Vacuum decay and quadratic gravity: the massive case
OOn analytic bounces at large radii
Silvia Vicentini ∗ and Massimiliano Rinaldi † Dipartimento di Fisica, Universit`a di Trento,Via Sommarive 14, I-38123 Povo (TN), Italy andTrento Institute for Fundamental Physics and Applications (TIFPA)-INFN,Via Sommarive 14, I-38123 Povo (TN), Italy
Abstract
We study analytic solutions of scalar field bounces near the false vacuum. We extend ourprevious work on massless or light fields to include massive scalar fields, scalar field theories withtime-dependent couplings, and higher-order kinetic terms, as they may have important physicalimplications. We consider also such theories in space dimensions other than three. We includeEinstein-Hilbert gravity when the false vacuum has a flat geometry. Finally, we improve ournumerical method, which now is based on maximization, to find the bounce. ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] F e b . INTRODUCTION Metastable states are separated from regions of lower energy by potential barriers withfinite height and are classically stable at zero temperature. However, they can decay dueto quantum tunnelling. The decay rate of metastable fields depends on the solution ofthe Euclidean equations of motion called bounce [1, 2]. This is a trajectory between thetunnelling point beyond the potential barrier and the false vacuum, which is reached withinan infinite Euclidean time. Once that the bounce is found, one can compute the decay ratefrom a combination of the on-shell action and a prefactor, called fluctuation determinant.For specific single scalar field theories an analytic bounce solution has been found long ago[4, 5]. This is seldom the case, though, as the equations of motion are highly non-linear.An approximate solution may be computed when the energy difference among the falsevacuum and the true one is sufficiently small, i.e. when the thin wall approximation is valid[1–3, 6, 7]. For most cases one needs to resort to numerical procedures to calculate thebounce trajectory, the most popular being the shooting method. The bounce is found as thesolution of the equations of motion that separates undershoot trajectories, i.e. trajectoriesin which the field makes multiple oscillations in the Euclidean potential well until it getsto rest in its bottom, and overshoot trajectories, in which the scalar field reaches the falsevacuum with finite, non vanishing velocity. Computational limits aside, one can determinesuch trajectory with arbitrary precision. There are also other techniques that work forsingle and multiple scalar field theories [8–15] but they have not been generalized to includegravity so far, an exception being [16, 17]. Moreover, also numerical methods to computethe fluctuation determinant have been recently investigated [18, 19]. The possibility of com-puting gravitational corrections to single scalar field bounces on a Euclidean backgroundhas been addressed in [20, 21], when the gravitational backreaction on the scalar field is small.In a preliminary work we noted that the O (4)-symmetric bounce of a single scalar fieldtheory is independent on the form of the potential near the false vacuum. This regionis probed for large values of the coordinate on which it depends, which is the radius ofa Euclidean four-dimensional spacetime [22]. We required the scalar field to be massless,or light, and cubic self-interactions to be small. Our calculation was extended to includeEinstein-Hilbert gravity, as long as the false vacuum has a flat geometry, which implies thatthe potential vanishes there. We were able to verify that the following existence conditionsfor the bounce are met:1. no inconsistency arises in the Einstein equations in the asymptotic region, i.e. giventhe asymptotic behaviour of the scalar field, the approximate solution to the Einsteinequations at arbitrary large radii on the bounce is flat space;2. the on-shell action exists and it is finite.Our results allowed also to introduce a novel numerical method to find the bounce, whichis based on minimization of some functional, closely related to the action. Due to technicaldifficulties, we overlooked some scalar theories which are actually of great physical interest,in particular massive scalar fields and higher-derivative kinetic terms. The former may beimportant in determining existence conditions for the bounce of the Higgs field, for examplewhen modified gravity is included. The latter has been found to be a candidate high-energycorrection to solve the hierarchy problem [24–27] and solves the unitarity problem in theagravity theory [29]. To determine the asymptotic bounce in both cases, it proves useful toextend first our results to a time-dependent potential. Another interesting possibility is tofind the asymptotic bounce in a number of space dimensions d other than three, as thereare recent proposals for analogue experiments with d < φ ( t ) ∼ t − . This result might prove useful for theories with n scalar fields, in some cases. Moreover, our method is helpful in Sec.IV, where we find theasymptotic bounce of a massive single scalar field theory. We also get a closer look to theminimization method discussed in [22], as it needs to be improved for the reasons explained3bove. Then, we turn to higher-order kinetic terms in Sec.V and prove that such terms donot influence the asymptotic bounce, that go over the massless or massive one, dependingon other terms in the potential. Finally, we look at the asymptotic bounce in a number ofspace dimensions other than three in Sec.VI. We find that, when larger than three, similarconsiderations hold, while when smaller than three a more careful analysis is needed. II. PREVIOUS RESULTS
In this section we quickly review the original idea explained in [22]. In the followingSections, we repeat the calculations from scratch, but it is useful to briefly recap our previousresults: in this way, we provide both a first example of our method and some basic formulasthat we will need for the rest of the paper.
