A novel method to compute bounce solutions
AA novel method to compute bounce solutions
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 investigate bounce solutions for single scalar field theories, with and without Einstein-Hilbertgravity, considering a false vacuum living on flat Euclidean spacetime. We find that, under appro-priate conditions, the behaviour of the scalar field far away from the bubble is independent on theform of the potential and it may be determined analytically. In particular, we need the scalar fieldto be massless, or light, and cubic self-interactions should be small. We mainly discuss Einstein-Hilbert gravity, and, in particular, the case of the Higgs decay, but generalizations to modifiedgravity are in principle possible. We use our findings to introduce a novel numerical procedurebased on minimization, as an alternative to the shooting method. ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] S e p . INTRODUCTION The phenomenon of metastability is of primary importance in many fields in physics, be-ing observed in nature in quite different contexts. As metastable states are local minima ofthe potential, they are classically stable at zero temperature. However, quantum processesmay allow a transition to states which have a lower energy. In particular, recent measure-ments of the Higgs and the top quark mass [1, 2] place the Standard Model in the metastableregion [3–9], opening up the possibility for vacuum decay to happen, along with all its catas-trophic consequences. Luckily, the typical lifetime of such metastable state is much longerthan the age of the Universe, which is in agreement with our cosmological history. However,the Standard Model Higgs potential is unbounded from below, so we expect that some newphysics enter at sufficiently high energy, possibly changing by several orders of magnitudethe lifetime of the false vacuum [10–20]. There are also other mechanisms that make suchstates unstable, for example the presence of black holes that may act as nucleation seeds[21–26]. The tunneling rate depends on the shape of the potential barrier that separates thefalse vacuum and the tunneling point; it does not depend on the shape of the potential faraway from it. Thus, if one wants to account for high-energy or quantum corrections (suchas one-loop corrections) the decay rate is expected to be affected only if these are localizedin this region. This behaviour is manifest, for example, in Fig.1 of [13], where an increase ofthe non-minimal coupling parameter between the Higgs field and gravity shifts the positionof the bounce to the region where the higher-order corrections are not important. Anyway,even relatively small changes in the potential may alter significantly the decay rate. This be-haviour has been observed in the case of Higgs decay [3, 18]. Small gravitational correctionsinstead do not change much the decay rate of the Higgs field [10, 14, 15].In field theory, the tunnelling rate at zero temperature can be computed in the semi-classical approximation by finding a solution to the Euclidean equations of motion thatinterpolates between some initial condition and the false vacuum [27, 28]. Such trajectoryis called ’bounce’. Once that the bounce is found, one can compute the decay rate froma combination of the on-shell bounce action and the fluctuation determinant. Since theequations of motion are highly non-linear, an analytical approach is usually impractical.Nevertheless, this has been done in a few cases [33–35]. This problem can be circumventedwhen the energy difference among the false vacuum and the true one is sufficiently small,2.e. when the thin wall approximation is valid [29, 30, 32–35]. When this is not the case, oneneeds to resort to numerical procedures, the most popular being the shooting method. Withthe latter, one solves numerically several times the equations of motion and see whetherthe boundary condition at infinity is met. In the case of a single real scalar field, if itsinitial value is smaller than the one of the bounce, the trajectory undershoots and the fieldmakes multiple oscillations in the well of the Euclidean potential (an example is reportedin Fig.