid:10) (cid:12)(cid:12) P ϕ ( x ) ϕ ( x ′ ) (cid:12)(cid:12) (cid:11) = ( G + ( x, x ′ ) when x ′ precedes xG − ( x, x ′ ) when x precedes x ′ , (4.3)where G ± ( x, x ′ ) are the Wightman functions given by eq. (3.15).Let us briefly comment on analyticity of G P ( x, x ′ ) (see appendix B for details). Thisfunction is singular when x and x ′ are null-separated. However, by assuming a small imag-inary part in the time coordinates along the in-in path in eq. (4.2), G P ( x, x ′ ) becomes ana-lytical. As a result, (cid:10) ϕ ( x ) ϕ ( x ) · · · ϕ ( x N ) (cid:11) is also analytical with respect to x i s. To calculate the bispectrum at the tree-level, we substitute the constraint equations into theoriginal action, keep O ( ϕ ) quantities, and obtain the reduced third-order action in the flatgauge (in which ζ ≡ O ( ǫ / ) andgiven by (see appendix A.2 for derivation) S = − Z dtd x √ γ a ˙ φ (cid:2) ( ∂ − − ˙ ϕ (cid:3) ˙ ϕ + Z dtd x √ γ ( a ¨ φ (cid:2) ( ∂ − − ϕ (cid:3) ˙ ϕ + 3 a ˙ φ H (cid:2) ( ∂ − − ϕ (cid:3) (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19)) + Z dtd x ( − a ˙ φ H (cid:20) ( ∂ − − (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19)(cid:21) − a H (cid:2) ( ∂ − − ˙ ϕ (cid:3) (cid:16) ˙ φ ˙ ϕ − ¨ φϕ (cid:17)) δ L δϕ , (5.1)where δ L /δϕ ≡ a √ γδL/δϕ , with δL/δϕ given in eq. (3.11), and ∂ ≡ γ ij ∂ i ∂ j . Let us checkthe correspondence between this action and that found by Clunan and Seery [31], who givethe third-order action with positive spatial curvature. • We need to replace their ∂ + 3 (denoted as ∆ + 3) with ∂ −
3, as we deal with negativecurvature. • The first line in eq. (5.1) agrees with the second term in their eq. (4.5). • The second line in eq. (5.1) does not appear in their action: the first term proportionalto ¨ φ is ignored in their action because it is a higher order in slow-roll in their model.While ¨ φ/ ( H ˙ φ ) is slow-roll suppressed in the usual inflation scenario, it is not so inopen inflation because ¨ φ/ ( H ˙ φ ) = O (1) soon after bubble nucleation. The second termproportional to 3 a ˙ φ/ (4 H ) does not appear in their action due to the difference inEOM. Had our EOM had an extra mass term like theirs, the second term would cancelout with a term in the second line. • The last line of eq. (5.1) agrees with the last term in their eq. (4.5) with f ( ϕ ) given intheir eq. (4.6), except for the sign of the second term in their eq. (4.6). This is becausetheir second-order action (eq. (2.15)) contains an extra mass term of V ′′ = − /a . • The first term in their eq. (4.5), ( aH/ ˙ φ ) ϕ , which comes from V ′′′ , is ignored in ouraction.As in the case of the second order action, we expect that this expression derived in the R -region can also be used outside the R -region by analytical continuation, except near thebubble wall, for which a non-zero mass term should be inserted into the action. Potentiallylarge self-interaction terms near the bubble wall do not contribute to the bispectrum in thesub-curvature approximation, as discussed in appendix C.The terms in the second line are proportional to the EOM; thus, we remove it by makingthe field redefinition [44], ϕ → ϕ c , where ϕ = ϕ c + ˙ φ H (cid:2) ( ∂ − − ∂ i ϕ c ∂ i ϕ c (cid:3) + · · · , (5.2)where · · · contains terms which vanish outside the horizon. The field redefinition does notaffect the second-order action, S , given in eq. (3.3). The positive-frequency function for ϕ c is given by eqs. (3.13) with (3.14). – 13 –e write the interaction Lagrangian density, L int , in terms of ϕ c as L int = √ γ n − a ˙ φ (cid:2) ( ∂ − − ˙ ϕ c (cid:3) ˙ ϕ c + a ¨ φ (cid:2) ( ∂ − − ϕ c (cid:3) ˙ ϕ c + 3 a ˙ φ H (cid:2) ( ∂ − − ϕ c (cid:3) (cid:18) ˙ ϕ c + 1 a ∂ i ϕ c ∂ i ϕ c (cid:19)) . (5.3)If we write eq. (5.2) schematically as ϕ = ϕ c + λϕ c , the 3-point function of ϕ is given by D ϕ ( x ) ϕ ( x ) ϕ ( x ) E = D ϕ c ( x ) ϕ c ( x ) ϕ c ( x ) E + λ hD ϕ c ( x ) ϕ c ( x ) ED ϕ c ( x ) ϕ c ( x ) E + (perms . ) i , (5.4)where (cid:10) ϕ c ( x ) ϕ c ( x ) ϕ c ( x ) (cid:11) is obtained by substituting L int (eq. (5.3)) into the expressionfor the N -point function given in eq. (4.2). Before we compute the bispectrum using the cubic action given by eq. (5.3), let us show howthe calculation proceeds using a simpler example. Consider L int given by L int ( x ) = √− g λ int ( t ) ϕ ( x ) . (5.5)Although L int ( x ) that we wish to use is not in this form, both eqs. (5.3) and (5.5) respectO(4)-symmetry of the background universe; thus, the calculations proceed essentially in thesame way.Using eq. (4.2), Wick’s theorem, and eq. (4.3), we obtain the tree-level 3-point functionfor space-like separated points x , x , and x as D ϕ ( x ) ϕ ( x ) ϕ ( x ) E =2Re (cid:20) − i Z V d x √− g λ int ( t ) G + ( x, x ) G + ( x, x ) G + ( x, x ) (cid:21) + (perms . ) , (5.6)As the direct evaluation of the real-space expression such as eq. (5.6) is often not practical,we shall move to the harmonic-space expression by operating Q i =1 R d x i √ γ Y ∗ p i l i m i ( x i ) onboth sides of eq. (5.6), and using eqs. (3.15), (3.13), (3.7), and (3.16). We shall only considerthe modes with p >
0. Orthogonality of Y p i l i m i then implies that super-curvature modeswith p < R d x √ γ Y p l m ( x ) Y p l m ( x ) Y p l m ( x ) = F l l l p p p G m m m l l l ,where F l l l p p p ≡ Z dr sinh rf p l ( r ) f p l ( r ) f p l ( r ) , (5.7) G m m m l l l ≡ Z d Ω Y l m (Ω) Y l m (Ω) Y l m (Ω) . (5.8)See appendix B for the issue on the integration outside the R -region in V in eq. (5.6). Thebispectrum, B ( p , p , p ), is given by (cid:10) ϕ p l m ϕ p l m ϕ p l m (cid:11) = B ( p , p , p ) F l l l p p p G m m m l l l . (5.9)– 14 –s we show in appendix C, the contribution to the bispectrum in the sub-curvature approxi-mation ( p ≫
1) mainly comes from the R -region, and the bispectrum at late time (i.e., η ≈ B ( p , p , p ) ≈ (cid:26) − iv p (0) v p (0) v p (0) (cid:20)Z −∞ a ( η ) dη λ int ( η ) u p ( η ) u p ( η ) u p ( η ) (cid:21) + (perms . ) o . (5.10)The contribution from outside the R -region can be included by extending the integrationdomain to the complex plane (see eq. (B.9) in appendix B). Whilst eq. (5.10) was obtainedfrom L int given in eq. (5.5), B ( p , p , p ) calculated from the true L int given in eq. (5.3) takesa similar form. In this section we calculate the bispectrum from open inflation in the sub-curvature approx-imation, p ≫