A. Settings
We consider a single scalar field theory on a four-dimensional Euclidean spacetime, withcanonical kinetic term S = (cid:90) d x √ g (cid:20) g µν ∇ µ φ ∇ ν φ + V ( φ ) (cid:21) . (1)Following the original Coleman’s prescription, we focus on O (4) symmetric scalar fields,namely, φ depend on spacetime coordinates ( τ, (cid:126)x ) only as a function of t ≡ (cid:112) τ + | (cid:126)x | . Theequation of motion is ¨ φ + 3 ˙ φt = dVdφ (2)where the dot indicates derivative with respect to t . As t is the coordinate that controlsthe equations of motion and in analogy with Eq.(21), in the following we will refer to suchcoordinate at “time”. We consider a scalar field potential with a local minimum at φ fv , thefalse vacuum, such that V ( φ fv ) = 0. This is separated by a potential barrier from a region V ( φ ) <
0, which we consider to lie at positive values of the scalar field φ >
0. The bouncetrajectory is a non-trivial solution to the equation of motion with boundary conditions φ ( ∞ ) = 0 , ˙ φ (0) = 0 (3)and finite on-shell action. The decay rate of the metastable state is given byΓ = A e − B , (4) That is, a solution with initial condition φ (0) = φ (cid:54) = 0 B = S E − S fv is the difference among the action computed on the bounce trajectoryand the false vacuum action, which vanishes in our case. A is the quantum fluctuationdeterminant. It can be estimated as [41] A = T U R − (5)and thus the decay time τ = Γ − may be written as τ = T U (cid:18) R T U (cid:19) e S E . (6)We will label the initial condition φ (0) for the bounce as φ , while φ in indicates any otherinitial condition for the scalar field. As φ is typically of order O (0 . G − / , we measuremasses in units of G − / (so the time unit is G / ). B. Previous results
As anticipated in the Introduction, there exist only a few analytic bounce solutions forsingle scalar field theories. In particular, in [4, 5] it was found that a scalar field withscale-invariant potential λφ has a family of bounces, labelled by a constant C φ ( t ) = 16 C t + C | λ | . (7) C is related to the bounce radius R ≡ φ (0)2 as C = 4 √ (cid:112) | λ | R . At large t we havelim t → + ∞ φ ( t ) = C t . (8)In [22], we found that the asymptotic time dependence of the scalar field in Eq.(8) is a quitegeneral feature of the bounce of single scalar field theories described by the action Eq.(1).Basically, one finds that the potential gives a negligible contribution to the equations ofmotion in that region, and so the solution is given by¨ φ + 3 ˙ φt + a ≈ φ fv vanishes V ( φ fv ) = 0 and, thus,the false vacuum lives on flat space. To prove this more rigorously, and get some insighton C , which is fixed by the dynamics, we considered a generic undershoot trajectory nearthe bounce, and we expanded the equation of motion of the scalar field around the turningpoint, namely, the time t ∗ at which the field inverts its velocity for the first time (which is5alled t tp in [22]). We give now a sketch of the proof, as the calculation is similar to theones that we will do in the following sections. The subscript ∗ in the following indicates aquantity evaluated at t ∗ . The bounce corresponds to the limit t ∗ → + ∞ . The equation ofmotion expanded around t ∗ is¨ φ + 3 t ˙ φ = + ∞ (cid:88) n =0 (cid:18) dVdφ (cid:19) ( n ) ∗ ( t − t ∗ ) n n ! (10)and may be approximated as ¨ φ + 3 t ˙ φ = (cid:18) dVdφ (cid:19) ∗ (11)for sufficiently large t ∗ , t , and t ≤ t ∗ , under some conditions on the potential. In Eq.(10)we indicated with ( n ) a time derivative of order n . In order for Eq.(11) to hold in a largeregion around t ∗ we need that (cid:88) n ≥ (cid:18) dVdφ (cid:19) ( n ) ∗ t ∗ n (cid:28) (cid:18) dVdφ (cid:19) ∗ . (12)If (see [22]), (cid:18) d j Vdφ j (cid:19) ∗ ¨ φ j − ∗ ( t ∗ + a ) j − (cid:28) j ≥ t ∗ (13)we have (cid:18) dVdφ (cid:19) ( n +1) ∗ = i =¯ ı − (cid:88) i =0 ˜ A i (cid:18) dV ¯ ı − i dφ ¯ ı − i (cid:19) ∗ ¨ φ ¯ ı − i − ∗ ( t ∗ + a ) i +1 even n , with n + 5 − ı = 3 (14) (cid:18) dVdφ (cid:19) ( n +1) ∗ = i =¯ ı − (cid:88) i =0 ˜ A i (cid:18) dV ¯ ı − i dφ ¯ ı − i (cid:19) ∗ ¨ φ ¯ ı − i − ∗ ( t ∗ + a ) i odd n with n + 5 − ı = 2 (15)which satisfies Eq.(12). Moreover, if derivatives of V ( φ ) are finite for φ →
0, andlim t ∗ → + ∞ (cid:18) dVdφ (cid:19) ∗ t ∗ = lim t ∗ → + ∞ ¨ φ ∗ t ∗ = 0 (16)Eq.(13) is always satisfied for j >
2. Instead, (cid:18) d Vdφ (cid:19) ∗ t ∗ (cid:28) t ∗ (17)should be separately imposed. For lim t ∗ → + ∞ ¨ φ ∗ t ∗ = 4 C (18)we should also require that (cid:18) d Vdφ (cid:19) ∗ C (cid:28) . (19)6s, in the end, we are interested in the limit φ ∗ → t ∗ → + ∞ , these conditions implythat the scalar field should be massless and cubic self-interactions gφ should be sufficientlysmall 24 gC (cid:28)
1. If the scalar field is not massless but light, we expect this approximationto be valid for large t such that t (cid:28) m . Under these assumptions, Eq.(11) giveslim φ ∗ → t ∗→ + ∞ ˙ φ ( t ) = − C t lim φ ∗ → t ∗→ + ∞ (cid:18) dVdφ (cid:19) ∗ = 4 C t ∗ lim φ ∗ → t ∗→ + ∞ φ ( t ) = C t (20)from sufficiently large t up to t → + ∞ . Notice that, despite the asymptotic bounce is as inEq.(7), we cannot use our method for a theory described by a pure quartic potential, as ithas no potential barrier and so no undershoot trajectory.This procedure may be extended to include Einstein-Hilbert gravity. We choose an O (4)symmetric line element with Euclidean signature ds = dt + ρ ( t ) d Ω . (21)The action is (we set G = 1) S = (cid:90) d x √ g (cid:20) − R π + 12 g µν ∇ µ φ ∇ ν φ + V ( φ ) (cid:21) . (22)and the equations of motion are ¨ φ + 3 ˙ ρ ˙ φρ = dVdφ (23)˙ ρ = 1 + 8 π ρ (cid:32) ˙ φ − V ( φ ) (cid:33) (24)where the dot indicates derivative with respect to the Euclidean time t . If the false vacuumlives on flat space (that is V ( φ fv ) = 0), we have˙ ρρ ≈ t + a a ∈ R , (25)sufficiently near the bounce at large times. We findlim φ ∗ → t ∗→ + ∞ ˙ φ = − C ( t + a ) (26)and analogously, Eq.s (20) hold with t ∗ → t ∗ + a , t → t + a .7e used this result to provide a numerical method to find the bounce of single scalarfield theories with Einstein-Hilbert gravity, via minimization of the functional S EC ≡ S EC, + S EC, (27) S EC, ≡ − π (cid:90) ¯ t ¯ ρ ( t ) V ( ¯ φ ) dt (28) S EC, ≡ π (cid:90) ∞ ¯ t t (cid:18) C t + V (cid:18) C t (cid:19)(cid:19) dt (29)where ¯ ρ , ¯ φ are computed numerically on-shell, and S EC, depends on C and some matchingtime ¯ t , which is determined by continuity and derivability. In particular, we find ¯ t as¯ t + 2 φ (¯ t )˙ φ (¯ t ) = 0 . (30)If such ¯ t do not exist (as it might be the case for undershoot trajectories) we consider thepoint of closes approach 3 − φ (¯ t )˙ φ (¯ t ) ¨ φ (¯ t ) = 0 . (31)With this choice, one can see that ¯ t → + ∞ on the bounce. Moreover, S EC is finite if S EC, isand, for some C = C , ¯ t → + ∞ we have δ SδC = π a . (32)Once that C and ¯ t are found we can compute the on-shell action as S E = − π (cid:32)(cid:90) ¯ t ρ ( t ) V ( φ ) dt + (cid:90) ¯ ∞ ¯ t t V ( φ + ∞ ) dt (cid:33) (33)where φ + ∞ is the asymptotic bounce, φ + ∞ ≡ lim φ ∗ → t ∗→ + ∞ φ ( t ) (34)which in this case is as in Eq.