(1)) until it gets to rest at its bottom. On the contrary, if the initial value is larger,the field overshoots and reaches the false vacuum in a finite time. The bounce can then befound as the solution of the equations of motion that separates undershoot and overshoottrajectories. This allows to determine it with arbitrary precision, computational limits aside.In general, using this method implies long integration times, as the bounce initial condition φ (0) can be large ( O (0 . M P )) and/or the friction term can be very effective in slowing downthe scalar field. Moreover, one should compute the bounce with sufficient precision to get agood estimate of the on-shell action S E . The Lagrangian must be integrated up to a cut-off,which should be carefully chosen. A possible way out is the gradient flow method [36], inwhich the trajectory can be found as a stable fixed point of some flow equation. This meansthat there is no need of choosing carefully the initial conditions. While the shooting methodcan be used in a wide variety of situations, the gradient flow method is (so far) specific ofthe scalar field case. The possibility of computing corrections of gravitational origin to thesingle scalar field bounces on Euclidean background has been addressed in [14, 15]. The newbounce is expanded in inverse powers of the Planck mass, of which the old bounce representsthe zeroth-order. The expansion is valid when the gravitational backreaction on the scalarfield is small.In this paper, we derive a quite general property of bounce solutions for theories withone real scalar field and gravity when the ‘parent’ (i.e. outside the bubble) spacetime isflat. In particular, we find that if the scalar field is massless, or light, and with a sufficientlysmall cubic self-interaction, the behaviour of the scalar field on the bounce is independent onthe form of the potential near the false vacuum. We will focus mainly on Einstein gravity,but results may be generalized to modified gravity to some extent. We also introduce anovel numerical method to determine the bounce for a given potential, with and withoutdynamical gravity. We consider off-shell profiles for the scalar field and the Euclidean scalefactor that depend on a parameter C , and that satisfy the asymptotic behaviour of the3 V ( ϕ ) ϕ fv ϕ tv ϕ L ϵ h FIG. 1. The Euclidean potential − V ( φ ). Green arrows indicate the scales that enter the definitionof the thin-wall approximation. The green shaded area shows the interval of possible φ allowedby energy conservation. bounce. The Euclidean action is minimized with respect to C in order to find the bounce,but such stationary point is a saddle point. If gravity is dynamical, we may consider analternative functional of the scalar field and gravity which is finite on such profiles, and thathas a minimum on the bounce. Computing this functional for different C allows to find thebounce, and thus to compute the decay rate.The paper is organized as follows. In the next section we derive the asymptotic behaviourof bounces for massless scalar fields with small cubic self-interactions, in the case of a singlescalar field theory with and without Einstein-Hilbert gravity. In Sect.III we use this propertyto provide a numerical method to find the bounce action for a given theory. We discuss itsvalidity when the scalar field is not massless, but light. In Sect.IV we test our method bycomparing our results with the ones obtained with the usual shooting method. We concludewith some considerations and future prospects. II. ASYMPTOTIC BEHAVIOUR OF THE BOUNCE
Following the original Coleman’s prescription, we consider a O (4)-symmetric Euclideanmetric ds = dt + ρ ( t ) d Ω (1)4here d Ω is the line element of the three-sphere and ρ ( t ) is the Euclidean scale factor. Theaction 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) (2)and the equations of motion are ¨ φ + 3 ˙ ρ ˙ φρ = dVdφ (3)˙ ρ = 1 + 8 π ρ (cid:32) ˙ φ − V ( φ ) (cid:33) (4)where the dot indicates derivative with respect to the Euclidean time t . The second term onthe left-hand side of Eq.(3) couples the differential equation to Eq.