Vacuum fluctuations in an ancestor vacuum: A possible dark energy candidate

We consider an open universe created by bubble nucleation, and study possible effects of an"ancestor vacuum"(de Sitter space in which bubble nucleation occurred) on the present universe. We compute vacuum expectation values of energy-momentum tensor for a minimally coupled scalar field, carefully taking into account the effect of the ancestor vacuum by the Euclidean prescription. In the study of the time evolution, an important role is played by the so-called supercurvature mode, which is non-normalizable on a spatial slice of open universe and decays in time most slowly. We point out that vacuum energy of a quantum field can be regarded as dark energy if mass of the field is of order the present Hubble parameter or smaller. We obtain preliminary results for the dark energy equation of state w(z) as a function of the redshift.


Introduction
In a theory with multiple vacua, nucleation of bubbles of a true vacuum can occur due to quantum tunneling. Bubble nucleation provides a mechanism for the creation of an FLRW(Fiedmann-Lemaître-Robertson-Walker) universe. Since superstring theory is expected to have a "landscape" of metastable vacua (see e.g., [1,2]), it is likely that our universe has been created by bubble nucleation.
Understanding of the characteristic features of a universe created by bubble nucleation is of great importance. One of such features is the negative spatial curvature. We know from the classical solution due to Coleman and De Luccia [3] which describes the creation and the evolution of the bubble, that the universe inside the bubble should have negative curvature. Even though our universe is known to be flat within the margin of error, it is logically possible that there is a finite radius R c of negative curvature. The bound is roughly R c 10H −1 0 , where H 0 is the current Hubble parameter [4,5].
Another feature would be the possible signatures in the cosmic microwave background radiation (CMB). The spectrum of the CMB temperature fluctuations is consistent with the nearly scale-invariant spectrum of primordial fluctuations predicted by inflation [6]. However, if the number of e-foldings of inflation is finite, we might be able to see deviations from scale invariance due to the evolution of the universe before inflation. Such effects are expected to affect the low-ℓ (angular momentum) modes of the power spectrum. The CMB spectrum in the universe created by bubble nucleation has been understood in the 1990's 1 [7,8,9,11,12,13,14]. The studies on the CMB spectrum after the advent of string landscape include [15,16,17,18,19]. Although no signature of bubble nucleation have been found in the CMB yet, theoretical study for seeking such a signature is undoubtedly important.
In this paper, we will focus on a third possible feature, related but different from the above two. We will consider how the quantum fluctuations generated before bubble nucleation affect vacuum energy in the present universe. Universe created by bubble nucleation is surrounded by a parent de Sitter space, which we call the ancestor vacuum (See Fig. 1). The Hubble parameter H A for the ancestor vacuum is typically larger than the value H I for inflation after tunneling. On dimensional grounds, large fluctuations will be generated in the ancestor vacuum. In this paper, we will compute the vacuum expectation value of the energy-momentum tensor of a free scalar field, carefully taking into account the effect of the ancestor vacuum.
To find the vacuum expectation value of the energy-momentum tensor, which is quadratic in the quantum field, we compute the two-point functions of the field and take the coincident-point limit. The two-point functions at early time (in our universe) are obtained by the Euclidean prescription, and their subsequent time-evolution is studied by using the equation of motion for the scalar field.
We will be particularly interested in the contribution from a special mode of fluctuations, the so-called "supercurvature mode 2 ," originally found by M. Sasaki, T. Tanaka and K. Yamamoto [7,8,9] 3 . This mode decays more slowly than e −R at large R (where R is the geodesic distance), and is not normalizable on the spatial slice H 3 in the open universe. Such a mode appears if the mass of the scalar field in the ancestor vacuum is small enough. For example, for exactly massless fields, two-point functions remain finite even when two points are infinitely separated on H 3 . Heuristically, such fluctuations can be considered as the super-horizon fluctuations in the ancestor (de Sitter) vacuum, seen from the inside of the bubble [23]. Mathematical reason that a non-normalizable mode can exist in the open universe is that the spatial slice H 3 is not a global Cauchy surface (see Fig. 1). To quantize the fluctuations, one need to take a complete set of normalizable modes on a Cauchy surface, such as the surface on the horizonal brown line in Fig. 1. Supercurvature mode appears as a result of analytic continuation of the correlator to the open universe, as we will review in Section 3.
Supercurvature modes decay more slowly than normalizable modes, not only in space but also in time. Thus, it may have a chance to affect our current 2 It is difficult to find a situation in which the supercurvature mode affects the CMB. The fluctuations of the tunneling field should have large mass in order for Coleman-De Luccia instanton to exist [3], thus they will not have a supercurvature mode. Multi-field models (the tunneling and inflaton fields being different) have a potential problem as pointed out in [10], but there is a recent attempt [20,21] at explaining a "dipolar anomaly" in the CMB using a supercurvature mode in curvaton models. Gravitons (tensor modes) are massless, but their supercurvature modes are pure gauge [11].
universe. In previous papers by a part of the present authors [24,25] 4 , the evolution of vacuum fluctuations generated during and before inflation has been studied, without taking into account the bubble nucleation. The emergence of the supercurvature mode is an interesting phenomenon, special to a universe created by bubble nucleation.
As long as the mass of the scalar field is smaller than the Hubble parameter in our universe, the field value for the supercurvature mode remains nearly constant. This is the well-known freezing of the super-horizon fluctuations; supercurvature modes can always be considered to be outside the horizon. The energy-momentum tensor for this mode behaves similarly to that for cosmological constant. After Hubble parameter decreases below the mass, the field begins to oscillate, and its energy decays 5 . If there is a field with mass of order the present Hubble H 0 or smaller, the supercurvature mode of this field will be essentially frozen until today. This gives us an interesting possibility for the realization of dark energy.
The primary purpose of this paper is to explain how to calculate the vacuum expectation value of the energy-momentum tensor in a background with a bubble nucleation. To the best of our knowledge, this calculation has not been done before. Another purpose is to show that dark energy can be obtained as a contribution from the supercurvature mode to the vacuum energy.
We consider the fluctuations of a minimally coupled scalar field φ which is different from the tunneling field Φ. The field φ does not have vacuum expectation value, and is treated as a free field in curved spacetime. We assume the mass m A of φ in the ancestor vacuum to be sufficiently small relative to Hubble parameter, m A ≪ H A , so that a supercurvature mode exists. We allow a possibility that mass m 0 of φ in the true vacuum is different from m A . (This can be realized e.g., if there is a coupling of the form Φ 2 φ 2 , and the expectation value of Φ is large in the false vacuum and small in the true vacuum.) It would be natural to assume m 0 ≪ m A , since the energy scale in the true vacuum will be typically lower than in the false vacuum.
The order of magnitude of the vacuum energy (derived rigorously below) can be estimated as follows. The expectation value of the field squared φ 2 in 4 See also [26,27,28,29,30] for related work. 5 The energy density of a homogeneous field oscillating in time decays at the same rate as matter energy density as universe expands, ρ ∼ a −3 . the ancestor vacuum goes as φ 2 ∼ H 4 A /m 2 A . This is essentially the same as the well-known expression in pure de Sitter space, which grows infinitely large in the massless and infinite e-folding limits (see e.g., [31,32]) 6 . If we assume m 0 H 0 , the field value φ 2 is nearly frozen until now (apart from a weak time dependence due to the non-zero m A , which may play an important role in observationally distinguishing our mechanism from others). The dominant part of energy-momentum tensor for the field φ is the mass term, ρ ∼ m 2 Then, it is possible to make this of order the dark energy, ρ ∼ , H A being the geometric mean of M P and m A ), which does not seem particularly difficult to satisfy.
In this scenario we need an ultra-light field with mass m 0 ∼ H 0 ∼ 10 −33 eV. Thus, we cannot call this a "natural" solution to the cosmological constant problem. Nevertheless, we believe a detailed study is worthwhile, especially in view of the proposal of the "string axiverse," which states that there is a large number of axions with mass ranging down to m 0 ∼ H 0 [33].
The idea of realizing dark energy as vacuum energy of an ultra-light field is not essentially new. See e.g., [33,34,35,26,25,29,36] for previous work, which do not take account of bubble nucleation of the universe during pre-inflationary epoch. At the end of the paper, we will comment on the possibility of observationally distinguishing our mechanism from others. This paper is organized as follows. In Section 2, we review Coleman-De Luccia instanton, which describes the nucleation and the evolution of a bubble in de Sitter space (ancestor vacuum). In Section 3, we review the calculation of correlation functions of the scalar field based on analytic continuation from Euclidean space. Using this prescription, we obtain the correlators in the early-time limit in the open universe. In Section 4, we compute the expectation value of the energymomentum tensor by taking the coincident-point limit of the two-point function obtained in the previous section. In particular, we study the mass term in the energy-momentum tensor in the limit of small mass. In Section 5, we consider the evolution of the energy density in the open universe. We first obtain the scale factor for the open FLRW universe in each era (curvature domination, inflation, radiation and matter domination). We then solve the equation of motion for the 6 In the case of an inflation with a finite e-folding N and a Hubble parameter H, we instead have φ 2 ∼ N H 2 . See, e.g., [24,25] and references therein. scalar field in each era, and find the wave function by smoothly connecting the solution to the one in the early-time limit. Using this wave function, we obtain the expectation value of the energy-momentum tensor at late time. In Section 6, we will summarize our results and comment on the directions for future work. In Appendix A, we describe the harmonics on H 3 . In Appendix B, we calculate φ 2 in pure de Sitter space using our method, and show that it agrees with the known value obtained by standard techniques. In Appendix C, we find the scale factor for an open universe which contains both matter and radiation. In Appendix D, we present the details on the matching of the wave functions of the scalar field across different eras.