(20). Both Eq.(27) and Eq.(33) are off-shell for t > ¯ t awayfrom the bounce, but this contribution is smaller and smaller the closer we get to it. Wechoose Eq.(27) instead of Eq.(33) for minimization because the latter has a saddle point onthe bounce. If the scalar field moves on a background flat spacetime, the on-shell actionmay be computed as S E = 2 π (cid:32)(cid:90) ¯ t t (cid:32) ˙ φ V ( φ ) (cid:33) dt + (cid:90) ¯ ∞ ¯ t t (cid:32) ˙ φ ∞ V ( φ + ∞ ) (cid:33) dt (cid:33) . (35)This has a saddle point for ¯ t → + ∞ , too. In this case we were not able to find a functionalas Eq.(27) that has a minimum or a maximum on the bounce. As we do not determine the bounce with infinite precision numerically, ¯ t will actually be finite. II. SCALAR FIELD WITH TIME-DEPENDENT POTENTIAL
In this Section we derive the asymptotic behaviour of the bounce for a single scalar fieldwith a time-dependent potential. Before entering the details of the calculation, we brieflycomment on the possible origin of such time dependence. Of course, a time-dependentcoupling may be put by hand, if the underlying physical phenomenon that the field theorywants to describe contains such feature. Another possibility is that it arises from a timeindependent potential if we are sufficiently close to the bounce. For example, if we considera n scalar field theory, in which all scalar fields but one are decoupled from the others, i.e. ∂V∂φ i is a function of φ i only, for i = 2 , . . . , n (36) ∂V∂φ is a function of all scalar fields (37)the equations of motion for the scalar fields ( φ , . . . , φ n ) are¨ φ i + 3 ˙ φ i t = ∂V∂φ i ( φ i ) for i = 2 , . . . , n ¨ φ + 3 ˙ φ t = ∂V∂φ ( φ , φ i ) . (38)We can choose initial conditions φ i (0) such that φ i for i = 2 , . . . , n are on the bounce and φ lies on a undershoot trajectory. Thus we can write ∂V∂φ ( φ , φ i ) = ∂V∂φ ( φ , φ i,b ( t )) ≡ ∂V∂φ ( φ, t ) (39)where φ i,b ( t ) are the bounces for the decoupled scalar fields and we renamed φ as φ . Inthis way, the potential depends on time both through the scalar field φ and explicitly.Now we derive the asymptotic behaviour of the bounce, in the case of a time-dependentpotential. We focus on under which conditions Eq.(8) holds. To do that, we expand theright-hand side of Eq.(2) in a Taylor series around the turning point t ∗ ∂V∂φ = (cid:18) dVdφ (cid:19) ∗ + (cid:88) n ≥ f n ( t − t ∗ ) n n ! (40) f n ≡ (cid:18) dVdφ (cid:19) ( n ) ∗ ( φ ∗ , t ∗ ) (41)and evaluate f n in the large t ∗ limit. The details of the calculation are reported in theAppendix. We find that, besides Eq.(17)-(19), we need (cid:18) ∂ i + j V∂φ i ∂t j (cid:19) ∗ t ∗ j (cid:28) i ≥ j ≥ (cid:18) ∂ i +1 V∂φ∂t i (cid:19) ∗ t ∗ i (cid:28) ¨ φ ∗ for i ≥ . (43)If this is the case, then Eq.(8) is satisfied. If instead the scalar field is massive, we expectEq.(61) to hold. As mentioned in the previous Section, if we include Einstein-Hilbert gravitywith a flat spacetime for φ = φ fv , and we are sufficiently close to the bounce, the frictionterm satisfies Eq.(25) and thus we obtain the same result as above, taking t → t + a . Wenow turn to massive scalar field theories. IV. MASSIVE SCALAR FIELD
The procedure outlined in Sec.II excludes massive scalar fields, as Eq.(17) is clearlyviolated at large t ∗ . Moreover, the action Eq.(1) would diverge on the bounce if we hadEq.(8), as (cid:90) + ∞ dt t φ ≈ ln( t ) | + ∞ . (44)Our calculations suggest that each order in the Taylor expansion is dominated by the massterm, and that every order is equally important. We don’t know though which term on theleft-hand side of Eq.(2) dominate on the bounce at large times. We show now that in thisregime ¨ φ ≈ m φ, (45)namely, the friction term gets subdominant with respect to the other contributions in theequation of motion . In this approximation we have φ ( t ) = C e − mt (46)which verifies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) ¨ φ ≈ m φ. (47)In order to see this, we repeat the analysis of the previous sections considering both thefriction term and the mass term as contributions to dVdφ . We suppose that other terms inthe potential are such that V ( φ ) ≈ m φ (48) This is the thin-shell condition as formulated by [1–3], but now it holds only in a small region around thefalse vacuum and thus it is not fitted to find the on-shell action analytically. φ ≈
0. We write ˙ φt = ddt (cid:18) φt (cid:19) + φt (49)and the equation of motion as¨ φ = m ( t ) φ − ddt (cid:18) φt (cid:19) m ( t ) = m − t . (50)We also define ∂V ∂φ ≡ m ( t ) φ ∂V ∂φ ≡ − φt (51)so we have ¨ φ = ∂V ∂φ + ddt (cid:18) ∂V ∂φ (cid:19) . (52)Our goal is to find which terms in the Taylor expansion ∂V ∂φ + ddt (cid:18) ∂V ∂φ (cid:19) = m φ ∗ + (cid:88) n ≥ (cid:32)(cid:18) ∂V ∂φ (cid:19) ( n ) + (cid:18) ∂V ∂φ (cid:19) ( n +1) (cid:33) ( t − t ∗ ) n n ! (53)dominate for t ≤ t ∗ and sufficiently large t ∗ , t For such values of t we have (cid:88) n ≥ (cid:32)(cid:18) ∂V ∂φ (cid:19) ( n ) + (cid:18) ∂V ∂φ (cid:19) ( n +1) (cid:33) ( t − t ∗ ) n = (cid:88) n ≥ (cid:18) ∂V ∂φ (cid:19) ( n ) t ∗ n + (cid:88) n ≥ (cid:18) ∂V (cid:48) ∂φ (cid:19) ( n ) t ∗ n (54)with V (cid:48) defined as ∂V (cid:48) ∂φ = − φt t ∗ (55)apart from numerical factors. For our purposes, Eq.(54) is tantamount to the Taylor expan-sion for a theory with equation of motion¨ φ = m ( t ) φ m ( t ) = m − t − t t ∗ (56)apart from the zeroth and the first-order term. So, if we find the time-dependent part to benegligible with respect to the constant mass term at each order in the Taylor expansion, weexpect that the same holds for our theory. We have ∂ j ( V + V (cid:48) ) ∂φ ∂t j ( φ ∗ , t ∗ ) t ∗ j ≈ t ∗− for j ≥ t ∗ → + ∞ , and ∂ i +1 ( V + V (cid:48) ) ∂φ∂t i ( φ ∗ , t ∗ ) t ∗ i ≈ t ∗− φ ( t ∗ ) for i ≥
1; (58) Equivalently, we may take t − t ∗ = − At ∗ with A some constant of order O (0 .