1. This approximation is justified because we have not yet detected spatialcurvature within our current horizon. As we show in appendix D, the power spectrum, P ( p ),and the bispectrum, B ( p , p , p ), reduce to P ( k ) and B ( k , k , k ) in the sub-curvature ap-proximation with p i → k i , where k i ’s denote Fourier wavenumbers defined in flat space.Therefore, we shall use k i for the indices of harmonics instead of p i .We calculate the bispectrum using the same method as was used to derive eq. (5.10),but replacing L int in eq. (5.5) with that given in eq. (5.3). ϕ First, let us compute the bispectrum from the field redefinition term. Recalling P ( k ) = H I / (2 k ) in the sub-curvature approximation, we find B (redef) ( k , k , k ) = ˙ φ H I k · k k + 4 H I k H I k + (perms . ) . (5.11)Second, we compute the bispectrum from the in-in integral. As we show in appendix C,the leading-order term in the sub-curvature approximation is the first term in eq. (5.3), L (1) int ≡ −√ γa ˙ φ (cid:2) ( ∂ − − ˙ ϕ c (cid:3) ˙ ϕ c , (5.12)and the other terms are sub-dominant. We thus compute B ( k , k , k ) = 2Re ( iv k (0) v k (0) v k (0) "Z −∞ dη a ˙ φk + 4 ˙ u k ( η ) ˙ u k ( η ) ˙ u k ( η ) + (perms . ) ) . (5.13)The mode functions, u k ( η ), ˙ u k ( η ), and v k (0), are given by (see eq. (3.14)) u k ( η ) ≈ − ia ( η ) √ k (cid:16) e − ikη − e − πk − iδ k e ikη (cid:17) ( −∞ < η . − H I √ k h (1 + ikη ) e − ikη − e − πk − iδ k (1 − ikη ) e ikη i ( − . η < , (5.14)˙ u k ( η ) ≈ a ( η ) r k (cid:16) e − ikη + e − πk − iδ k e ikη (cid:17) , (5.15) v k (0) = u ∗ k (0) ≈ H I √ k (1 − e − πk − iδ k ) , (5.16)– 15 –here we have kept the terms proportional to e − πk e ikη , despite that they are suppressed by e − πk . This is because these terms represent the effect of a non-Bunch-Davies initial state,and we shall discuss the effect of these terms toward the end of this subsection.Ignoring the non-Bunch-Davies terms for the moment, we obtain B ( k , k , k ) = 2Re " i X i =1 H I k i ! k k k Z −∞ dη ˙ φe − i ( k + k + k ) η . (5.17)Using eq. (2.21) for ˙ φ , the integration during the curvature-dominated era (i.e., −∞ < η < −
1) is given by Z − −∞ dη ˙ φe − i ( k + k + k ) η = − √ ǫH I e − i ( k + k + k ) − i ( k + k + k ) , (5.18)and that during the inflationary era (i.e., − < η <
0) is given by Z − dη ˙ φe − i ( k + k + k ) η = −√ ǫH I − e i ( k + k + k ) − i ( k + k + k ) . (5.19)We thus find, in k ≫ B ( k , k , k ) = √ ǫH I k k k ( k + k + k ) X i =1 k i ! h − . k + k + k ) i , (5.20)where the factor 0 . − e − ≈ − .
9. This result is based upon the approximateform for ˙ φ given in eq. (2.21), which is discontinuous at η = −
1. We find that the secondoscillatory term in eq. (5.20) is an artifact due to this discontinuity. To show this, we usethe numerical solution of eq. (2.17) for ˙ φ and evaluate the integral. In figure 4, we show thatthe numerical solution does not contain any oscillations. Henceforth we shall neglect theoscillatory terms coming from the discontinuity of the approximate solution of ˙ φ . We obtain B ( k , k , k ) = 2 √ ǫH I k k k ( k + k + k ) X i =1 k i ! , (5.21)which agrees with the result for the single-field slow-roll inflation models with the canonicalkinetic term [44]. In the squeezed limit, k ≪ k ≈ k ≡ k , we find B ( k , k , k ) → √ ǫH I k k = √ ǫP ( k ) P ( k ) . (5.22)The bispectrum from the field redefinition term (eq. (5.11)) in the squeezed limit goes as B (redef) ( k , k , k ) → − √ ǫH I
16 1 k k = −√ ǫ k k P ( k ) P ( k ) . (5.23)Thus, the field redefinition term is sub-dominant in this limit.Next, let us discuss an enhancement of the bispectrum due to a non-Bunch-Daviesvacuum initial state, B (NBD) ( k , k , k ). The relevant term in eq. (5.15) is the part of ˙ u k ( η ) We would like to thank T. Tanaka for pointing this out. – 16 –
I k T Figure 4 . We calculate the integral given by I ≡ (1 / √ ǫH I )2Re [ i R −∞ dη ˙ φe − ik T η ] with k T ≡ k + k + k in three ways. First, we use the numerical solution of eq. (2.17) for ˙ φ in the integral[solid (red) line]. Second, we use the approximate solution for ˙ φ given in (2.21) [dot-dashed (green)line]. This gives the term in the square bracket in eq. (5.20) divided by k T . Finally, we remove theoscillatory term from the second case [dashed (blue) line]. This result is in good agreement with thefull solution (solid line) for sub-curvature modes (1 ≪ k T ), justifying our ignoring the oscillatory termin eq. (5.21) and thereafter. that is proportional to e ikη (i.e., ˙˜ v k ( η )) instead of e − ikη (i.e., ˙˜ u k ( η )). We then do this flippingin each of ˙ u k i in the integral of eq. (5.13). For example, flipping ˙ u k we obtain B (˜ v k ) ( k , k , k )= 2Re " i X i =1 H I k i ! e − πk − iδ k k k k Z −∞ dη ˙ φe − i ( − k + k + k ) η = √ ǫH I e − πk k k k X i =1 k i ! Re " e − iδ k − e i ( − k + k + k ) − k + k + k − i e − i ( − k + k + k ) − i ( − k + k + k ) ! . (5.24)Flipping ˙ u k or ˙ u k will similarly give B (˜ v k ) or B (˜ v k ) ; hence B (NBD) = P i =1 B (˜ v ki ) to theleading order of e − πk . We are interested in a peculiar behavior of the bispectrum froma non-Bunch-Davies initial state in the squeezed limit, k ≪ k ≈ k ≡ k [38–40], whilestill satisfying the sub-curvature approximation, 1 ≪ | − k + k + k | . Again ignoring theoscillatory term which is an artifact of using the approximate form of ˙ φ in eq. (2.21), we find B (NBD) ( k , k , k ) → r ǫ H I cos(2 δ k ) e − πk k k = 4 √ ǫ cos(2 δ k ) e − πk kk P ( k ) P ( k ) . (5.25)There is an enhancement factor in the squeezed limit, k/k ≫
1, compared to the usualsingle-field model with a Bunch-Davies initial state, B ( k, k, k ) ∝ k − k (see eq. (5.22)).This result is in agreement with the previous work on the effect of modified initial states– 17 –n the bispectrum, in which modifications are given by hand [38–40]. Despite the enhance-ment factor by k/k , however, B (NBD) ( k , k , k ) is always subdominant in the sub-curvatureapproximation due to the exponential suppression factor, e − πk .Let us now consider the exact folded limit, k i = k j + k l . In the previous study on theeffect of modified initial states (given by hand) on the bispectrum, the bispectrum divergesin this limit [35–37]. However, a self-consistent computation presented here gives a finiteresult. There is one subtlety: in order for us to use Fourier wavenumbers defined in flatspace, k i , we need to make sure that k i + k j − k l for any combinations of i , j , and k aregreater than unity (i.e., sub-curvature approximation). This requirement is not met in theexact folded limit. We thus go back to the open harmonics with p i . Also taking the squeezedlimit, p ≪ p ≈ p ≡ p , for simplicity, we obtain a compact formula, B (NBD) ( p, p − p , p ) → − . r ǫ H I sin(2 δ p ) e − πp p p , (5.26)which remains finite, and is suppressed by e − πp . ζ To compute the late-time observables such as the temperature and polarization anisotropy ofthe CMB, we need to compute the curvature perturbation on a uniform-density hypersurface, ζ , which is equal to a “comoving” ( ϕ = 0) hypersurface and is conserved outside the horizon.We thus perform the following non-linear transformation relating the scalar-field fluctuationin the flat gauge, ϕ , to ζ [65] (also see eq. (2.66) of ref. [66]) ζ = Z φ ∗ φ ∗ + ϕ ∗ dφ H ˙ φ = − H ∗ ˙ φ ∗ ϕ ∗ − ∂∂φ (cid:18) H ˙ φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) φ = φ ∗ ϕ ∗ + O ( ϕ ∗ )= − H ∗ ˙ φ ∗ ϕ ∗ + 12 H ∗ ¨ φ ∗ ˙ φ ∗ + 12 ! ϕ ∗ + O ( ϕ ∗ ) , (5.27)where the subscript ∗ indicates the moment of the horizon exit, and H ∗ ≈ H I during slow-roll inflation. The first term in eq. (5.27) implies that the bispectrum of ϕ computed inthe previous section times − H ∗ / ˙ φ ∗ yields the bispectrum of ζ . In addition, the second-termyields an additional contribution to the bispectrum of ζ .In the sub-curvature approximation, the power spectrum of ϕ given in eq. (3.17) mul-tiplied by H ∗ / ˙ φ ∗ = 1 / (2 ǫ ∗ ) with p → k yields P ζ ( k ) = H ∗ ǫ ∗ k . (5.28)Using the relation k = a ∗ H ∗ and taking into account the slow-roll time-dependence of H ∗ and ǫ ∗ , we obtain the power spectrum index as n s − ≡ d ln k P ζ ( k ) d ln k = − φ ∗ H ∗ ˙ φ ∗ − φ ∗ H ∗ . (5.29)– 18 –gnoring the non-Bunch-Davies terms and taking the squeezed limit, k ≪ k ≈ k ≡ k ,we obtain from eqs. (5.22) and (5.27) B ζ ( k , k , k ) → − H I ˙ φ ∗ H I √ ǫ ∗ k k + 2 H I ˙ φ ∗ H I ¨ φ ∗ ˙ φ ∗ + 12 ! H I k H I k = (1 − n s ) P ζ ( k ) P ζ ( k ) , (5.30)where we have used √ ǫ ∗ = − ˙ φ ∗ /H I (recall ˙ φ < B ζ → (1 − n s ) P ζ ( k ) P ζ ( k ) as k →
0, holds. However, note that we are not taking the exact squeezed limit, k →
0; rather,we are working in the sub-curvature approximation, and thus k has to satisfy k ≫
1. Also,once again, eq. (5.30) is valid at the leading order in the sub-curvature approximation. Seeappendix C for sub-leading contributions.
In this paper, we have computed the bispectrum from a single-field open inflation model, inwhich a single scalar field is responsible for both quantum tunneling (nucleation of a bubbleinside of which we live) and slow-roll inflation inside a bubble after tunneling. We assumethat the potential energy inside the bubble at the onset of slow-roll inflation is approximatelyequal to the false vacuum energy density of the background de Sitter space.A comoving observer in de Sitter space charted by open coordinates detects particleswith respect to the Euclidean vacuum state (which is equivalent to the Bunch-Davies vac-uum state in de Sitter space charted by flat coordinates), in the same sense that a comovingobserver in Minkowski space charted by Rindler coordinates detects particles with respectto the Minkowski vacuum state. Moreover, quantum tunneling modifies an initial state forquantum fluctuations away from the Euclidean vacuum state. Therefore, open inflation nat-urally provides a “non-Bunch-Davies” initial state for quantum fluctuations. The occupationnumber of particles detected inside the bubble has a thermal spectrum, exponentially fallingoff at large momenta.Most of the previous work on the effect of modified initial states for quantum fluctuationsputs mode functions characterizing a non-Bunch-Davies initial state by hand at an artificialinitial time. Instead of doing this, we compute the effect of a non-Bunch-Davies vacuum onthe bispectrum self-consistently within the framework of open inflation. An “initial time”naturally emerges because slow-roll inflation follows the period of a curvature-dominated eraright after quantum tunneling. The mode functions (hence the initial state) are uniquelyfixed given a model of open inflation.We find that the bispectrum of the curvature perturbation on a uniform-density hyper-surface, ζ , has a term going as B ζ ( k, k, k ) ∝ e − πk k − k − in the squeezed limit, k ≪ k ≈ k ≡ k . (The units are such that k ≈ k than that of thestandard single-field inflation model, B ζ ( k, k, k ) ∝ k − k − , by a factor of k/k ≫
1. Thisbehaviour agrees with phenomenological studies done by the previous work [38–40]. However,the amplitude of the bispectrum is exponentially suppressed by e − πk in the sub-curvatureregion, k ≫
1, i.e., for wavelengths shorter than the present-day curvature radius. Given that– 19 –e do not see evidence for spatial curvature within our current horizon, we conclude thatthe non-Bunch-Davies effect of open inflation on the observable bispectrum is exponentiallysuppressed. This is a consequence of the occupation number of particles detected in ourbubble falling off exponentially at large momenta. We also find that the bispectrum in theexact folded limit, e.g., k = k + k , remains finite.The leading-order bispectrum from open inflation in the sub-curvature approximationis similar to that of the standard single-field model with a Bunch-Davies initial state. Thebispectrum specific for open inflation arises only in the sub-leading order in the sub-curvatureapproximation, which is suppressed at least by the factor 1 /k i ( ≪
1) compared to the leading-order bispectrum.Open inflation provides an attractive framework within which we can discuss the originof our universe without its initial singularity. It is thus worthwhile to find any observablesignatures of open inflation. We have shown that the bispectrum picks up some signaturesof open inflation, although they are exponentially suppressed in the model we have explored.On the other hand, it is possible that the false vacuum energy density is much greater thanthe potential energy inside the bubble. In such a case, the high false vacuum energy mayovercome the exponential suppression factor, e − πk . Also, while we have ignored the effect ofsuper-curvature modes (for which p < p is the wavenumber of the open harmonics),they may become important when the false vacuum energy density is high [34]. Acknowledgments
KS would like to thank J. White, D. Yamauchi, T. Tanaka and M. Sasaki for useful discussionsand valuable comments. KS would also like to thank Max-Planck-Institut f¨ur Astrophysik forhospitality, where this work was initiated and completed. This work was supported in partby Monbukagaku-sho Grant-in-Aid for the Global COE programs, “The Next Generation ofPhysics, Spun from Universality and Emergence” at Kyoto University. KS was supported byGrant-in-Aid for JSPS Fellows No. 23-3437.