The CDL geometry
As a model of bubble nucleation, we consider a scalar field Φ with a potential which has two minima, like the one depicted in Fig. 1. This field Φ goes through tunneling from the false vacuum to a true vacuum (Φ should not be confused with the field φ, which will be introduced later and has zero expectation value). For phenomenological reasons, we need inflation after tunneling. Thus, we assume the potential has a plateau region on the true vacuum side, on which slow-roll inflation occurs. We ignore the present cosmological constant (dark energy), since we want to derive it from the vacuum energy of a quantum field, rather than treating it as a background given by the tunneling/inflaton field 7 .
In a theory with gravity, the geometry for a false vacuum with positive vacuum energy is de Sitter space. We assume that there is a global surface without bubbles at some time in de Sitter space 8 . Then, bubbles of true vacuum will form inside the "ancestor vacuum" (parent de Sitter). This process is described by the Coleman-De Luccia (CDL) instanton [3].
Even though our goal is to explore the evolution of the universe that results from the scalar potential in Fig. 1, we first study a thin-wall limit, in which the transition occurs sharply from a false vacuum with positive vacuum energy to a true vacuum with zero vacuum energy. The purpose of this analysis is to un-derstand the early-time behavior in the FLRW universe inside the bubble. The universe at early time is dominated by the spatial curvature, thus vacuum energy is unimportant and can be set to zero. In principle, one should be able to obtain the entire evolution of our universe by analytic continuation from Euclidean. However, for that purpose one needs to know the corresponding Euclidean space to an infinite precision, which is clearly an intractable task. Thus we will use analytic continuation to obtain the early-time behavior only, and solve the equation of motion for the scalar field to obtain the late-time behavior in the FLRW universe. The latter analysis will be done in Section 5.

Causal structure
The Penrose diagram for a spacetime which contains one bubble is depicted in Fig. 1. Our Universe is an open FLRW universe inside the bubble (Region I). The beginning of the FLRW time is the 45-degree line shown in red. Even though the scale factor vanishes there, it is merely a coordinate singularity.
The constant-time slices in Region I are 3-hyperboloid, H 3 , represented by blue lines. The bubble wall is represented by the green line in Region III. On the right of the domain wall is the ancestor vacuum. We have depicted the timereversal symmetric Penrose diagram to make the symmetry orbits clearer, but the lower half part of the diagram should be considered to be unphysical, and replaced by pure de Sitter space.
The Penrose diagram in Fig. 1, which has null future infinity in Region I, is the one for the case of zero final cosmological constant (c.c.). We do not know whether the present c.c. (dark energy) persists into infinite future. If dark energy is due to vacuum energy of a quantum field as proposed later in this paper, it will relax to zero in the future, thus we draw the Penrose diagram for this case. If the final c.c. is positive, the 45 degree line for the null infinity is replaced by a more horizontal curve which represents spacelike infinity.
The whole geometry and the configuration of the field Φ have the SO(3, 1) symmetry. The surfaces of constant Φ are the slices on which the SO(3, 1) symmetry acts as isometry. In Region III, these surfaces are timelike. They are (2+1) dimensional de Sitter spaces. In particular, the world volume of the bubble has this symmetry; bubble wall is the "vacuum domain wall," which has no structure, and is invariant under boost. On the other hand, in Region I the SO(3, 1) symmetric surfaces are spacelike, H 3 . In the thin-wall limit with zero c.c., Region I The potential for the field Φ. The local minimum at Φ = Φ A is the false vacuum, whose geometry is a de Sitter space (the ancestor vacuum). On the true vacuum side, it is assumed that a plateau region exists, on which slow-roll inflation occurs. The true minimum of the potential is taken to be zero, since we expect the present cosmological constant to be realized as vacuum energy of a quantum field on this background. Right panel: Penrose diagram for the Coleman-De Luccia geometry. The spacetime is divided into five regions by the red lines. Region I is an open FLRW universe. Green line represents the bubble wall. On its left (right) is the true (ancestor) vacuum. The blue curves indicate orbits of the SO(3, 1) symmetry. The directions of the coordinates (η, R for Region I; τ , X for Region III) are indicated by arrows. Region I is drawn with the future null infinity, because the present cosmological constant will relax to zero in the future if it is due to vacuum energy of quantum field. is nothing but a part of Minkowski space in the open slicing (known as the Milne universe).