11o Eq.(43) holds in the same limit. Thus, as regards the time-dependent part of the potential,the zeroth-order term of the Taylor expansion dominates in the large t ∗ limit. The constantmass term instead should be accounted for at each order in the Taylor expansion. Thus, theequation of motion reduces to ¨ φ ≈ m φ (59)for large t ∗ near the bounce. For finite t ∗ a solution to Eq.(59) is φ ( t ) = φ ∗ e mt ∗ ( e − mt + e − mt ∗ e mt ) (60)with integration constants such that ˙ φ ( t ∗ ) = 0 and φ ( t ∗ ) = φ ∗ . Thus, taking the limit t ∗ → + ∞ we have φ ( t ) = C e − mt ¨ φ ∗ = m C e − mt ∗ . (61)This implies that other terms in the potential become negligible with respect to the quadraticone for t ≥ m − , in agreement with our estimation in [22]. Thus, we expect that theexponential behaviour becomes important only in a narrow region around the false vacuum,namely, when the mass term dominates. For t (cid:28) m − and sufficiently large t the asymptoticbounce is as in Eq.(8). Moreover, we can verify thatlim t ∗ → + ∞ ¨ φ ∗ t ∗ = 0 (62)so the same applies at each order in the Taylor expansion. Using Eq.(61) in Eq.(22) andEq.(24) we can verify that conditions 1. and 2. of Sec.I hold. Notice also that our resultagrees with the ones in [23], in which is stated that massless fields dominate the bouncefor R m (cid:28)
1, and the mass is important only for large values of t , thus giving just a smallcontribution to the on-shell action. At large times, we have that energy is approximatelyconserved, as ˙ φ − V ( φ ) = 0 for φ ( t ) = C e − mt and V ( φ ) = m φ . (63)At earlier times, we expect that φ ( t ) ∝ t − for sufficiently small t such that the right-handside of Eq.(2) is negligible with respect to the left-hand side. So, one may think that if weadd a mass term to a scalar field theory with quartic potential the field tends to overshootfor all possible values of φ in , as without the mass term we have φ ( t ) ∝ ( t + R ) − forall times, and if Eq.(61) holds at some point the energy loss is reduced. Actually, the fieldundershoots: the reason is that the scalar field decays as a power of time even when the massterm dominates over the quartic one, for sufficiently small times such that their contributionto the equation of motion is negligibly small with respect to the friction term and ¨ φ . Suchmass term induces additional loss of energy in the system, and the field cannot climb thehill to reach the false vacuum. In this way, we recover the well known result that a massivescalar field theory with a quartic potential does not have a bounce [23].12 . Numerical method We would like to extend the numerical method that we derived in our previous workto include more general scenarios. While in [22] we focused on finding a functional thatresembles the action, it is better to relax this requirement, as it suffers of multiple issues.Besides the ones mentioned above, the chosen functional may have other minima than thebounce which are not stationary points of the action , and they partially spoil the bisectionmethod. Moreover, while our choice of the matching time is particularly simple, it entersnon-trivially in the functional, and thus sometimes making it difficult to asses whether it is aminimum, a maximum or a saddle point. We can get rid of all these issues considering max-imization of the matching time in place of minimization of some functional. In fact, if thematching time reflects the property of the bounce being a limiting undershoot (overshoot)trajectory, we should have that it is a monotonically increasing (decreasing) function of theinitial condition near it, and thus suitable for maximization. Of course this depends on ourability of determining such time (which depends on the particular asymptotical behaviour),but its monotonicity is a general property. It does not depend on gravity being dynamical,so we expect our method to hold equally well for single scalar field theories. In the case ofmassless fields, such matching time has already been determined in [22] and it is reportedin Sec.II.We derive now the matching time for a massive field. We set ¯ t ≥ m , as, for ¯ t (cid:28) m ,matching conditions are as in Sect.II. As Eq.(63) holds at asymptotically large times on thebounce, then we may choose ¯ t as the time that satisfies it for nearby trajectories. Undershoottrajectories have typically smaller values of | ˙ φ | than the bounce, so we expect that, at somefinite time ¯ t , Eq.(63) is fulfilled. This might not be the case for overshoot trajectories. Ifthis happens, we may find the point of closest approach¨ φ (¯ t ) − m φ = 0 . (64)The bounce may be found as a limiting undershoot trajectory as above, but it is also alimiting overshoot trajectory with vanishing velocity when the false vacuum is reached.Thus, we expect to have that ¯ t on neighbouring trajectories increases as we get closer andcloser to the bounce. Thus, ¯ t is a monotonically increasing function of the initial condition φ in for φ in < φ , while it is decreasing for φ in > φ . If there are multiple matching timesfor a given trajectories, we are guaranteed that only the largest one shows the appropriatebehaviour. Nonetheless, this is extremely inconvenient if our purpose is to reduce the typ-ical time for which we need to solve the equations of motion of the scalar field to find the Such points should have finite ¯ t . φ in .We expect that larger matching times show the same behaviour. If, instead, the matchingtime we chose (for example, the first matching time encountered for a given trajectory)seems regular for all φ in there might be a larger one that shows the appropriate behaviour.Regarding this matter, choosing m − as limiting value for the validity of Eq.