(4) and acts as a frictionterm for the scalar field. The Ricci scalar is R = 16 π (cid:32) ˙ φ V ( φ ) (cid:33) , (5)and it can be used to compute the on-shell action, which reads S bounce = − π (cid:90) + ∞ ρ ( t ) V ( φ ) dt. (6)The potential − V ( φ ) is qualitatively as in Fig.1, where the false vacuum value of the scalarfield is φ fv and the true one φ tv . As we focus on Euclidean false vacua, we have V ( φ fv ) = 0.We also assume that φ fv = 0 and φ tv > to Eq.s (3) (4) with boundary conditions φ ( ∞ ) = 0 , ˙ φ (0) = 0 , ρ (0) = 0 (7)and finite on-shell action. We define a as the boundary condition at infinity on the bouncefor the Euclidean scale factor, that is a = lim t → + ∞ ( ρ ( t ) − t ) . (8)Both a and φ are determined once that the bounce is found. In the limit in which gravitycan be ignored, the spacetime is Euclidean at all times. Thus, in this case, ρ ( t ) = t and a = 0. The decay rate is given by Γ = A e − B , (9) That is, a solution with initial condition φ (0) = φ with φ (0) (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 and can be estimated as A ≈ T U R [37] where R is such that φ ( R ) = φ T U is the age of the Universe. In the following, we will focus on the exponent B , andassume that we may use such approximation to evaluate Γ, or, in alternative, that Γ has aweak dependence on A . We will label the initial condition φ (0) for the bounce as φ , while φ in indicates any other initial condition for the scalar field. Note that energy conservationimplies φ fv < φ < φ tv , so that φ tv is never reached during the bounce. In fact, the potential V ( φ ) may as well be unbounded from below, the most prominent example being the Higgspotential. A. Asymptotic behaviour of the bounce
In this Section, we derive the asymptotic behaviour of the bounce for a theory as inEq.(2) with metric (1). If ρ ( t ) = t + a for t → + ∞ on the bounce, we have that there existsome time beyond which ˙ ρρ ≈ t + a . (10)Assuming that the right-hand side of Eq.(3) is negligible with respect to the left-hand side, (cid:12)(cid:12)(cid:12)(cid:12) dVdφ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) φt tp + a ≈ ¨ φ (11)we get ˙ φ ( t ) = − C ( t + a ) (12)where C > φ ( t ) = C t + a ) . (13)We can see from Eq.s (3) (4) that, in order for Eq.(11) to hold, a polynomial potentialaround the false vacuum should depend on the scalar field as φ n with n >
3. Moreover, weexpect that, if Eq.(13) holds for the bounce at very large times, trajectories arbitrary closeto it (either undershoot or overshoot ones) satisfy Eq.(11) for an arbitrary long time. AsEq.(12) is negative definite at all times and monotonically increasing, and Eq.(13) is positivedefinite at all times and monotonically decreasing, we expect that deviations from Eq.(13) for6ndershoot (or overshoot) trajectories occur approximately when ˙ φ (or φ ) vanishes duringthe field evolution towards the false vacuum. We will indicate such time as t tp . Thisprocedure is trivially valid also in the case of background Euclidean spacetime, as it simplyamounts to taking a = 0.We would like to give a more rigorous derivation of Eq.(13), beyond the consistencyrequirements described here. Firstly, because we might have C = 0: in this case higher orderterms in Eq.(13) are important. Secondly, we will never be able to determine numericallythe bounce trajectory with infinite precision; thus, it is important to understand how nearbytrajectories behave. The remainder of this Section is dedicated to find Eq.(13) from thevery definition of bounce as a limiting undershoot trajectory.If the field undershoots, it means that at some finite time it turns around and startsoscillating in the well of − V ( φ ) until it settles in the minimum (Fig. 1). The value of thescalar field at such turning point will be labelled as φ tp , and it is reached at some finite t tp ,for which ˙ φ ( t tp ) = 0, as anticipated above. The relation among the initial condition φ in forsuch trajectories and t tp , φ tp is in general complicated, due to the friction term, as the fieldcan experience many accelerations and decelerations. However, we expect that if φ − φ in ispositive and sufficiently small, increasing φ in towards φ gives larger and larger oscillationsin the well: the smaller is φ tp , the larger is t tp , and the bounce corresponds to the limit φ tp → t tp → ∞ , as φ in → φ (Fig.3) [27]. We now look for an approximate solutionto the equations of motion Eq.(3) at large t , t ≤ t tp , in order to take the limit t tp → ∞ andthus find the asymptotic behaviour of the bounce. We can always choose the undershoottrajectory sufficiently close to the bounce so that t tp lies in the region where Eq.(10) holds.The right-hand side of Eq.(3), expanded around t tp , reads dVdφ ( φ ( t )) = dVdφ ( φ tp ) (cid:32) (cid:88) n ≥ f n ( φ tp , t tp + a )( t − t tp ) n (cid:33) (14)with f n ( φ tp , t tp + a ) = (cid:18) dVdφ (cid:19) ( n ) ( φ tp , t tp + a )where the index ( n ) indicates the n -th order time derivative and the coefficients f n s aregiven in Appendix A. We require that f n s are such that we can safely take dVdφ ( t ) ≈ dV ( φ tp ) dφ (15) We will call ’nearby trajectories’ solutions to the equation of motion of the scalar field with initial condition φ (0) close to the bounce one. t ≤ t tp and sufficiently large t , t tp . As explained in Appendix A, in order to do that weneed d j Vdφ j ¨ φ j − t j − tp (cid:28) j ≥