Metric in Euclidean and at early time
The geometry containing one bubble of true vacuum is obtained by analytic continuation from the Euclidean geometry (Coleman-De Luccia [CDL] instanton), which is believed to contribute dominantly to the path integral of quantum gravity.
Let us first describe the Euclidean geometry. The Euclidean version of the 3+1 dimensional de Sitter space is a sphere S 4 , while the CDL instanton is a deformed sphere described by the metric of the following form, This geometry is topologically S 4 , but is deformed in the X direction. It has an S 3 factor parametrized by θ and Ω 2 , so it preserves the SO(4) subgroup of SO (5). The scale factor a(X) behaves as a(X) = c ± e ±X with some constant c ± for X → ∓∞ due to the fact that the geometry is smooth (i.e., locally flat) at both ends. We will mostly consider the case where the true vacuum has zero c.c., since this is sufficient for the study of the early time limit in the open FLRW universe (as explained below). In the thin-wall limit with a zero c.c. true vacuum, the geometry is S 4 patched with a flat disk at the domain wall located at X = X 0 , and scale factor takes the form, The natural length scale associated with this scale factor is the inverse Hubble H −1 A of the ancestor vacuum.  Figure 2: Left panel: The Euclidean geometry for a CDL instanton in the thinwall limit. A flat space for the true vacuum (whose vacuum energy is assumed to be zero) is patched to a piece of a sphere S 4 for the false vacuum with positive vacuum energy. The S 3 part is represented by S 1 in the figure. Right panel: The potential a ′′ /a in the massless scalar equation of motion (3.3) on a CDL background in the thin-wall limit. Asymptotically, a ′′ /a → 1 for X → ±∞. On the true-vacuum side (X < X 0 ), the potential is flat. At the bubble wall (X = X 0 ), there is a negative delta function. If mass is non-zero, the potential will be lifted by the additional term +m 2 a 2 .
We can analytically continue the geometry from the reflection symmetric surface, namely an equator of the S 3 . By taking we obtain the metric which covers Region III in Fig. 1, This has a factor of (2 + 1)-dimensional de Sitter space, parametrized by τ and Ω 2 . The SO(4) symmetry of the Euclidean space becomes SO(3, 1) after analytic continuation. This symmetry acts as the isometry of the (2+1) dimensional de Sitter space. The coordinate X parametrizes the spatial direction transverse to these de Sitter spaces. The X-direction for τ = 0 is represented by the horizontal line in Fig. 1. Region III is not geodesically complete. The spacetime can be extended past the horizons (X → ±∞, τ → ±∞), represented by 45 degree red lines in Fig. 1, which are locally equivalent to the Rindler horizon. By continuing past X → −∞, τ → +∞, we obtain an open FLRW universe (Region I) described by the metric The spatial sections are H 3 , parametrized by R and Ω 2 with the isometry SO(3, 1). This metric (2.5) is most easily obtained by an analytic continuation, from the Euclidean metric (2.1). The scale factor behaves as in the early-time limit η → −∞. In the thin-wall limit with the zero final c.c., the scale factor is exactly a(η) = const. × e η . One way to understand the relation between the coordinates in (2.4) and (2.5) is to note that the geometry near the light cone is locally flat. One can use the coordinatest,r to cover the global Minkowski space, ds 2 flat = −dt 2 + dr 2 +r 2 dΩ 2 2 . The metric (2.4) with a(X) = e X is obtained bŷ t = e X sinh τ,r = e X cosh τ, (2.8) whereas the metric (2.5) with a(η) = e η is obtained bŷ t = e η cosh R,r = e η sinh R. (2.9)

FLRW universe
The FLRW universe (Region I) is entirely inside the bubble, and the CDL instanton sets its initial condition. The scale factor has to behave as (2.7) in the early-time limit. The evolution afterwards can be found by solving the Friedmann equation, where the prime denotes the derivative with respect to the conformal time η, and M P is the reduced Planck scale. The last term in (2.10) is the contribution from the negative spatial curvature. The energy density ρ and the pressure p satisfy the conservation equation If the universe is dominated by a classical scalar field Φ(η) which depends only on time, we have Just after the beginning of the FLRW time, there is a curvature dominated era in which the metric is approximately (2.7). (The spacetime curvature is zero in the early-time limit, and the universe is dominated by the spatial curvature.) This era will continue until vacuum energy for inflation ρ = V I takes over the contribution from the curvature V I ∼ M 2 P /a 2 . The importance of this curvature dominated era in understanding the late time physics has been emphasized in [15].
Then, slow-roll inflation occurs. For simplicity, we ignore the gradient of the potential in the plateau region in Fig. 1, and assume that there is a constant energy density ρ = V I ≡ 3M 2 p H 2 I from the beginning of the FLRW time until the end of slow-roll inflation. We will also ignore a possible fast-rolling phase after tunneling and before slow-roll inflation 9 . We expect these features will give only small corrections to our main result.
The solution of (2.10) with ρ = 3M 2 p H 2 I is up to a shift of η. We could replace η in (2.13) by η −η 1 where so that (2.13) becomes a(η) = H −1 A e η−X 0 / cosh X 0 in the early-time limit η → −∞, to match the normalization of the Euclidean scale factor (2.2), but we will not do this in the following. Instead, we will use (2.13) in the FLRW universe, and replace η by η +η 1 in the correlators obtained by analytic continuation from Euclidean. See eq. (4.4) below.
After the slow-roll inflation, reheating occurs, and the radiation and matter dominated eras will follow. The evolution through these eras until present will be studied in Section 5.

Correlation functions
We will now calculate the two-point functions in the early-time limit in the FLRW universe. We will first find the correlator in the Euclidean space (2.1), and analytically continue it to the Lorentzian spacetime. The essential part of the calculation is the decomposition of the field into a complete set of states in the X direction (along the horizontal line in Fig. 1). For more details of this calculation, see Appendix of [22].