(63) is somehowarbitrary, as actually we need ¯ t (cid:29) m − . This choice should be irrelevant on the bounce,as ¯ t → + ∞ , but it may be determinant far away from it. In absence of a proper matchingtime in such region this condition may be slightly changed, depending on the individualsystem considered. More on this matter will be explained in the following, where we givean example of the determination of the bounce and calculation of the decay rate.As described above, the matching time - as a function of the initial condition φ in - divergeson the bounce. The bisection method, as well as the improved shooting method (describedin [22]), are thus suitable to determine it, by finding the initial condition φ . At each stepof the maximization procedure, we can compute the action Eq.(33) (if gravity is dynamical)or Eq.(35) (on a flat spacetime) with φ + ∞ as φ + ∞ = C t t < mCe − mt t > m . (65)where C is found as C = − ˙ φ (¯ t ) ¯ t t < me m ¯ t φ (¯ t ) t > m . (66)Our accuracy in determining the on-shell action may be set as Eq.(33) or Eq.(35) not varyingmore than 10 − in a further step. We won’t compare our results to the shooting methodones, as we expect it to have similar properties of the minimization method outlined in [22].We report in Fig.(1),(2) the matching time ¯ t as a function of the initial condition φ in fora theory V ( φ ) = m φ − φ + φ with m ∈ [2 × − , − ] and Einstein-Hilbert gravity(67)and V ( φ ) = λ φ − φ + φ with λ ∈ [4 × − , × − ]. (68)Each curve displays a divergence at some value of φ in , which marks the initial conditionof the bounce ( φ = φ in ), and its position varies with the coupling. Overshoot trajectories14 G - / ] [ G / ] FIG. 1. Matching time ¯ t against the initial condition φ in , for a massive scalar field with potentialEq.(67). The mass squared ranges from 10 − (green dots) to 10 − , (pink dots). The matching timediverges on the bounce, and it is monotonically increasing (decreasing) for φ in < φ ( φ in > φ ). are not displayed in the massive case as convergence to the bounce is much slower thanthe undershoot ones - for a given distance from the bounce, ¯ t is much smaller for overshoottrajectories than for undershoot ones. The reason lies in how we chose ¯ t . In fact, for sometrajectories (that we label by their initial condition φ in ) we had multiple choices for thematching time. We found that the matching given by Eq.(63) occurs at a larger time thanthe ones given by Eq.(30), and this last condition is sometimes satisfied multiple times. Insome cases instead we found no matching given by Eq.(63), and only one given by Eq.(30).In order to avoid ambiguities, we used Eq.(63) for matching when possible, and Eq.(30)otherwise. Imposing Eq.(63) in a region resulted in a much higher ¯ t than in the other.Anyway, both trajectories showed a ever-increasing ¯ t as we got closer to the bounce, up toaccuracy.For completeness, we report the on-shell action computed on the bounce for both theories.We set the accuracy as described above. The age of the universe is estimated as T U ∼ √ G . We found S E = 36 . S E = 47 .
4) for the massless theory with λ = 10 − ( λ =2 × − ) corresponding to a decay time log ( τ T − U ) ≈ −
226 ( log ( τ T − U ) ≈ − S EC = 6 .
69 ( S EC =7 .
02) for m = 2 × − ( m = 10 − ), corresponding to a decay time log ( τ T − U ) ≈ − ( τ T − U ) ≈ − G - / ] [ G / ] FIG. 2. Matching time ¯ t against the initial condition φ in , for a scalar field with potential Eq.(68). λ ranges from 4 × − (red dots) to ≈ × − ] (bluedots). V. HIGHER-ORDER KINETIC TERMS
Quantum tunnelling through an energy barrier has an exponentially small probability tooccur in the semi-classical approximation. The smallness of some numbers, for example theratio among the Higgs vacuum expectation value and the Planck mass, may be viewed interms of such exponential suppression, thus alleviating the hierarchy problem [24–28] (cid:104) φ (cid:105) = (cid:90) D φ φ e − S = e − W b (69)where the generating functional W b is computed on the bounce. It is found by solving theequations of motion of the original theory with a pointlike source, which is φ (0) ≡ e ψ (0) = exp (cid:18)(cid:90) d xδ ( x ) ψ ( x ) (cid:19) . (70)The source generates a singular instanton at t = 0. The singularity drives W b to infinity. Ithas been shown that a possible way to make it finite is to add higher order kinetic terms( ∂ψ ) n n > n (cid:3) ψ ( ∂ψ ) n − n > (cid:3) ψ (cid:0) ∂ψ ) n − (cid:1) (cid:28) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ ¨ ψ for n > ψ ( t ) ∝ t − (73) (cid:3) ψ (cid:0) ∂ψ ) n − (cid:1) (cid:28) m ψ ≈ ¨ ψ for n > ψ ( t ) ∝ e − mt . (74)16o a naive reasoning would suggest that such term does not influence the asymptotic be-haviour of the bounce. We test this result giving a proof as in the previous section: first weTaylor expand dVdψ + n (cid:3) ψ ( ∂ψ ) n − , (75)we consider t ≤ t ∗ and large t ∗ and see if the zeroth-order term dominates over the others.Here V ( ψ ) contains non-derivative terms that generate a potential barrier, through whichthe scalar field can tunnel.Eq.