2. If all derivatives of the potential in the scalar field are finite for φ → dV ( φ tp ) dφ depends on t tp as C t ktp with k < −
4, we only need to impose that d V ( φ tp ) dφ t tp (cid:28) . (17)If k = − C d Vdφ ( φ tp ) (cid:28) . (18)Solving ¨ φ + 3 ˙ φt + a = dV ( φ tp ) dφ (19)we get ˙ φ ( t ) = dV ( φ tp ) dφ t + a − dV ( φ tp ) dφ ( t tp + a ) t + a ) . (20)so φ ( t ) = φ tp − dV ( φ tp ) dφ ( t tp + a ) dV ( φ tp ) dφ ( t + a ) dV ( φ tp ) dφ ( t tp + a ) t + a ) . (21)As this is to be valid for undershoot trajectories arbitrarily close to the bounce, we considerthe limit φ tp → t tp → + ∞ . Then, the bounce should satisfylim φ tp → ttp → + ∞ ˙ φ ( t ) = − C ( t + a ) , lim φ tp → ttp → + ∞ dV ( φ tp ) dφ = 4 C ( t tp + a ) , lim φ tp → ttp → + ∞ φ ( t ) = C t + a ) (22)from sufficiently large t up to t → + ∞ . Eq.(17) suggest that Eq.(13) does not hold atarbitrarily large t tp if the scalar field is massive, with mass m . Moreover, a polynomialpotential with cubic self-interactions gφ , should have gC (cid:28) ρ ( t ) ≈ t and Eq.(13) in Eq.(6) for a quadratic potential: the integralis not convergent. Anyway, from Eq.(17) we see that the asymptotic behaviour is valid upto some time t ∗ , such that t ∗ (cid:28) m .We have (cid:12)(cid:12)(cid:12)(cid:12) dVdφ (cid:18) C t + a ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) C ( t + a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φt + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ ¨ φ ( t ) . (23)8ndershoot and overshoot trajectories sufficiently near the bounce should satisfy the sameinequality and φ ( t ) ≈ C ( t + a ) for large t , approximately up to some finite, large t tp , such that ˙ φ ( t tp ) = 0 ( φ ( t tp ) = 0)) forundershoot (overshoot) trajectories. We expect these results to hold also when gravity isneglected, as our procedure holds also for a = 0.The t − behaviour of the scalar field for large t near the bounce has been already observedby [19] in the case of the Higgs potential with loop corrections and Einstein-Hilbert gravity,and it appears also in the bounce h ( t ) for the scale invariant theory λφ on flat spacetime h ( t ) = (cid:114) λ R t + R (24)although our method is not valid in this case, as the theory is scale invariant, and thus wecannot distinguish among undershoot trajectories and the bounce. This is manifest in thesecond equation of (22), as we need dVdφ to be positive. Anyway, if we slightly modify thescale invariant potential (for example, by adding quantum corrections) so that the methodapplies, the asymptotic time dependence of the scalar field Eq.(13) holds for sufficientlylarge t . For some time, the scalar field, satisfying Eq.(13), may be in the region where dVdφ is negative, that is, where the scale invariant potential is virtually the same as the scale-dependent one. If we smoothly modify one potential into the other, we find that Eq.(13)holds also in the scale invariant case.When dV ( φ tp ) dφ increases, t tp decreases, thus it has a minimum at φ tp such that dV ( φ tp ) dφ is maximum. Moreover, by expanding Eq.(4) in the vicinity of the bounce, we get ρ ( t ) = t + a + O (cid:18) t (cid:19) (25)which is consistent with the boundary conditions for gravity when the false vacuum isreached. We will check the validity of the second equation in Eq.(22) against numericalcalculations in Sect.IV.Notice that we determine the asymptotic behaviour of the bounce as a limiting under-shoot trajectory, and separately we can verify that the on-shell action is convergent. Inmore generic scenarios, knowing the asymptotic behaviour of the bounce has importantconsequences in the determination of the decay rate of the false vacuum. In particular,9t might be the case that the system indeed reaches the false vacuum for arbitrary large t , but the integral of the Lagrangian, the on-shell action, does not converge in the upperlimit. Another possibility is that the asymptotic behaviour of scalar fields is such that theappropriate boundary conditions for gravity are not met. This would have been the case ifwe would have found that φ ( t ) ≈ t at large t , as clearly Eq.(4) does not give ρ (cid:48) ( t ) = 1, ifwe take ρ ( t ) = t . Such properties are difficult to infer by numerical evaluation. III. DETERMINING THE BOUNCE BY MINIMIZATION