Calculation in Euclidean space
We consider the Euclidean CDL geometry, and compute correlation functions of a minimally coupled scalar field φ, whose action is The field φ is different from the tunneling field Φ. We assume φ to have zero expectation value, and can be treated as a free field on the curved (CDL) background. It is convenient to define a field χ(X, Ω 3 ) = a(X)φ(X, Ω 3 ), for which the kinetic term is independent on a(X), S = dXdΩ 3 (∂ X χ) 2 + · · · . We would like to obtain the two-point function that satisfies the equation of motion with a delta-function source, where ∇ 2 S is the Laplacian on S 3 , and δ 3 (Ω 3 ) is the delta function on S 3 . We will obtain the correlator as an expansion in the complete basis in the X direction. In the Lorentzian geometry, the X direction lies along a global Cauchy surface (See Fig. 1). The complete basis is formed by the eigenfunctions u I k (X) of the following differential operator, where k labels the eigenvalue, and I is either L (R) for waves coming from the left (right), or B for the bound state. This equation is of the form of the timeindependent Schrödinger equation in one dimension. The potential a ′′ /a + m 2 a 2 approaches +1 for X → ±∞ (see Fig 2). Thus, the modes for real k(> 0) with the eigen-energy larger than +1 are the waves oscillating in X. The orthogonal set consists of the waves coming from the left u L k (X), and those from the right u R k (X), which satisfy and R(k) and T (k) are the reflection and transmission coefficients for the scattering from the left, and R R (k) and T R (k) are those for the scattering from the right. They are related by (See e.g., [37] for basic facts about the scattering problem in one dimension.) These modes satisfy the orthogonality property where I, J are either L or R.
If there are bound states in (3.3), one has to include them in the complete set. Bound states occur at discrete imaginary values of momentum k = k B with Im(k B ) > 0. At k = k B , both R(k B ) and T (k B ) have a pole. This means that e ±ik B X in u L/R k (X) is negligible compared to R(k B )e ∓ik B X and T (k B )e ±ik B X , thus the wave function (3.4) with k = k B decays at both ends X → ±∞, and is normalizable.
Let us first consider the massless case. In this case, one can easily see that An example of the potential a ′′ /a for the Schrödinger equation (3.3) is depicted in Fig. 2. The bound state is essentially supported at the dip of the potential. If the geometry is compact in the X direction (which is the case for de Sitter space or the CDL geometry, but not for anti-de Sitter space or Minkowski space), this mode is normalizable and should be included in the complete set. It can be shown that there is at most one bound state in 3+1 dimensions [13,22] 10 . Normalization factor N in the thin-wall limit (2.2) is In the limit of X 0 → −∞ (the limit of the small bubble), we get a finite value N → 1/ √ 2, which agrees with the one for the pure de Sitter. When the scalar field has mass, the additional term +m 2 a 2 (X) lifts the potential, and the dip in the potential becomes shallow. The energy of the bound state (as long as exists) increases, shifting the pole from k B = i in the massless case to If mass is larger than a certain value (which is of order H A ), the bound state disappears.
We will consider the possibility that mass of φ is different in the true and the false vacuum, where m A and m 0 are assumed to be constant. The case of our interest is m A ≪ H A and m 0 ∼ H 0 ≪ m A . In this case we can set m 0 = 0 in the analysis on the early-time behavior performed in this section, since such a tiny m 0 as compared to the natural scale (the Hubble parameter at the time in question) will not affect the dynamics. It is not easy to calculate ǫ (i.e., the bound state energy) in general 11 , but when m A H −1 A is small, as in the case of our interest, it can be evaluated by the first order perturbation theory. We take (3.7) as the zeroth order wave function and m 2 a 2 (X) as perturbation Hamiltonian in the one-dimensional Schrödingerlike equation (3.3). The eigen-energy E = (k 2 B + 1) = 1 − (1 − ǫ) 2 is zero at the zeroth order. The first order eigen-energy, E (1) = 2ǫ, is with the zeroth order (massless) wave function, Thus, for the case of m 2 0 = 0, we obtain The reflection coefficient R(k) for (3.3) with m A = m 0 = 0 in the thin-wall limit (2.1) can be obtained exactly [22], where γ is given by and takes a value 0 < γ < 1. The k = i pole corresponds to the bound state. The pole in the lower half plane (k = −iγ) does not correspond to a physical mode for (3.3), since the wave function blows up at X → ±∞ and is not normalizable.
Using this complete set, the correlation function satisfying (3.2) can be expressed as where G k (θ) is the Green's function on S 3 for a field with the effective mass (k 2 + 1), is a function of the geodesic distance θ on S 3 , and given by We can see that G k (θ) is the correct Green's function on S 3 by noting the following facts. G k (θ) satisfies the Laplace equation, and is regular everywhere on S 3 except for the source θ = 0 with the correct singularity There is a subtlety when the mass of the scalar field φ is exactly zero. We have a bound state with k = i, and the effective mass on S 3 becomes zero, k 2 + 1 = 0. But there does not exist such a Green's function on a compact space, since the flux cannot extend to infinity and the Gauss law cannot be satisfied. (In other words, the differential operator in the equation of motion cannot be inverted in this case 12 , since the equation is unchanged under the constant shift of φ). See [22] for how to deal with this case. In this paper, we will not have this problem, since we will always consider the massive field.
The correlation function in the X → −∞ limit (i.e., on the true vacuum side) when m 0 = 0, can be written as If we take the thin-wall limit, (3.22) is valid throughout the X < X 0 region. The integration contour C 1 is defined in Fig. 3. This contour is basically along the real axis but is deformed near the origin so that it goes above the bound state pole. This deformation amounts to adding a discrete mode due to the bound state. Note that the normalization factor 1/ sinh kπ for the S 3 Green's function G k (θ) introduces poles at integer multiple of k = i. Thus, if the mass is exactly 12 In non-compact spaces, this does not cause a problem, since the zero mode is measure zero.  (3.19). Right panel: The contour C 2 used at the end of Section 4 to show the finiteness of φ 2 in the early-time (η, η ′ → −∞) limit. By deforming C 1 to C 2 , the k-integral is expressed as a sum over the residues at k = in (n = 1, 2, . . .).
zero, the pole at k = i becomes a double pole. As mentioned above, we will always consider the massive case, so the contour C 1 passes between k = i(1 − ǫ) and k = i. We can take massless limit starting from this expression, if necessary.