(72) gives a vanishing contribution to the zero-order term, as ˙ φ ( t ∗ ) = 0. Moreover, itappears in the Taylor expansion of φ n − at n -th order, namely (cid:0) ψ n − (cid:1) ( n ) = (cid:3) ψ (cid:0) ∂ψ ) n − (cid:1) + . . . (76)If there aren’t any incidental cancellations among the first term on the right-hand side andthe ones in . . . , we can trade the Taylor expansion of the higher-order term with the onefor ( ψ n − ) ( n ) . Moreover we can write (cid:88) m =1 (cid:0) ψ n − (cid:1) ( n + m ) ∗ ( t − t ∗ ) m m ! = t − t ∗ ≈− At ∗ (cid:88) m = n +1 (cid:18) ψ n − t ∗ n (cid:19) ( m ) ∗ t ∗ m m ! (77)apart from numerical factors. Thus our considerations are the same as for a theory with W ( ψ ) = V ( ψ ) + ψ n t ∗ n n > t ∗− n is a time independentcoupling which vanishes in the limit t ∗ → + ∞ . As we saw in Sect.II, for n ≥ n = 3we need to impose Eq.(19) which gives t ∗− C → t ∗ → + ∞ (79)and thus also this contribution is negligible in the large t ∗ limit. We conclude that higher-order kinetic terms do not affect significantly the Taylor expansion. Moreover, their con-tribution to zero order vanishes. Thus, we conclude that Eq.(8) holds for massless ψ whensuch terms are added to the Lagrangian. If ψ is massive, then Eq.(61) holds. Our proof in the Appendix does not account for numerical coefficients, and so it does not for incidentalcancellations either. So even if there are incidental cancellations, if we find that higher order terms inthe Taylor expansion are subdominant with respect to the zeroth-order for (cid:0) ψ n − (cid:1) ( n ) we expect to be thesame for (cid:3) ψ (cid:0) ∂ψ ) n − (cid:1) G - / ] [ G / ] FIG. 3. Matching time ¯ t against the initial condition φ in , for a scalar field with potential Eq.(68)in d = 4 space dimensions. λ ranges from 4 × − (blue dots) to 4 × − ] (red dots). VI. CHANGING THE NUMBER OF SPACETIME DIMENSIONS
We consider here whether our findings may be extended to a theory defined on a spacetimeof arbitrary dimension d + 1 ( d space dimensions) with a O (1 + d ) metric ds = dt + ρ ( t ) d Ω d . (80)Now the equation of motion for the scalar field is¨ φ + 3 dt ˙ φ = dVdφ . (81)Our considerations on the Taylor expansion in four spacetime dimensions ( d = 3) do notdepend on d apart from numerical factors. We need only to verify when Eq.(13) holdsfor arbitrary d , and thus find the conditions on the potential such that the zeroth-orderterm of the Taylor expansion dominates over higher order terms. We start by imposing themasslessness condition Eq.(17), which does not depend on d . We will deal with massivescalar fields later. If the field is massless and the zeroth-order term dominates, the equationof motion have a d dependent solution that, in the limit of large t ∗ giveslim φ ∗ → t ∗→ + ∞ ˙ φ ( t ) = − Ct d , lim φ ∗ → t ∗→ + ∞ (cid:18) dVdφ (cid:19) ∗ = C (1 + d ) t ∗ d +1 . (82)From the discussion in Sect.II, we see that for d +1 >
4, that is d > , no additional conditionsneed to be imposed. The equation of motion for the scale factor if we include Einstein-Hilbertgravity is as in Eq.(24) for d ≥
2, with the factor 3 replaced with a d − dependent factor.Thus the equations of motion at large times on the bounce is˙ ρ = 1 + O ( t − d ) for ρ ( t ) ≈ t and φ ( t ) ∝ t − d +1 (83)which consistently gives ˙ ρ ≈ t → + ∞ for d ≥
2. The matching time may be determinedas a solution of t + ( d − φ ˙ φ = 0 or ( d − − ( d − φ ˙ φ ¨ φ = 0 . (84)18hus, for d >
3, the asymptotic behaviour of the scalar field is as in Eq.(82) if the scalarfield is massless. An example is shown in Fig. 3. We plotted the matching time ¯ t againstthe initial condition φ in for a theory as in Eq.(68), in d = 4 space dimensions, and matchingtime determined by Eq.(84). The on-shell action is S E = 615 ( S E = 618) for λ = 10 − ( λ = 10 − ) corresponding to a decay time log ( τ T − U ) ≈
26 ( log ( τ T − U ) ≈ d <
3. In d = 2 we have additional conditions onthe potential (cid:18) d Vdφ (cid:19) ∗ t ∗ C (cid:28) (cid:18) d Vdφ (cid:19) ∗ C (cid:28) d = 1, we have (cid:18) d j Vdφ j (cid:19) ∗ ¨ φ j − ∗ ( t ∗ + a ) j − ∼ t ∗ for j > t ∗ (86)and thus Eq.(82) does not hold for any j . In fact, the on-shell Euclidean Lagrangian inarbitrary dimension d is L = t d (cid:34) ˙ φ V (cid:18) C ( d − t d − (cid:19)(cid:35) and thus the on-shell action S E = (cid:90) d d +1 x L is divergent for d = 1 if Eq.(82) holds.Massive scalar fields instead dominate the Taylor expansion. Our proof in Sect.IV isindependent on the number of spacetime dimensions and so we expect that Eq.(61) holdsfor arbitrary d (cid:54) = 1, if other terms in the potential satisfy Eq.(17) in d >
3, Eq.(17) andEq.(19) in d = 3 and Eq.(17) and Eq.(85) in d = 2. Including Einstein-Hilbert gravity doesnot affect this result as an exponentially small scalar field does not spoil ρ ≈ t at large times.An example is shown in Fig.4, in which we plotted the matching time ¯ t against φ in for atheory V ( φ ) = m φ − φ + φ for m ∈ [10 − , − ] with Einstein-Hilbert gravity (87)with d = 2. Our choice for the matching time is as in the example reported in Sec.IV. Thematching time for overshoot trajectories is typically smaller than the undershoot one alsoin this case, and it corresponds to the line in the bottom of the plot. The on-shell actionis S E = 0 .