A Expansion of action in a FLRW universe with negative spatial curvature
In this appendix, we derive the reduced action for scalar-type perturbations by perturbativelyexpanding the full action, which consists of the Einstein-Hilbert action, the York-Gibbons-Hawking term, and the scalar-field action. With the metric given in eq. (3.2), the full actionis given by S = 12 Z dtd x √ h n N h R (3) − V − h ij ∂ i φ ( t, x ) ∂ j φ ( t, x ) i + 1 N (cid:18) E ij E ij − E + h ˙ φ ( t, x ) − N i ∂ i φ ( t, x ) i (cid:19)(cid:27) , (A.1)where E ij ≡ (cid:16) ˙ h ij − ∂ i N j − ∂ j N i (cid:17) , E = h ij E ij , (A.2)and R (3) is the 3-dimensional Ricci scalar.In an open FLRW universe, where the curvature of the universe is given by K = − K = 1, which has– 20 –een studied by Clunan and Seery [31]. However, there is one big difference. In open inflation,where ˙ φ ( t ) = 0 and H ( t ) ≡ ˙ a ( t ) /a ( t ) = ∞ at the initial time, t = 0, ǫ ( ≡ ( dV /dφ ) / (2 V )) ≪ φ / (2 H ) ≪ H ( t ) = 0 at the time ofbounce, ˙ φ / (2 H ) ≪ ǫ ≪
1. ref. [31] thus imposes a different conditionon V ( φ ) in order to satisfy ˙ φ / (2 H ) ≪ A.1 Second-order action
As in section 3, taking the uniform curvature (flat) gauge ( ζ = 0), we expand φ around theuniform background, φ ( t ), as φ ( t, x ) = φ ( t ) + ϕ ( t, x ), and hereafter denote φ ( t ) as φ . Weobtain the reduced action by solving the constraint equations after gauge fixing. For thesecond- or third-order action, it is enough to solve the constraint equations up to O ( ϕ ) [44].By writing N = 1 + δN and N i = γ ij ∂ j χ , we obtain the Hamiltonian constraint andthe momentum constraint up to O ( ϕ ), respectively, as2 dVdφ ϕ + 2 δN ( − H + ˙ φ ) − H∂ χ − φ ˙ ϕ = 0 , (A.3) ∂ i h HδN − Kχ − ˙ φ ˙ ϕ i = 0 . (A.4)From these equations, we obtain δN and χ up to O ( ϕ ), respectively, as δN = ˙ φ H ϕ + KH χ , (A.5) χ ≈ − H ( ∂ + 3 K ) − (cid:16) ˙ φ ˙ ϕ − ¨ φϕ (cid:17) . (A.6)Here, we have kept only the leading-order terms in the expansion in ˙ φ / (2 H ) in eq. (A.6).This expansion is equivalent to the expansion in the slow-roll parameter, ǫ . Both δN and χ are O ( ǫ / ) quantities. We keep the term proportional to ¨ φ eq. (A.6), as ¨ φ/ ( H ˙ φ ) is an O (1) quantity in the curvature-dominated era. It is however slow-roll suppressed during theslow-roll inflationary era.By substituting eqs. (A.5) and (A.6) into eq. (A.1) and keeping quantities up to O ( ϕ ),we obtain the second-order action as S = Z dtd x a √ γ " − ( d V /dφ )2 − ˙ φ ( dV /dφ )2 H ! ϕ + 12 a ∂ i ϕ∂ i ϕ + K∂ i χ∂ i χ + 12 ˙ ϕ − ˙ φ H ϕ ˙ ϕ − K ( dV /dφ ) H χϕ − K ˙ φH χ ˙ ϕ + − φ H ! ˙ φ ϕ + K ˙ φχϕ + χ ! ≈ Z dtd x a √ γ (cid:20)
12 ˙ ϕ − a ∂ i ϕ∂ i ϕ (cid:21) , (A.7)where the sub-leading terms in the slow-roll expansion are neglected in the last expression.This is eq. (3.3) in section 2. A.2 Third-order action
We also keep only the leading-order quantities in the slow-roll expansion in the third-orderaction. By substituting eqs. (A.5) and (A.6) into eq. (A.1) and keeping quantities up to– 21 – ( ϕ ), we obtain S = Z dtd x √ γ " − a ˙ φϕ H + Ka χ H ! (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19) − a ˙ ϕ∂ i χ∂ i ϕ , (A.8)where S is an O ( ǫ / ) quantity.The last term in eq. (A.8) can be integrated by parts to give − Z dtd x √ γa ˙ ϕ∂ i χ∂ i ϕ = Z dtd x √ γ (cid:18) − a Hχ∂ i ϕ∂ i ϕ − a ˙ χ∂ i ϕ∂ i ϕ + a χ ˙ ϕ∂ ϕ (cid:19) . (A.9)The last term in this expression can be integrated by parts again to give Z dtd x √ γa χ ˙ ϕ∂ ϕ = Z dtd x (cid:26)(cid:20) √ γ (cid:18) a H χ ˙ ϕ − a χ ˙ ϕ (cid:19)(cid:21) + a χ ˙ ϕ δ L δϕ (cid:27) , (A.10)where the Lagrangian density, L , is defined by the second-order action as S = R d x L .Substituting this back into eq. (A.9) we get − Z dtd x √ γa ˙ ϕ∂ i χ∂ i ϕ = Z dtd x (cid:26)(cid:20) √ γ (cid:18) a H χ (cid:0) ˙ ϕ − ∂ i ϕ∂ i ϕ (cid:1) − a χ (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19)(cid:19)(cid:21) + a χ ˙ ϕ δ L δϕ (cid:27) . (A.11)The time derivative of χ can be written as˙ χ = − Ka H χ − Hχ − ˙ φ Ha ϕ + 3 K ˙ φ Ha ( ∂ + 3 K ) − ϕ + ˙ φ H ( ∂ + 3 K ) − δLδϕ , (A.12)where δL/δϕ is given in eq. (3.11). ( L is related to L by L = a √ γL .) We have ignoredthe term proportional to ... φ using | ... φ / ( H ˙ φ ) | ≪
1, which can be derived from eq. (2.21). Bysubstituting eq. (A.12) into eq. (A.11), we obtain − Z dtd x √ γa ˙ ϕ∂ i χ∂ i ϕ = Z dtd x (" √ γ a Hχ ˙ ϕ + Ka χ H + a ˙ φ H ϕ − Ka ˙ φ H ( ∂ + 3 K ) − ϕ ! (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19)! + − a ˙ φ H (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19) ( ∂ + 3 K ) − + a χ ˙ ϕ ! δ L δϕ ) . (A.13)Finally, substituting eq. (A.13) into eq. (A.8), we obtain the third-order action S = Z dtd x √ γ " − a ( ∂ + 3 K ) − (cid:16) ˙ φ ˙ ϕ − ¨ φϕ (cid:17) ˙ ϕ − Ka ˙ φ H ( ∂ + 3 K ) − ϕ (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19) + Z dtd x " − a ˙ φ H (cid:18) ˙ ϕ + 1 a ∂ i ϕ∂ i ϕ (cid:19) ( ∂ + 3 K ) − − a H ( ∂ + 3 K ) − ˙ ϕ (cid:16) ˙ φ ˙ ϕ − ¨ φϕ (cid:17) δ L δϕ . (A.14)This is eq. (5.1) in section 5 after moving ( ∂ + 3 K ) − in the first term in the second line tothe left by integration by parts, and replacing K with K = − Analytical structure of the universe in open inflation
B.1 Coordinate systems and Wightman functions
The coordinate system defined by eqs. (2.6) to (2.9) is suitable for studying one of the modefunctions forming a complete set, u k ( x ), which multiplies the annihilation operator, ˆ a k , ineq. (3.4). Let us call this coordinate system “ K .” The K is suitable for u k ( x ) becausethe R -, L - and C -regions are connected via E , and u k ( x ) is analytical in and across all theregions.On the other hand, K is not suitable for studying another mode function, v k ( x ), whichmultiplies the creation operator, ˆ a † k , in eq. (3.4). We thus introduce a new set of coordinatesystems, which we shall call “ K .” The K is defined by ( τ ( − π/ (2 H I ) ≤ τ ≤ π/ (2 H I )) χ (0 ≤ χ ≤ π ) , ( t R = − i ( τ − π/ (2 H I )) (0 ≤ t R < ∞ ) r R = i ( χ − π ) (0 ≤ r R < ∞ ) , ( t L = − i ( − τ − π/ (2 H I )) (0 ≤ t L < ∞ ) r L = i ( χ − π ) (0 ≤ r L < ∞ ) , ( t C = τ ( − π/ (2 H I ) ≤ t C ≤ π/ (2 H I )) r C = i ( χ − π/