Analytic continuation to Lorentzian
We perform the analytic continuation (2.6) on (3.22), and obtain the correlation function in the FLRW universe 13 , This is valid in the early-time limit η, η ′ → −∞ (i.e., curvature dominated era), and can be regarded as the initial condition for the correlator in the FLRW universe. The correlator at later times will be obtained in Section 5. A subtle prescription for analytic continuation has been used to obtain the second term in (3.23). If we naively made the substitution (2.6) in the second term in (3.22), we would have gotten but the first term in the numerator on the right hand side of (3.24) diverges for k → ∞, so the k integral along the real axis does not converge. To get a convergent integral, we should make a replacement sinh k(π − θ) → e −kπ sinh θ For the calculation of the correlator starting from this representation of discrete sum, and using Watson-Sommerfeld transformation to convert it into an integral, see e.g., [38,39].
The correlator (3.23) is a function of the geodesic distance R on H 3 . In other words, one point is set at the origin and the other point is at radial distance R from the origin. When two points are at general positions on H 3 , we can rewrite the correlator using the relation where R 1 and R 2 are the radial coordinates of the two points, and ψ is the angle on S 2 between them. The correlator can also be expressed as a sum over products of harmonics f (ℓ,m) k (R, Ω 2 ) at the two points. Harmonics on H 3 are the eigenfunctions of Laplacian, The modes with real k are normalizable on H 3 , The ℓ = 0 mode, which is homogeneous on S 2 , is a function of only the radial coordinate f 29) where N = 1/( √ 2π) is a normalization constant. When one point is at the origin R 1 = 0 or R 2 = 0, only the ℓ = 0 modes contribute to the two-point functions (as is familiar in the harmonic expansion in quantum mechanics in flat space). The factor sinh R in the denominator of (3.29) compensates for the exponential growth of the volume for large R on the hyperboloid. For the explicit form of the harmonics with ℓ = 0, see Appendix A and [7,8,9].
Note that a constant function on H 3 is not normalizable. To express nonnormalizable functions, which decay more slowly than e −R as R → ∞ (with arbitrary dependence on S 2 ), we need the modes with imaginary k. As indicated by the integration contour C 1 , the correlator (3.23) contains a discrete nonnormalizable mode on H 3 , which is called the "supercurvature mode" [7,8,9]. As we have seen, a bound state in the Euclidean problem is in one-to-one correspondence with a supercurvature mode on H 3 .
The first term in (3.23), which is a function of η − η ′ , is nothing but the correlator in global Minkowski space written in open slicing. This can be seen by rewriting the massless correlator in terms of Minkowski coordinatest,r, and performing the coordinate transformation (2.9), where we have put one point at the originr ′ = 0. Carrying out the k integral for the first term in (3.23) as a sum over residues from the poles, we recover the r.h.s. of (3.30). The second term, which is a function on η + η ′ and depends on the reflection coefficient R(k), carries the information about the ancestor vacuum. This term is finite in the coincident-point limit R → 0, η − η ′ → 0. the previous section.

General remarks
Energy-momentum tensor for a minimally coupled scalar field is The energy density ρ = − T η η and the pressure p = T i i , where i's are not summed over, are obtained by taking the vacuum expectation value of (4.1): They can be computed by taking the coincident-point limit of the two-point function.
The two-point function (3.23) consists of two terms. The first term is independent of the ancestor vacuum. In the early-time limit, it coincides with the correlator in global Minkowski space. We will touch on this term only briefly in this paper, since the analysis of this term is essentially contained in the previous work [24,25].
The second term in (3.23) depends on the properties of the ancestor vacuum. The supercurvature mode is contained in this term. We will mostly focus on this term because this term is expected to be larger than the first term in the case of interest. Since this term is finite in the coincident-point limit, the contribution from this term is independent of the renormalization prescription for the UV divergence.
The wave functions in the ancestor vacuum are of order H A , which is the natural mass scale. As we will see in detail in Section 5, the continuous modes have time dependence e −η , and they decreases to order H I (which is the natural magnitude of fluctuations in the slow-roll inflation with Hubble parameter H I ) by the end of the curvature domination [15]. The supercurvature mode has time dependence e −ǫη , and decays more slowly than continuous modes. Furthermore, its contribution to φ 2 is enhanced by an extra factor ǫ −1 ∼ H 2 A /m 2 A , as we will see shortly.
The continuous modes are different from the modes in the flat FLRW cosmology, in the sense that the infrared modes are cut off by the spatial curvature. We can see this e.g., in (3.29) where the mode (with real k) decays exponentially for R 1.

Mass term in the limit of small mass
Let us study the mass term in the limit of small mass (ǫ), which will be the most dominant term in the energy density of the scalar field.
The correlation function in the early time limit η, η ′ → −∞ is where we have made the shift η → η +η 1 withη 1 defined in (2.14) (similarly for η ′ ) in (3.23), to account for the difference in the definition of η mentioned in Section 2.
Let us explicitly evaluate φ 2 (η) (s.c.m.) in the ǫ → 0 limit. In this limit, one can use the residue of R(k) for ǫ = 0, given by (3.17), and obtain where A(X 0 ) depends on X 0 (i.e., the size of the bubble), We have kept ǫ in the denominator and in the time-dependent function e −2ǫ(η+η 1 ) , but have replaced the term e 2(1−ǫ)X 0 by e 2X 0 , since this difference will not have a large effect. Using the value of ǫ obtained by the first order perturbation theory in (3.16), the expectation value (4.6) becomes (4.8) In the limit of the small bubble X 0 → −∞, (and ǫ → 0), (4.8) approaches . This agrees with the well-known result in pure de Sitter space obtained by standard techniques (see e.g., [31,32]). For the convenience of the reader, we summarize the calculation of φ 2 for the pure de Sitter case in Appendix B.

Evolution of vacuum energy
Having obtained the vacuum expectation value of energy-momentum tensor in the early-time limit in the FLRW universe, we now study its time evolution. First, in Section 5.1, we determine the scale factor for the background open universe. Then, in Section 5.2, we obtain the wave function by solving the equation of motion for the scalar field, and in Section 5.3, we study the evolution of the vacuum expectation value of energy-momentum tensor. Although our analysis is general, we will mostly focus on the supercurvature mode. Finally, in Section 5.4, we consider the possibility that the vacuum energy gives the present dark energy.

Scale factor
The scale factor for an open universe can be found by solving the Friedmann equation, with the energy density ρ appropriate for each era. The universe at early time is curvature dominated (CD). As we mentioned in section 2.3, when the vacuum energy of inflation takes over the curvature contribution, the slow-roll inflation occurs. The scale factor in the CD and Inflation eras is given in (2.13). After reheating, the universe becomes radiation dominated (RD). For simplicity, we assume the reheating occurs instantaneously. In fact, the reheating process takes place for producing the particles, mainly of sub-horizon modes, which the inflation field is coupled to. Thus the detailed dynamics of particle production will be irrelevant to our main concern on the energy density given by the supercurvature mode. Then, after the matter-radiation equality, the universe becomes matter dominated (MD).
The scale factors in each era are obtained by solving (5.1) as where ρ = 3M 2 p α 2 /a 4 for the RD era and ρ = 3M 2 p β/a 3 for the MD era are used 14 . We will connect them by requiring a(η) and a ′ (η) are continuous across each era. Here, we closely follow the convention of ref. [24] 15 . Namely, we introduce the different shift of η in each era, to keep the expression for a(η) simple. The conformal time at present is η = η 0 . We ignore the present cosmological constant (dark energy) and assume the MD era continues until now.
The continuity conditions of a and a ′ give the following relations among the parameters in (5.2): The Hubble parameter at each era is calculated from (5.2) as . (5.7) By comparing the Hubble parameters at the beginning and the end of each era, we obtain a relation where H 0 is the present Hubble parameter. This can also be obtained by multiplying (5.4), (5.6), and (5.7) at present. From observations [4], we know The ratio of the curvature radius R c = a (Note that curvature radius is unity in the comoving coordinate) to the Hubble radius H −1 = a 2 /a ′ is . (5.11) This ratio grows during inflation, and decreases during the RD and the MD eras. r c is a huge number at the end of inflation, and remains large until now due to a bound at present time [4], r c0 10 (now). (5.12) The redshift parameter at the matter-radiation equality is z eq ≃ 3.4 × 10 3 , (5.13) thus we have 1 + z eq = a 0 a 4 = sinh 2 (η 0 /2) sinh 2 (η 4 /2) . (5.14) Using the above relations, we now estimate the values of the parameters in (5.2), η 1 , η 2 , η 3 , η 4 , η 0 , α and β. From (5.11) at present and (5.14), the values of η 0 and η 4 are constrained as where the observational bound (5.12) and value (5.13) are used. Then, from (5.6), and (5.7) at present, the values of α and β are obtained, Finally, using (5.4), we find the value of η 2 : where the lower bound for r c0 and the upper bound for H I are applied in the last equality. Slow-roll inflation starts when inflaton potential takes over the curvature, i.e., at η ∼ −1. The e-folding N e of the slow-roll inflation is where we have used (5.19) for the value of η 1 = −η 2 . If we take larger values for r c0 (the ratio between the curvature radius and Hubble radius at present), N e becomes larger. Since the value (5.20) roughly coincides with the value derived from an anthropic bound for the spatial curvature [15], it would be reasonable to adopt these minimal values for r c0 and N e .