005 ( S E = 0 . m = 2 × − ( m = 10 − ) corresponding to a decay timelog ( τ T − U ) ≈ −
233 ( log ( τ T − U ) ≈ − G - / ] [ G / ] FIG. 4. Matching time ¯ t against the initial condition φ in , for a massive scalar field with potentialEq.(87) in d = 2 space dimensions. The square mass ranges from 10 − (green dots) to 10 − (pinkdots). VII. CONCLUSIONS AND FUTURE PROSPECTS
Bounce solutions for scalar field theories with dynamical gravity are usually found withnumerical methods, unless the gravitational backreaction is very small, or the thin-wallapproximation is valid. Necessary (but not sufficient) existence conditions for the bounceare the finiteness of the action and the consistency of boundary conditions for all fields.To verify whether such stabilization happens, we do not need the full analytic solution:the asymptotic behaviour suffices. In this work, we found it for single scalar field theorieswith and without dynamical gravity that were overlooked in our previous work. First, weapplied our method to massive scalar fields and compared our results to the ones of [23] asa consistency check. We also found that higher-order kinetic terms do not play a relevantrole on the asymptotic behaviour of the bounce. Moreover, we determined the asymptoticbehaviour of the bounce for both massless and massive scalar fields in a number of spacedimensions different than three. In both cases, we found a similar result for any d (cid:54) = 1,while in one dimension our analysis fails. We improved the numerical method previouslydeveloped: the bounce is found by maximizing the matching time, as it is positive anddivergent there. This is a general property of the matching time, and thus should be easilyapplied to other settings also.In the simple setting that we analyzed, conditions 1., 2. of Sec.I were always satisifed.This might not be the case in more complicated scenarios, for example in modified gravity.In particular, gravity terms beyond the Einstein-Hilbert one are required by perturbativerenormalizability, and it would be interesting to analyze their effect on the bounce. We willreport soon on these issues [43]. 20inally, we note that one important limitation of our method is that dynamical gravity isincluded only when the false vacuum lives on flat space. To study cosmological applications,however, it is better to introduce de Sitter false vacua in our analysis, which will be ournext goal. ACKNOWLEDGMENTS
S. V. acknowledges the financial support of the Italian National Institute for NuclearPhysics (INFN) for her Doctoral studies. This work has been partially performed using thesoftware Mathematica.
Appendix A: Time-dependent Taylor expansion of the potential
Here we compute the coefficients f n of Eq.(40). We denote (total) time derivatives ofarbitrary order with the index ( n ), while time derivatives of first and second order of thescalar field are denoted by one dot or two dots respectively. Partial derivatives in the scalarfield and in time are denoted with the symbot ∂ . Partial derivatives of order i are indicatedas ∂ i . The equations are implicitly evaluated at time t ∗ (such that ˙ φ ( t ∗ ) = 0) and field value φ ∗ . Using the equation of motion for the scalar field we can write ∂ i V∂φ i ( n +1) = (cid:18) ∂ i +1 V∂φ i +1 ˙ φ + ∂ i +1 V∂t∂φ i (cid:19) ( n ) , φ ( n ) = (cid:18) ∂ V∂φ ˙ φ + ∂ V∂φ∂t (cid:19) ( n − + n − (cid:88) i =2 B i φ ( i ) ( t + a ) n − i (A1)where B i s are numerical factors, whose value is not relevant for the following discussion.As the explicit time dependence of the potential makes the calculation more involved withrespect to the time independent case (reported in [22] ), we omit numerical coefficients forsimplicity in the following. Using the first equation in (A1), we can write the (n+1)-thderivative of dVdφ as ∂V∂φ ( n +1) = ∂ V∂φ φ ( n +1) + · · · + (cid:18) ∂ V∂φ (cid:19) ( n − ¨ φ + ∂ V∂φ∂t ( n ) = ¨ φ (cid:32)(cid:18) ∂ V∂φ (cid:19) ( n − ¨ φ + · · · ++ ∂ V∂φ φ ( n − (cid:19) + φ (3) (cid:32)(cid:18) ∂ V∂φ (cid:19) ( n − ¨ φ + · · · + ∂ V∂φ φ ( n − (cid:33) + · · · + ∂ V∂φ φ ( n +1) ++ ∂∂t (cid:32) ∂ V∂φ φ ( n ) + · · · + ∂ V∂φ n − ¨ φ + ∂ V∂φ∂t ( n − (cid:33) + n − (cid:88) j =1 ∂ V∂t∂φ n − j − φ ( j +1) , (A2)21hich can be further expanded using again Eq.(A1). We obtain ∂V∂φ ( n +1) = ∂V∂φ∂t n +1 + (cid:88) j ∂ V∂φ ∂t j φ ( n +1 − j ) + (cid:88) j ∂ V∂φ ∂t j ( ¨ φ φ ( n − − j ) + φ (3) φ ( n − − j ) + · · · ++ φ ( n +1) / φ ( n +1) / − j ) + (cid:88) j ∂ V∂φ ∂t j ( ¨ φ φ ( n − − j ) + φ (3) ¨ φφ ( n − − j ) + · · · ++ φ ( n +1) / φ ( n +1) / φ ( n +1) / − j ) + . . . . (A3)where sums on j run from j = 0 to some upper limit, for which derivatives of the scalarfield are of order two. The result is similar to the time independent case: each term ∂ i + j V∂φ i ∂t j in Eq.(A3) is multiplied by i − φ . Such derivatives are oforder n + 5 − i or lower, thus these terms are non-vanishing only if n + 5 − i >