2) (0 ≤ r C < ∞ ) , (B.1)with the coordinates of a 2-sphere Ω i commonly used for all regions. For simplicity, weassume that the spacetime is approximately de Sitter with the metric given by eq. (2.10).In the K system, the R -, L - and C -regions are connected via E , and v k ( x ) is analyticalin and across all the regions. The difference between K and K is not a mere coordinatetransformation, but they define the relations between the E -, R -, L -, and C -regions in adifferent way.The two Wightman functions coincide for space-like separated x and x ′ , i.e., G + ( x, x ′ ) = G − ( x, x ′ ) ( ≡ G ( x, x ′ )), where G + ( x, x ′ ) = P k u k ( x ) v k ( x ′ ) and G − ( x, x ′ ) = P k u k ( x ′ ) v k ( x );however, they are different for time-like separated x and x ′ . The Wightman functions divergewhen x is on the past- or future-light cones of x ′ . This divergence is the so-called UVdivergence of the Wightman functions.In performing calculations it is necessary to choose one of the two sets of coordinatesystems, K or K , and to choose one of the two Wightman functions, G + ( x, x ′ ) or G − ( x, x ′ ).We shall discuss the origin of these possible options, in terms of the analytical structure of deSitter space charted by these coordinates. We shall then explain as to why u k ( x ) is analyticalin K , whilst v k ( x ) is not; and why v k ( x ) is analytical in K , whilst u k ( x ) is not [18].To discuss the analytical structure of de Sitter space charted by these coordinates, let uschart the 5-dimensional Minkowski spacetime, ds = η ab dx a dx b with η = diag( − , , , , x = ( x , x , x , x , x ) . (B.2)The 4-dimensional de Sitter spacetime is embedded as a hyperboloid defined by − ( x ) + X i =1 ( x i ) = 1 H I . (B.3)– 23 – E RE Figure 5 . Analytical continuation. (Left) The R -region is connected to the C -region via E by x + ,where Im x >
0. (Right) The R -region is connected to the C -region via E by x − , where Im x < G ( x, x ′ ), is uniquely determined in the space-like region of x ′ , but it divergeson the light-cone of x ′ shown by the wavy lines. Analytically extending G ( x, x ′ ) into the past-lightcone of x ′ using x + and x − , we obtain G − ( x, x ′ ) in the shaded area in the left panel and G + ( x, x ′ ) inthe shaded area in the right panel, respectively. The Euclidean region is given by imposing x to be pure imaginary, and it is related to thecoordinates given in eqs. (2.6) to (2.9) or eq. (B.1) as [51] ix = cos τ cos χ , x = sin τ , x x x = cos τ sin χ Ω , (B.4)where Ω = (Ω , Ω , Ω ) is a 3-dimensional unit vector. With the Cartesian coordinates, wecan discuss the analytical structure of de Sitter spacetime without worrying about coordinatesingularities.In QFT on an open inflation background, we encounter two types of singularities. Oneis a coordinate singularity and the other is a UV divergence of the Wightman functions. Thefirst issue is the coordinate singularity. Let us consider the coordinates ( t C , r C , Ω) definedin the C -region (i.e., ( − π/ (2 H I ) ≤ t C ≤ π/ (2 H I ) and 0 ≤ r C < ∞ ) with the metric givenby the fourth line of eq. (2.10). The coordinates in the C -region are analytical (i.e., themetric is analytical and non-singular), which means that even if the coordinates are givenonly in a finite part of the C -region, we can analytically extend them to the whole C -region.However, coordinate singularities appear on the boundaries between the C - and R -regionsand the C - and L -regions, and the analytical extension of the coordinate system beyond theseboundaries seems not possible, even though the spacetime is extended.The second issue is the UV divergence of the Wightman functions. Let us fix x ′ andregard the Wightman function, G ( x, x ′ ), as a function of x . As G ( x, x ′ ) is analytical in thespace-like region of x ′ , we can analytically extend it to the whole space-like region of x ′ ,even if G ( x, x ′ ) is given only in a finite part of the space-like region of x ′ . However, G ( x, x ′ )diverges on the past- or future-light cones of x ′ , and again, the analytical extension of G ( x, x ′ )beyond the past- or future-light cones of x ′ seems not possible. There is no difference between K and K in the C -region. – 24 – m t Re tRRLL C, EC, E Figure 6 . Correspondence between the do-mains of t and the E -, R -, L -, and C -regions.The upper (red) lines show the integrationpath V charted by K , while the bottom (green) lines show V charted by K . Im r Re rL, RL, RCCE E Figure 7 . Same as figure 6, but for r . Thebottom (red) dashed lines show the originalintegration path V charted by K , while theupper (green) dashed lines show V chartedby K . The integral can be evaluated by de-forming the paths to lie along the real axis of r , as shown by the solid lines. Those apparent issues regarding the analytical extension of the coordinates and theWightman functions can be overcome by taking a path through the complex plane. Byadding an infinitely small complex number, ± iǫ ( ǫ > x , we obtain x ± = ( x ± iǫ, x , x , x , x ) . (B.5)Since x ± no longer passes right through the singularities, the coordinates and the Wightmanfunctions can be analytically extended to all the regions charted by K or K .Let us take x + first, and see how the analytical extension is performed. The coordinatesystem defined in the C -region can be analytically extended with x + beyond the coordinatesingularities on the boundaries between the C - and R -regions and the C - and L -regions. AsIm x > E , the resulting set of coordinate systems corresponds to K , which connects the R -, L -, and C -regions via E (see figure 5). The Wightman function G ( x, x ′ ), considered asa function of x for a given x ′ ( x is in the space-like region of x ′ ), can be analytically extendedby shifting x + beyond the past-light cone of x ′ , where G ( x, x ′ ) diverges (see section 5.4 ofref. [53]). As a result, G − ( x, x ′ ) is obtained inside the past-light cone of x ′ as shown infigure 5. Since the analyticity of v k ( x ) and G − ( x, x ′ ) with respect to x are equivalent, theanalyticity of v k ( x ) with respect to x + is also guaranteed, which leads to the analyticity of v k ( x ) with respect to K and the regularity of v k ( x ) on E .Similarly, the analytical extension using x − gives K , which connects the R -, L -, and C -regions via E . As a result, G + ( x, x ′ ) is obtained inside the past-light cone of x ′ . Then u k ( x ) is analytical with respect to K and is regular on E . B.2 Deformation of integration path using analyticity