Equation of motion
We now consider a minimally coupled scalar field φ in the open FLRW universe. The equation of motion for χ, which is defined as φ = χ/a, is where H denotes the coordinates on H 3 , and ∇ 2 H is the Laplacian on H 3 . Mass that appears in (5.21) is the mass m 0 after tunneling.
We expand the field into harmonics on H 3 , defined in (3.27), where the first term is the contribution from the continuous modes and the second term is from the supercurvature mode. v k (η) and v * (η) are the time-dependent part of the wave functions for the continuous modes and supercurvature mode.
(The corresponding parts in φ will be called ϕ k = v k /a and ϕ * = v * /a.) The harmonics are the eigenfunctions of Laplacian with eigenvalues ∇ 2 H → −(k 2 + 1).
For the supercurvature mode, we have k = k B ≡ i(1 − ǫ), where ǫ depends on the mass m A in the ancestor vacuum, and is of order m 2 A H −2 A in the small ǫ limit, as explained in Section 3.1.
Each mode on H 3 are independent at the linearized level. Thus, we will concentrate on the supercurvature mode. The wave function v * (η) satisfies This is of the form of the time-independent Schrödinger equation, like the Euclidean equation of motion studied in Section 3. Note that a ′′ /a − 1 ≥ 0 in the cases of interest: From (2.10) and (2.11), we find Thus, as long as w ≤ 1/3, which is the case for the CD era, inflation, the RD and the MD eras, the "potential" is non-negative, a ′′ /a − 1 ≥ 0. The "eigen-energy" of the supercurvature mode, k 2 = −(1 − ǫ) 2 , is negative and is always below the potential. Thus the massless wave function is not oscillating. In Sections 5.2.2 and 5.2.3 below, we will study Eq. (5.23) by making an approximation, retaining the relevant two terms from the following three terms, a ′′ /a, 1 − (1 − ǫ) 2 = 2ǫ − ǫ 2 , and m 2 0 a 2 . Since a ′′ /a > 1 is satisfied in all the eras of interest (as shown in (5.24) and also in (5.27) below), a ′′ /a > 2ǫ − ǫ 2 is always satisfied. Thus, there are the following three cases to be considered: As time passes and the scale factor a increases, the universe undergoes the periods (i), (ii) and (iii) in turn. Before starting the analysis, let us recall the well-known qualitative properties of Eq. (5.23). In the periods (i) and (ii), the term a ′′ /a is the largest in comparison with 2ǫ − ǫ 2 and m 2 0 a 2 . If we keep only this term in (5.23), the general solution is 25) where c 1 and c 2 are integration constants. The first term shows frozen behavior of the wave function while the second term describes a decaying mode. Thus, for general cases with c 1 = 0, after a sufficiently long duration, the wave function becomes frozen. On the other hand, in the period (iii), the term m 2 0 a 2 is the most relevant. If we only keep this term in (5.23), the solution is Wave function is oscillating and decreasing. Owing to the frozen behavior of the wave function in the periods (i) and (ii), vacuum fluctuations generated in the ancestor vacuum remain until late times, and may provide the present dark energy. At later times, in the period (iii), the mass term will be relevant, and the vacuum energy will diminish. In the following, we will study Eq. (5.23) in more detail by taking into account not only the first but also the second largest of the three terms, a ′′ /a, 2ǫ − ǫ 2 , and m 2 0 a 2 .

The massless approximation
In the period (i), we approximate (5.23) by neglecting the mass term. The scale factor (5.2) gives rise to the following potential in the Schrödinger-like equation (5.23) in each era, .

(5.27)
Then, the solutions are given by , (5.28) where the normalization of the wavefunction will be considered later. These solutions can also be obtained simply by the replacement k → i(1 − ǫ) from the wave solutions of the continuous modes, . (5.29) We require the solution (5.28) to match smoothly onto (4.8) found from the CDL geometry. This selects the solution with e +(1−ǫ)η in the CD-Inflation era in (5.28). The wave functions in the RD and the MD eras are obtained by requiring continuity conditions of v * and v ′ * across each era. We thus have the following normalized wave functions: .
(5.30) To obtain the two-point function φφ s.c.m. , we replace the factor e −ǫη in the early time expression (4.8), by the solution v * /a at late times with a suitable normalization explained below. We determine the coefficients A to D and obtain the wave functions v * in the RD and MD eras in Appendix D.
We now summarize the wave functions ϕ * = v * /a. In the CD and Inflation eras, the wave function is given by (5.30) with a = −(H I sinh(η)) −1 . Choosing In the early time limit (η → −∞), we have ϕ * ≃ 2−ǫ 2 e −ǫη . Therefore, the twopoint function is obtained by replacing the factor e −ǫη in (4.8) by 2 2−ǫ ϕ * . Near the end of inflation (η → −0), (5.31) approaches a constant ϕ * ≃ 1. The two-point function in this limit (and in the small ǫ limit) becomes 16 with an X 0 -dependent constant This has been obtained by using the expression (4.8) for ǫ in the first-order perturbation theory.
In the RD era, v * is given by (D.6). Using (5.2) and (5.4), and setting N * = H −1 In the MD era, ϕ * = v * /a can be found from (D.7), (5.2), (5.4) and (5.6), Therefore, even though non-zero ǫ introduces some time dependence for the wave function, we can conclude that this effect is not large in the case of interest (small ǫ and small η).