1. So, thehighest-order derivative ∂ ¯ ı V∂φ ¯ ı that appears in Eq.(A3) is the one satisfying n + 5 − ı = 3 foreven n and n + 5 − ı = 2 for odd n . The difference with respect to the time independentcase is that now the potential contains also partial derivatives in time and such terms aremultiplied by i − φ , the only exception being time derivativesof ∂V∂φ . For example, the sixth-order time derivative ∂V∂φ (6) ( n = 5) is expanded in terms of ∂ V∂φ , ∂ V∂φ and ∂ V∂φ as: ∂V∂φ (6) = ∂ V∂φ φ (6) + ∂ V∂φ ∂t φ (5) + ∂ V∂φ ∂t φ (4) + ∂ V∂φ ∂t φ (3) + ∂ V∂φ ∂t φ (2) + ∂ V∂φ ( ¨ φ φ (4) ++ φ (3) φ (3) ) + ∂ V∂φ ∂t ¨ φ φ (3) + ∂ V∂φ ∂t ¨ φ + ∂ V∂φ ¨ φ + ∂V∂φ∂t . (A4)We expand time derivatives of ¨ φ in Eq.s(A1) using Eq.(A3). We find φ ( n +1) = (cid:88) j ∂ V∂φ ∂t j φ ( n − − j ) + (cid:88) j ∂ V∂φ ∂t j ( φ ( n − − j ) ¨ φ + φ ( n − − j ) φ (3) + . . . )++ n (cid:88) i =2 B i φ ( i ) ( t + a ) n − i +1 + ∂V∂φ∂t n − (A5)As a result, using Eq.(A5), we can express Eq.(A3) in terms of partial derivatives of thepotential with respect to the scalar field and time, ¨ φ and t ∗ only. We order such termsaccording to the order of the derivative of the potential in the scalar field, which will belabelled with i in the following. Partial derivatives in time of the potential should becompensated with appropriate powers of t ∗ with respect to the j = 0 term. Moreover, time22erivatives of ∂V∂φ compensate for some factors ¨ φt j with respect to the time independent case,as can be seen from Eq.(A5). We can carry out the calculation in the time independent case,as explained in [22] , and then add terms ∂ i + j V∂φ i ∂t j t j to ∂ i V∂φ i and ∂V∂φ∂t j to ¨ φt j . All such termsthat are multiplied by a negative or vanishing power of t ∗ contribute. It is easier to see howthis works with some examples. The highest-order derivative (the ¯ ı -th term) is multipliedonly by time derivatives of the scalar field of order 2 or 3 and thus it contributes as ∂ ¯ i V∂φ ¯ ı ¨ φ ¯ ı − odd n , (A6)¨ φ ¯ ı − ( t ∗ + a ) (cid:18) ∂ ¯ ı V∂φ ¯ ı + ∂ ¯ ı V∂φ ¯ ı ∂t ( t ∗ + a ) (cid:19) + ¨ φ ¯ ı − ∂V∂φ∂t ∂ ¯ ı V∂φ ¯ ı even n (A7)to f n . The first term for even and odd n is present also in the time independent case: twoadditional terms appear, one that replaces ¨ φt ∗ and a time derivative which is compensatedby an additional power of time. As positive powers of time cannot appear, there should beno other terms in the highest-order derivative.The second-highest derivative ¯ ı − , , ,
5. Using Eq.(A5), derivatives of order 4 and 5 may be expressed in terms oflower order derivatives. We getodd n ¨ φ ¯ ı − ( t ∗ + a ) (cid:20) ∂ ¯ ı − V∂φ ¯ ı − (cid:18) ∂ V∂φ ( t ∗ + a ) (cid:19) + ∂ ¯ ı V∂φ ¯ ı − ∂t ( t ∗ + a ) + ∂ ¯ ı +1 V∂φ ¯ ı − ∂t ( t ∗ + a ) (cid:21) ++ ¨ φ ¯ ı − ( t ∗ + a ) (cid:18) ∂ ¯ ı − V∂φ ¯ ı − ∂V∂φ∂t + ∂ ¯ ı − V∂φ ¯ ı − ∂V∂φ∂t ( t ∗ + a ) + ∂ ¯ ı V∂φ ¯ ı − ∂t ∂V∂φ∂t ( t ∗ + a ) (cid:19) ++ ∂ ¯ ı − V∂φ ¯ ı − ¨ φ ¯ ı − ∂V∂φ∂t ∂V∂φ∂t (A8)even n ¨ φ ¯ ı − ( t ∗ + a ) (cid:20) ∂ ¯ ı − V∂φ ¯ ı − (cid:18) ∂ V∂φ ( t ∗ + a ) + ∂ V∂φ ∂t ( t ∗ + a ) (cid:19) + ∂ ¯ ı V∂φ ¯ ı ∂t ( t ∗ + a ) (cid:18) ∂ V∂φ ( t ∗ + a ) (cid:19) + ∂ ¯ ı +1 V∂φ ¯ ı − ∂t ( t ∗ + a ) + ∂ ¯ ı +2 V∂φ ¯ ı − ∂t ( t ∗ + a ) (cid:21) ++ ¨ φ ¯ ı − ( t ∗ + a ) (cid:20) ∂ ¯ ı − V∂φ ¯ ı − ∂V∂φ∂t (cid:18) ∂ V∂φ ( t ∗ + a ) (cid:19) + ∂ ¯ ı − V∂φ ¯ ı − ∂V∂φ∂t ( t ∗ + a )++ ∂ ¯ ı V∂φ ¯ ı − ∂t ∂V∂φ∂t ( t ∗ + a ) + ∂ ¯ ı +1 V∂φ ¯ ı − ∂t ∂V∂φdt + ∂ ¯ ı V∂φ ¯ ı − ∂t ∂V∂φ∂t + ∂ ¯ ı − V∂φ ¯ ı − ∂V∂φ∂t (cid:21) ++ ¨ φ ¯ ı − ( t ∗ + a ) (cid:20) ∂ ¯ ı − V∂φ ¯ ı − ∂V∂φ∂t ∂V∂φ∂t + ∂ ¯ ı V∂φ ¯ ı − ∂t ∂V∂φ∂t ∂V∂φ∂t ( t ∗ + a )+ ∂ ¯ ı − V∂φ ¯ ı − ∂t ∂V∂φ∂t ∂V∂φ∂t ( t ∗ + a ) (cid:21) (A9)23he first term for even and odd n are the same as in the time independent case, and othercontributions appear with additional time derivatives, following the rules described above.Terms that appeared as ∂ i V∂φ i t i − ¨ φ i − (A10)in the time independent case, are now ∂ j + i V∂φ i ∂t j t i − j ¨ φ i − (A11)and thus they are much smaller than one if ∂ j + i V∂φ i ∂t j t j (cid:28) j ≥ ∂V∂φ ( n +1) = ∂ n +1 V∂φ∂t n +1 + i =¯ ı − (cid:88) i =0 ¯ ı − i − (cid:88) m =0 (cid:88) j ,...j m i +1 (cid:88) n =0 ∂V ¯ ı − i ∂φ ¯ ı − i ∂t j ¨ φ ¯ ı − i − − m ( t + a ) i +1 − n ∂V∂φ∂t j × · · · × ∂V∂φ∂t j m (A13)where j + · · · + j m = n , while in the time independent case we had ∂V∂φ ( n +1) = i =¯ ı − (cid:88) i =0 ∂V ¯ ı − i ∂φ ¯ ı − i ¨ φ ¯ ı − i − ( t + a ) i +1 (A14)which is negligle with respect to the zeroth-order term under conditions described in Sec.II.Thus we need also to impose that ∂ j V∂φ∂t j t j (cid:28) ¨ φ ∗ . (A15)In conclusion, if ∂ j + i V∂φ i ∂t j t j (cid:28) ∂ j V∂φ∂t j t j (cid:28) ¨ φ for j ≥ t ≤ t ∗ and large t ∗ , besides Eq.(17) and Eq.(19), then Eq.(8) is satisfied in thetime-dependent case. As in Sec.II we can repeat the calculation for a scalar field theorywith Einstein-Hilbert gravity which gives a flat spacetime on the bounce trajectory at largetimes. The same result holds, replacing t ∗ with t ∗ + a . [1] S. R. Coleman, Phys. Rev. D (1977), 2929-2936 doi:10.1103/PhysRevD.16.1248
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