Thanks to the analyticity of u k ( x ) and v k ( x ) with respect to the set of coordinate systems K and K , respectively, we can deform the path of integration given in eq. (4.2). For– 25 –implicity, we shall consider the tree-level calculation and write the integral schematically as R ( C + C ) × Σ λ dλd x √− gf ( x ).Let us consider the integral over V = C × Σ λ in the coordinate system K . We can usethe coordinates in the R -region, t and r , in the whole V by allowing t and r to be complexvariables. Using the relation between coordinates given by eq. (B.1), V is given by the sumof E = { ( t, r, Ω) | t ∈ (0 , iπ/H I ) , r ∈ (0 , − iπ/ , Ω ∈ S } ,C = { ( t, r, Ω) | t ∈ (0 , iπ/H I ) , r ∈ ( − iπ/ , − iπ/ ∞ ) , Ω ∈ S } ,R = { ( t, r, Ω) | t ∈ (0 , ∞ ) , r ∈ (0 , ∞ ) , Ω ∈ S } ,L = { ( t, r, Ω) | t ∈ ( iπ/H I , −∞ ) , r ∈ (0 , ∞ ) , Ω ∈ S } , (B.6)where, for simplicity, we additionally assume that Σ is in the far future of the nucleationsurface and that the Lorentzian region of V can be approximated as the whole de Sitterspacetime after the nucleation time. We can then simplify the integration domain by de-forming the integration path as Z C × Σ λ dλd x √− gf ( x ) = Z ∞ iπ/H I −∞ dt Z ∞ dr Z d Ω √− gf ( x ) . (B.7)The upper (red) lines in figure 6 show the integration path along t , while the bottom (red)lines in figure 7 show how we deform the integration path to lie along the real axis of r .The deformation shown in figure 7 is possible because the integrand √− gf ( x ) (which is afunction of the background quantities and v k ( x ), both of which are analytical) is analyticalwith respect to t and r given by eq. (B.6), and √− gf ( x ) falls off rapidly as Re r → ∞ forsub-curvature modes, i.e., p > f pl ( r ) ∝ e − r (seeeq. (3.6)). However, this deformation would not be valid for super-curvature modes, whichwe ignore in this paper.Similarly, with the set of coordinate systems K given by eqs. (2.6) to (2.9), the domainof integration V = C × Σ λ is given by the sum of E = { ( t, r, Ω) | t ∈ (0 , − iπ/H I ) , r ∈ (0 , iπ/ , Ω ∈ S } ,C = { ( t, r, Ω) | t ∈ (0 , − iπ/H I ) , r ∈ ( iπ/ , iπ/ ∞ ) , Ω ∈ S } ,R = { ( t, r, Ω) | t ∈ (0 , ∞ ) , r ∈ (0 , ∞ ) , Ω ∈ S } ,L = { ( t, r, Ω) | t ∈ ( − iπ/H I , −∞ ) , r ∈ (0 , ∞ ) , Ω ∈ S } , (B.8)and we can simplify the integration domain by deforming the integration path as Z C × Σ λ dλd x √− gf ( x ) = − Z ∞− iπ/H I −∞ dt Z ∞ dr Z d Ω √− gf ( x ) , (B.9)as is shown in the bottom (green) lines in figure 6 and the upper (green) lines in figure 7.In the single-field open inflation model that we study in this paper, the mode functionscontributing to the bispectrum in the R -region vanish outside the bubble. Thus, eqs. (B.7)and (B.9) further simplify to Z C × Σ λ dλd x √− gf ( x ) = Z ∞ iR W dt Z ∞ dr Z d Ω √− gf ( x ) , (B.10) Z C × Σ λ dλd x √− gf ( x ) = − Z ∞− iR W dt Z ∞ dr Z d Ω √− gf ( x ) , (B.11)respectively, where R W is the bubble radius measured in the coordinates of the E -regions.– 26 – Order-of-magnitude estimate for sub-leading contributions
In section 5.3, we have computed the leading-order contribution to the bispectrum, L (1) int , inthe R -region, but have ignored the other contributions. In this appendix, we show that thecontributions from the other terms in L int are sub-dominant in the sub-curvature approxima-tion. We also show that the contributions from outside the R -region are sub-dominant in thesub-curvature approximation, which implies that the effects of possibly large self-interactionnear the bubble wall are also sub-dominant. For simplicity, we shall show the result only inthe squeezed configuration, k ≪ k ≈ k ( ≡ k ).In the following calculations, we change the integration variable from t to the conformaltime, η , in the integration given in eq. (B.11). Accordingly, the integration domain changesfrom t : − iR W → → ∞ to η : η W → −∞ →
0, where η W is given by η W ≡ −
12 log (cid:18) cosh H I t + 1cosh H I t − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = − iR W (cid:18) ⇔ e η W = (cid:18) H I R W √ (cid:19) e − iπ/ (cid:19) . (C.1)The approximate forms of a , H , ˙ φ , ¨ φ , and u k given in eqs. (2.20), (2.21), and (5.14) originallyobtained with the condition −∞ < η . − η ∈ ( η W , −∞ ). This isbecause the approximate forms for −∞ < η . − < t . H − I ) are obtained using | tH I | ≪
1, which is also satisfied for η ∈ ( η W , −∞ ) (or t ∈ ( − iR W , H I R W . C.1 Contributions from outside the R -region We calculate the contribution to the leading-order bispectrum, B (1) ( k , k , k ), from L (1) int outside the R -region, η ∈ ( η W , −∞ ). Using the same methods as were used to obtaineq. (5.17), B (1) ( k , k , k ) is given by B (1) ( k , k , k ) = 2Re " iv k (0) v k (0) v k (0) Z −∞ η W dη a ˙ φk + 4 ˙ u k ( η ) ˙ u k ( η ) ˙ u k ( η ) ! + (perms . ) , (C.2)where only the integration domain is different from eq. (5.17). Substituting eqs. (2.21), (5.14),(5.15), and (C.1) into eq. (C.2), we obtain B (1) ( k , k , k ) = 2Re (cid:20) i (cid:18) H I k (cid:19) k k k Z −∞ η W dη ˙ φe − i ( k + k + k ) η + (perms . ) (cid:21) ≈ √ ǫH I e − π (2 k + k ) / k k H I R W , (C.3)where we do not show an O (1) pre-factor, which does not affect the order-of-magnitudeestimate.The exponential suppression factor, e − π (2 k + k ) / , appears because the mode functions ofsub-curvature modes are exponentially suppressed outside the R -region. (One may check thisby substituting η ∈ ( η W , −∞ ) into u k ( η ) given in eq. (3.9).) This factor makes eq. (C.3) muchsmaller than B ( k , k , k ) in eq. (5.21) in the sub-curvature approximation. This suppression– 27 –s universal for any integrations outside the R -region. We thus conclude that the integrationof all terms in L int outside the R -region is negligible in the sub-curvature approximation.This also implies that the contribution from the possibly large self-interaction near the bubblewall (which is in the C - and E -regions) is negligible in the sub-curvature approximation. C.2 Contributions from the sub-leading terms in Lagrangian