The ǫ = 0 approximation
In the periods (ii) and (iii), the term 2ǫ − ǫ 2 is the least relevant, and Eq. (5.23) is approximated by neglecting this term. In terms of ϕ * = v * /a, this equation is rewritten as where we use the physical time t instead of conformal time η (where dt/dη = a), and H = ∂ t a/a is the Hubble parameter. This equation has the same form as the one for the zero-modes in the flat-space case. Throughout the RD and MD eras (until today and perhaps much later), the spatial curvature can be neglected, as one can see from (5.11). In the case of spatially flat universe, the Hubble parameter behaves as H ∝ t −1 . For H = p/t with p being a constant parameter, solutions of (5.36) are expressed in terms of the Bessel function as and F and G are arbitrary constants. In the MD period, p = 2/3 and ν = 1/2. Then (5.37) becomes By smoothly connecting it to the frozen wave function in the previous subsection, we find In the period (ii), we have m 0 t 1 (or m 0 H −1 1), and the wave function (5.40) is frozen. In the period (iii), m 0 t 1 (or m 0 H −1 1), and the wave function starts oscillating. Note that at later times of η 1, we have to take into account spatial curvature, which will modify the wave function (5.37).

Time evolution of energy density and pressure
Let us now consider the expectation value of the energy-momentum tensor. First note that the continuous modes are expected to give negligible contribution. As mentioned in Section 4, the wave functions for continuous modes decreases to order H I by the end of curvature domination, due to the time dependence e −η , while the supercurvature mode decays only as e −ǫη . The continuous modes do not receive the enhancement factor ǫ −1/2 either. Thus, they are smaller by a factor One may worry about the divergence in the coincident-point limit. Such a divergence appears in the continuous mode. Its renormalization will introduce finite terms in the energy momentum-tensors, but they can be safely ignored. This is due to the fact that these terms will be a combination of curvature tensors because the renormalization is done in a local and covariant manner. In the present universe, such terms in the energy-momentum tensor will be of order H 4 0 , which is smaller than the term from the supercurvature mode.
Thus, we concentrate on the contribution from the supercurvature mode. We will estimate the energy density (4.2) and the pressure (4.3), by comparing the magnitude of each term. The spatial-derivative term is (∇φ In the period (i), the time-derivative term is ϕ ′2 * = O(ǫ 2 ), since the wave function is almost frozen and ϕ ′ * = O(ǫ). Hence, the spatial-derivative term is dominant in (4.2) and (4.3), and we obtain where c * is given in (5.33). The equation of state is w ≃ −1/3. In the period (ii), the wave function is almost frozen, and the time-derivative term in (4.2) and (4.3) is smaller than the mass term by a factor of order (m 0 t) 2 (or (m 0 /H) 2 ). From (5.40), we see ϕ * ≃ 1 − (m 0 t) 2 /6 and ϕ ′ * /a = ∂ t ϕ * ≃ −(m 0 t)/3 · m 0 ϕ * . Then, the mass term is dominant in (4.2) and (4.3), and we find The equation of state is w ≃ −1, and it gives a dark energy candidate. The wave function (5.40) is valid over the periods (ii) and (iii). With this wave function, the energy density (4.2) and the pressure (4.3) are estimated as · cos(2m 0 t) − 1 m 0 t sin(2m 0 t) + 1 2(m 0 t) 2 (1 − cos(2m 0 t)) . We now summarize the time evolution of the energy density and pressure.
(ii) When 2ǫ − ǫ 2 < m 2 0 a 2 < a ′′ /a (which is assumed to be satisfied in the present universe), the energy density is given by (5.43), and w = −1.
(iii) When a ′′ /a < m 2 0 a 2 is satisfied (which occurs when m 0 /H 1), the energy density and the pressure oscillate as (5.45) and (5.46). If we take an average over the period of oscillation (using the Virial theorem), we obtain w = 0. Thus, the energy density will decay as fast as the one for matter 17 .
We finally give some comments about the continuous modes. The wave functions with k 1 (or k phys = k/a R −1 c ) do not change much from those in the flat-space geometry, studied in [24,25], where modes with k ∼ 1/η (or k phys ∼ H) mainly contribute to the derivative terms of the energy-momentum tensor. Then, as long as η 1 (or H −1 R c ) is satisfied (i.e., throughout the RD and MD eras until today and later times), the results for the flat-space geometry obtained in [24,25] can be applied:

Vacuum energy as dark energy
To interpret the vacuum energy in the period (ii) as the present dark energy, the following three conditions are necessary. One is that the vacuum energy has the same order of magnitude as dark energy. Using (5.43), it is written as with Ω Λ ∼ 0.7. The second and third conditions are that the present moment is in the period (ii): With the scale factor (5.2) in the MD era, they become where we assume ǫ ≪ 1, and c ǫ is a constant of order one, defined by ǫ = c ǫ m 2 A H −2 A . From (3.16). we have To obtain (5.53), we have used (5.11) and (5.27). Using (5.15), the conditions (5.53) can be further rewritten as Let us heuristically explain the conditions in (5.55). The first inequality requires that the eigenvalue of Laplacian is smaller than the mass. Laplacian is associated with the inverse scale factor squared, and we have ǫa −2 0 < m 2 0 . Recalling the fact that the present curvature radius is equal to a 0 , we rewrite the inequality using a 0 = r c0 H −1 0 . The second inequality is the condition for non-oscillation of the wave function, and gives m 0 < H 0 .
The conditions (5.51) and (5.55) can be satisfied by physically acceptable values of the parameters 18 . For instance, satisfy the condition. One could also consider a situation where the second inequality of (5.55) is barely satisfied, m 0 ∼ H 0 . In this case, we need which means H A is the geometric mean of M P and m A . The transition from the period (i) (in which w = −1/3) to the period (ii) (in which w = −1) is an interesting signature of our mechanism for the realization of dark energy. The transition occurs at the time when 2ǫ − ǫ 2 = m 2 0 a 2 , i.e., at Let us compute w(z) as a function of the redshift z, assuming we can ignore the time derivative terms in ρ and p. This should be a good approximation in the period (i) and (ii). (It depends on the choice of parameters how good the approximation is quantitatively.) In this case, we get a very simple expression, (5.60) Here we have used a = r c0 H −1 0 (1 + z) −1 to obtain the right hand side of (5.59). When this approximation is valid, w(z) depends only onǫ, as one can see in (5.59). The deviation of the present equation of state w 0 = w(0) from −1 is where the last expression is forǫ ≪ 1, which will be satisfied in most cases that we are interested in. The derivative of w with respect to the scale factor (evaluated at present), which is sometimes called w 1 , (see e.g., [4]) is If this approximation (negligible time derivative term, andǫ ≪ 1) is valid, there is a simple relation between w 0 and w 1 , namely, w 1 = −2(w 0 + 1). In Figure 4, we show the plot of w(z) for two choices of the set of parameters ǫ, m 0 /H 0 , r c0 . These parameters have been chosen so that the time derivative terms in ρ and p can be safely ignored. (