Next, we calculate the contributions from the other terms of L int given in eq. (5.3). Inaddition to the leading-order term, L (1) int , there are three terms: L (2) int = √ γa ¨ φ ( ∂ − − ϕ c ˙ ϕ c , L (3) int = √ γ a ˙ φ H ( ∂ − − ϕ c ˙ ϕ c , L (4) int = − √ γ a ˙ φ H ( ∂ − − ϕ c a ∂ i ϕ c ∂ i ϕ c . (C.4)As the integral outside the R -region is negligible in the sub-curvature approximation, weshall consider the contributions from the integral inside the R -region.Let us start with L (2) int . The relative size of L (2) int compared to L (1) int is L (2) int / L (1) int ≈ ¨ φϕ c / ˙ φ ˙ ϕ c .According to eqs. (2.21), (5.14), and (5.15), this ratio is ¨ φϕ c / ˙ φ ˙ ϕ c ≈ aH/k for −∞ < η . − φϕ c / ˙ φ ˙ ϕ c ≈ O ( ǫ ) for − . η <
0. The ratio in − . η < −∞ < η . − k/ ( aH ) ≫
1. The horizon sizeand the curvature radius are comparable, i.e., aH ≈
1, during the curvature-dominated era, −∞ < η . −
1. Therefore, the ratio in −∞ < η . − k ≫ L (3) int . The bispectrum is B (3) ( k , k , k ) = 2Re " iu k (0) u k (0) u k (0) Z −∞ dη a ˙ φ H ( η )( k + 4) u k ( η ) ˙ u k ( η ) ˙ u k ( η ) ! + (perms . ) i . (C.5)Using eqs. (2.20), (2.21), (5.14), and (5.15), we estimate the contribution from η ∈ ( − ,
0) as √ ǫH Re (cid:20) i k k k Z − dη (1 + ik η ) e − i ( k + k + k ) η (cid:21) + (perms . ) ≈ √ ǫH k k , (C.6)where the integral is estimated as O ( k /k T ) with k T ≡ k + k + k . Using eqs. (2.20), (2.21),(5.14), and (5.15), we estimate the contribution from η ∈ ( −∞ , −
1) as √ ǫH k k k Re (cid:20)Z − −∞ dη e η e − ik η e − ik η e − ik η (cid:21) + (perms . ) ≈ √ ǫH k k , (C.7)where the integral is estimated as O ( k − T ). We thus find B (3) ( k , k , k ) ≈ √ ǫH k k , (C.8)up to a factor of O (1). This term is negligible compared to the leading term given in eq. (5.21)in the sub-curvature approximation, k ≫ L (4) int . The bispectrum is B (4) ( k , k , k )= 2Re " iu k (0) u k (0) u k (0) Z −∞ dη a ˙ φ ( − k + k + k + 1)8 H ( η )( k + 4) u k ( η ) u k ( η ) u k ( η ) ! + (perms . ) i , (C.9)where we have used the property of open harmonics Z d x √ γ Y p l m ∂ i Y p l m ∂ i Y p l m = − p + p + p + 12 Z d x √ γ Y p l m Y p l m Y p l m . (C.10)Using eqs. (2.20), (2.21), and (5.14), we estimate the contribution from η ∈ ( − ,
0) as √ ǫH ( − k + k + k ) k k k Re (cid:20) − i Z − dηη (1 + ik η )(1 + ik η )(1 + ik η ) e − i ( k + k + k ) η (cid:21) + (perms . ) ≈ √ ǫH k k , (C.11)where the integral is estimated as O ( k k k /k T ). Using eqs. (2.20), (2.21), and (5.14), weestimate the contribution from η ∈ ( −∞ , −
1) as √ ǫH ( − k + k + k ) k k k Re (cid:20)Z − −∞ dηe η e − i ( k + k + k )) η (cid:21) + (perms . ) ≈ √ ǫH k k , (C.12)where the integral is estimated as O ( k − T ). We thus find B (4) ( k , k , k ) ≈ √ ǫH k k , (C.13)up to a factor of O (1). This term is negligible compared to the leading term given in eq. (5.21)in the sub-curvature approximation, k ≫ D Open harmonics in the sub-curvature approximation
D.1 Correspondence with flat harmonics
In this appendix, we show the correspondence between the open and flat harmonics in thesub-curvature approximation, where the wavelength of the modes and the separation be-tween points are assumed to be smaller than the curvature radius, i.e., 1 ≪ p and r ≪ γ ij is given by eq. (2.15), the open harmonics Y plm ( x ) are defined by eq. (3.6). On a 3-dimensional flat space, whose metric is given by˜ γ ij dx i dx j = dr + r d Ω , (D.1)the flat spherical harmonics ˜ Y klm ( x ) are defined by [67]˜ Y klm ( x ) = Ψ kl ( r ) Y lm (Ω) , Ψ kl ( r ) = r π kj l ( kr ) , (D.2)– 29 – r f, ψ Figure 8 . Comparison between f kl ( r ) (solid lines) and Ψ kl ( r ) (dashed lines) for l = 10 as a functionof r . The red, blue, and green lines show l = 1, 3, and 5, respectively. The two functions agree wellin r ≪ where j l ( x ) is the spherical Bessel function and Y lm (Ω) is the usual spherical harmonics ona 2-sphere. While Y plm ( x ) satisfies the relations given by eq. (3.7), ˜ Y klm ( x ) satisfies thefollowing relations ∂ ˜ Y klm ( x ) = − k ˜ Y klm ( x ) , ˜ Y ∗ klm ( x ) = ( − m ˜ Y kl − m ( x ) , Z d x p ˜ γ Y ∗ k l m ( x ) ˜ Y k l m ( x ) = δ ( k − k ) δ l l δ m m , Z ∞ dk X lm ˜ Y klm ( x ) ˜ Y klm ( x ′ ) = δ (3) ( x − x ′ ) . (D.3)To begin with, let us find the correspondence between Y plm ( x ) and ˜ Y klm ( x ). As theangular parts Y lm (Ω) are the same for the open and flat harmonics, we focus on the cor-respondence between the radial parts, i.e., f pl ( r ) in eq. (3.6) and Ψ kl ( r ) in eq. (D.2). Thecorrespondence between f pl ( r ) and Ψ kl ( r ) with identification p → k in the sub-curvatureapproximation, 1 ≪ p and r ≪ f pl ( r ) (solid lines) and Ψ kl ( r ) (dashed lines) as afunction of r for p = k = 10 and l = 1, 3, and 5.The asymptotic behaviours of these functions are useful for understanding their corre-spondence: Ψ kl ( r ) → r π sin (cid:0) kr − π l (cid:1) r ( l ≪ kr ) k l +1 r l l +1 / Γ( l + 3 /
2) ( kr ≪ l ) , (D.4) f pl ( r ) → r π sin (cid:0) pr − π l (cid:1) sinh r ( l ≪ pr , ≪ p ) p l +1 r l l +1 / Γ( l + 3 /
2) ( pr ≪ l , ≪ p ) . (D.5)– 30 –n the sub-curvature approximation, where sinh r ≈ r , f kl ( r ) and Ψ kl ( r ) agree with eachother in the two opposite asymptotic regions, 0 < r ≪ l/k and l/k ≪ r ≪ Y plm ( x )and ˜ Y klm ( x ). The correspondence in the square integrals is exact, as both of them form anorthonormal set. We write the cubic integrals for Y plm ( x ) as Z d x √ γ Y p l m ( x ) Y p l m ( x ) Y p l m ( x ) = F l l l p p p G m m m l l l , (D.6)where F l l l k k k is given in eq. (5.7) and G m m m l l l is the Gaunt integral given in eq. (5.8). Wewrite the cubic integrals for ˜ Y klm ( x ) as Z d x p ˜ γ ˜ Y k l m ( x ) ˜ Y k l m ( x ) ˜ Y k l m ( x ) = Ψ l l l k k k G m m m l l l , (D.7)where Ψ l l l k k k is defined by Ψ l l l k k k ≡ Z ∞ drr Ψ k l ( r )Ψ k l ( r )Ψ k l ( r ) . (D.8)As the angular parts are given by the Gaunt integral, G l l l m m m , we shall focus on the corre-spondence between F l l l p p p and Ψ l l l k k k with the identification of p i → k i for all i = 1, 2, and3 in the sub-curvature approximation, 1 ≪ p i .Strictly speaking, however, 1 ≪ p i is not enough to show the correspondence in thecubic integral of harmonics. We thus additionally impose l i ≪ p i for all i and 1 ≪ p min where p min ≡ min( | ± p ± p ± p | ). In the following, we shall use k not only for the indicesof the flat harmonics, but also for those of the open harmonics.Now, there exists an r ∗ ( ≪
1) satisfying both 1 /k min ≪ r ∗ where k min ≡ min( | ± k ± k ± k | ) and l i /k i ≪ r ∗ for all i . Using r ∗ we can divide the integral in eq. (D.8) into twoparts, corresponding to 0 < r < r ∗ and r ∗ < r < ∞ . In the second part, by using the l ≪ kr case of eq. (D.4) and the addition theorem for the sin functions, we can rewrite the integrandin the form of sin[ kr + (phase)] /r , where k is one of ± k ± k ± k for all combinations of ± .Using the inequality (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r ∗ sin[ kr + (phase)] r dr (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) − cos[ kr + (phase)] kr (cid:21) ∞ r ∗ + Z ∞ r ∗ cos[ kr + (phase)] kr dr (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) kr ∗ (cid:12)(cid:12)(cid:12)(cid:12) , (D.9)and 1 ≪ k min r ∗ , we find that the integral over r ∗ < r < ∞ is small and Ψ l l l k k k mainlycomes from the integral over 0 < r < r ∗ . Similarly, we find that F l l l k k k mainly comes fromthe integral over 0 < r < r ∗ . The integrals over 0 < r < r ∗ for F l l l k k k and Ψ l l l k k k are infact the same, as the integrands coincide in this region; hence the correspondence between F l l l k k k and Ψ l l l k k k . (To see this, use eq. (D.5) for pr ≪ l and eq. (D.4) for kr ≪ l , and recallsinh r ≈ r for r ≪ F l l l k k k and Ψ l l l k k k . – 31 – l l l k k k (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Ψ l l l k k k F l l l k k k Table 1 . Some numerical values of Ψ l l l k k k and F l l l k k k . They agree well when 1 /k min ≪ r ∗ (where k min ≡ min( | ± k ± k ± k | )) and l i /k i ≪ r ∗ for all i are satisfied. D.2 Correspondence with Fourier modes in flat space
Having established the correspondence between the open and flat harmonics, let us nowestablish the correspondence between coefficients of the open harmonics and those of Fourierexpansion in flat space.Let φ ( x ) be a real function in a 3-dimensional flat space. We shall determine therelation between the multipole expansion coefficients of the flat harmonics, φ klm , and theFourier coefficients, φ ( k ). We expand φ ( x ) in two way as φ ( x ) = Z ∞ dk X lm Ψ kl ( r ) Y lm (Ω) φ klm , (D.10)= Z d k (2 π ) e i kx φ ( k ) , (D.11)where r = | x | and Ω is the direction of x . Using eq. (D.2), reality of φ ( x ), and the followingidentity e i kx = X lm i l j l ( kr ) Y ∗ lm (Ω) Y lm (Ω k ) , (D.12)where k = | k | and Ω k is the direction of k , we find φ ∗ klm = i l (2 π ) / k Z d Ω k Y lm (Ω k ) φ ( k ) . (D.13)By using eq. (D.2) and eq. (D.12), we can derive the useful relation i l k (2 π ) / Z d Ω k Y lm (Ω k ) e − i kx = ˜ Y klm ( x ) , (D.14)which will be used below. D.3 Power spectrum and bispectrum
We derive the correspondence between the power spectra and bispectra computed with theFourier coefficients and the coefficients of flat harmonics. The power spectrum and bispec-trum for the Fourier modes are given, respectively, by (cid:10) φ ( k ) φ ( k ∗ ) (cid:11) = (2 π ) δ ( k − k ) P ( k ) , (D.15)and (cid:10) φ ( k ) φ ( k ) φ ( k ) (cid:11) = (2 π ) δ ( k + k + k ) B ( k , k , k ) . (D.16)– 32 –sing eqs. (D.3), (D.13), (D.14), (D.15), and (2 π ) δ ( k − k ) = R d x e − i ( k − k ) x , we canre-write the power spectrum for the coefficients of flat harmonics as D φ k l m φ ∗ k l m E = (cid:18) i l k (2 π ) / Z d Ω k Y l m (Ω k ) (cid:19) (cid:18) ( − i ) l k (2 π ) / Z d Ω k Y ∗ l m (Ω k ) (cid:19) (2 π ) δ ( k − k ) P ( k )= δ ( k − k ) δ l l δ m m P ( k ) . (D.17)Similarly, using eqs. (D.3), (D.13), (D.14), (D.16), and (2 π ) δ ( k + k + k ) = R d x e − i ( k + k + k ) x ,we can re-write the bispectrum for the coefficients of flat harmonics as D φ k l m φ k l m φ k l m E = Ψ l l l k k k G m m m l l l B ( k , k , k ) , (D.18)where Ψ l l l k k k G m m m l l l are defined in eq. (D.8).Finally, let us find the correspondence between the power spectra and bispectra inopen and flat universes. In an open universe, P ( p ) is defined by (cid:10) φ ∗ p l m φ p l m (cid:11) = δ ( p − p ) δ l l δ m m P ( p ), as in section 3, where φ p i l i m i are themultipole expansion coefficients with respect to the open harmonics Y p i l i m i . Comparing thisequation with eq. (D.17), we find the correspondence between P ( p ) and P ( k ) withidentification of p → k . Similarly, B ( p , p , p ) is defined by (cid:10) φ p l m φ p l m φ p l m (cid:11) = B ( p , p , p ) F l l l p p p G m m m l l l , as in section D. Comparing thisequation with eq. (D.18), and using the correspondence between F l l l p p p and Ψ l l l k k k in thesub-curvature approximation, we find the correspondence between B ( p , p , p ) and B ( k , k , k ) with the identification of p i → k i . References [1] J. R. Gott,
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