Conclusions
Let us summarize our results. We have calculated the vacuum expectation value of the energy-momentum tensor for a scalar field in an open universe created by bubble nucleation. We pay particular attention to the contribution from the supercurvature mode, a non-normalizable mode on H 3 , which appears when the mass m A in the ancestor vacuum is small enough. The vacuum expectation value of energy-momentum tensor in the early-time limit is obtained by the Euclidean prescription. Then, its time evolution is studied using the equation of motion for the scalar field. The supercurvature mode decays more slowly than the continuous modes, thus it gives the most important contribution at late times. We have shown that the vacuum energy for a minimally coupled scalar field can be regarded as dark energy. For this interpretation, it is needed that there is a field with mass (in the true vacuum) m 0 of order the present Hubble parameter H 0 or smaller, and the ratios of m 0 , m A and the Hubble parameter in the ancestor vacuum H A satisfy certain inequalities. The latter condition does not seem difficult to satisfy. As long as m 0 H, the field value is essentially frozen (though there is weak time dependence due to non-zero m A ). In the future, after Hubble parameter decreases so that we have m 0 H, the energy density decays.
A nice point about our analysis is that the main result is free of theoretical uncertainties in the following senses. First, our result does not depend on the renormalization prescription for the UV divergence, since the supercurvature mode is non-singular in the coincident-point limit. Vacuum energy is often considered to be ambiguous due to the UV divergence, but we believe our finite result has intrinsic physical meaning. Second, the change of mass from m A to m 0 , which is assumed to occur during the tunneling, does not give rise to complicated non-equilibrium physics. This change occurs in the spatial direction in Region III in Fig. 1, with its effect essentially encoded in the initial condition in the FLRW universe, such as the position of the pole for the bound states in the Euclidean problem. Third, we treat the field φ fully quantum mechanically, and do not have to assign a random classical value for the field φ. In the study of axions, one sometimes has to set the misalignment angle by hand, but we do not have that kind of ambiguity.
We do not expect the fields other than scalars to give contributions of the type studied in this paper. Massless vectors (spin 1) are Weyl invariant in 3+1 dimensions, and so are massless spinors (spin 1/2) in any dimensions. A supercurvature mode appears when there is a bound state in the complete set in the X direction (see Fig. 1). For the scalar case, the presence of the bound state is due to the non-trivial potential a ′′ /a in the equation of motion arising from the coupling with the curved background. Weyl-invariant fields have the same equation of motion as in the flat spacetime, and there is no supercurvature mode.
Gravitons (spin 2) behave similarly to the massless scalar, but the counterpart of the k = i supercurvature mode in the scalar case is known to be pure gauge in the spin 2 case [11]. Furthermore, we cannot give mass to the gravitons, so there is no mechanism for generating the vacuum energy from the mass term. The remaining field is gravitino (spin 3/2). This field has not been studied in the context of bubble nucleation, and it is an interesting question how the correlation functions and vacuum energy of gravitino behave. However, it is not likely that the mass of gravitino is of order H 0 , since its mass is related to the scale of supersymmetry breaking, and as we know, there is no supersymmetry at such a low energy scale.
In this paper, we have not discussed the origin of the scalar field φ with mass of order m 0 ∼ H 0 ∼ 10 −33 eV. We expect that it could be one of the many axions, proposed in "string axiverse" [33]. One interesting possibility is that the field φ simultaneously gives dark energy and dark matter. To serve as dark matter, axions should have mass m DM 10 −22 eV (See e.g., [35,40]). If the mechanism for generating the mass m DM is given by local dynamics inside our universe, it could be that while the continuous modes get mass m DM , the supercurvature mode is unaffected to still have mass m 0 , because the latter mode is essentially determined in the ancestor vacuum. This point needs further study.
It is highly important to test observationally whether dark energy has been produced by our mechanism or not. According to our proposal, the equation of state of dark energy will deviate from w = −1 as we go back in the past, and will approach w = −1/3. The transition occurs when the spatial derivative term (of order ǫ) becomes dominant over the mass term m 2 0 a 2 in the energy density (4.2) and the pressure (4.3). The time of this transition depends on the parameters r c0 (the ratio of the curvature radius to H −1 0 ), m 0 /H 0 and ǫ, but there is a characteristic behavior of w(z), which seems to be rather general as mentioned at the end of Section 5. Even though the observational determination of the dark energy equation of state w(z) as a function of z would be a great challenge 19 , (and it would be especially difficult to obtain w(z) for high redshift, since dark energy will be less and less important than matter at early times), confirmation (or rejection) of the pattern like the one shown in Figure 4 might be within reach of the observations in the near future. One of such observational projects would be the multi radio telescope, Square Kilometre Array (SKA) [42], which is expected to deliver precise cosmological measurements through the survey of a large number of distant galaxies using the 21 cm hydrogen line [43,44]. Giving detailed theoretical predictions for w(z) for comparison with observations is an important subject for future studies.
will be performed in the Euclidean space. We will see that the contribution from the supercurvature mode (i.e., the bound state in the Euclidean problem) reproduces the result obtained by the standard techniques.

(B.2)
The first term is singular in the θ → 0 limit, but the coefficient of this singularity is independent of ǫ. This term represents the UV divergence which exists generally in quantum field theory, thus can be discarded 20 . Renormalization of this term will give rise to finite terms expressed in terms of local curvature tensors, but we will not consider such terms here. The second term is regular in the θ → 0 limit, but diverges as 1/ǫ in the ǫ → 0 limit. This term gives the dominant contribution that we are interested in, To compute the factors multiplying (B.3) to obtain (B.1), we can substitute k = i for the bound state wave function u k=i (X), since this function is regular in the ǫ → 0 limit. The wave function with k = i is proportional to the scale factor a(X), as explained in (3.7). For pure de Sitter space, we have a(X) = H −1 A / cosh X, and u B k=i (X) = N 1 cosh X , (B.4) 20 If we analytically continue to open FLRW universe following the prescription described in Section 3.2, this UV divergence dissapears from the supercurvature mode. See (4.5).
The parameter ǫ can be obtained by the first-order perturbation (2ǫ being the eigen-energy) as explained in Section 3, Putting these factors together, the expectation value (B.1) becomes which agrees with the known value (apart from the finite contribution due to renormalization) in pure de Sitter space (see e.g., [31,32]).

C Scale factor for open universe with both matter and radiation
We consider cases where the energy density contains both radiation and matter component, such as We are mostly interested in the case of small ǫ (i.e., m A ≪ H A ). Also note that η 2 and η 3 , which were determined in Section 5.1, are tiny numbers. Expanding the upper element of (D.1) with respect to ǫ and η 2 , we obtain