We now use the results found in the previous section to determine numerically the bouncetrajectory for a given potential. We introduce two one-parameter family of functions φ ( t ) C and ρ ( t ) C that in general do not satisfy Eq.(3), (4) but such that • they match the bounce for t → ∞ and for some choice of C , that will be indicated as C ; • if we define the functional S EC ≡ S EC, + S EC, (26)where S EC, ≡ − π (cid:90) ¯ t ¯ ρ ( t ) V ( ¯ φ ) dt (27) S EC, ≡ π (cid:90) ∞ ¯ t t (cid:18) C t + V (cid:18) C t (cid:19)(cid:19) dtS EC, is finite when computed on φ ( t ) C and ρ ( t ) C , if the scalar field is massless. Here, S EC, is the dominant contribution to S EC while S EC, , as we will see later, is fundamentalin determining the bounce trajectory; • S EC has a minimum on the bounce. This will be proven in Sect.III A.Explicitly, these functions are given by ( C > φ C ( t ) = ¯ φ ( t ) 0 < t < ¯ tC t ¯ t < t < ∞ (28)10nd ρ C ( t ) = ¯ ρ ( t ) 0 < t < ¯ tt ¯ t < t < ∞ (29)where ¯ φ ( t ) and ¯ ρ ( t ) are on-shell and are matched assuming continuity and differentiabilityat the matching point ¯ t . This matching condition is crucial to fix the second derivative of S EC when it is evaluated on C (see Sect.III A), and thus the type of the stationary point in S EC .Moreover, as we consider the t → + ∞ behaviour of the bounce, our ansatz is indipendenton a , which is not a priori known.Once that we have proven that the functional S EC behaves properly, the procedure to findthe bounce is very simple: solve the equations of motion numerically as regards the on-shellpart, match it with the off-shell one by continuity of φ ( t ) and ˙ φ ( t ) (this allows to find C and ¯ t ) and evaluate S EC . Repeating this procedure for different C s allows to find C , witha given approximation. Then, we can write the action as S E = − π (cid:34)(cid:90) ¯ t ρ ( t ) V ( φ ) dt + (cid:90) ¯ ∞ ¯ t t V (cid:18) C t (cid:19) dt (cid:35) . (30)Such action is on-shell when C = C , but it has a saddle point there. We consider S EC in placeof the action (30) for minimization because in general it is easier to determine the positionof minima or maxima with respect to saddle points. In fact, quasi-scale invariant theorieshave a nearly flat direction along C , which makes it difficult to determine the position of thebounce numerically, if we were to use S E instead of S EC . Similarly, if the scalar field moveson a fixed Euclidean background, we can consider an off-shell profile as in Eq.(28) and writethe action as S E = 2 π (cid:34)(cid:90) ¯ t ρ ( t ) (cid:32) ˙ φ V ( φ ) (cid:33) dt + (cid:90) ¯ ∞ ¯ t t (cid:18) C t + V (cid:18) C t (cid:19)(cid:19) dt (cid:35) . (31)It also has a saddle point on the bounce. We were not able to find a functional that turnssuch saddle point into a maximum or a minimum.The matching point ¯ t , as we will see, grows larger and larger the closer we get to the bouncetrajectory, so we need to integrate the equations of motion for an arbitrary long time todetermine φ with arbitrary small precision, as it happens with the shooting method. In We expect that an exact matching cannot always be done in some cases, in particular, for undershoottrajectories. More details are given in Sect.III A C . Moreover, dS EC dC gets smaller and smaller the closer we get to the bounce, thus wecan control the precision to which we determine S E ( C ). In the remainder of this sectionwe would like to justify the matching conditions, and describe how these affect S EC and itsderivatives in C on the bounce. A. Matching conditions
As stated before, we match the off-shell and on-shell parts of the profiles φ ( t ) C and ρ ( t ) C by continuity and derivability. This means we are looking for some finite ¯ t such that¯ t + 2 φ (¯ t )˙ φ (¯ t ) = 0 . (32)Then, we find C as C = − ¯ t ˙ φ (¯ t ). Close to t tp , overshoot trajectories satisfy¯ t + 2 φ (¯ t )˙ φ (¯ t ) > t < t tp that satisfies Eq.(32) always exist in this case. Undershoot trajectories insteadsatisfy ¯ t + 2 φ (¯ t )˙ φ (¯ t ) < t tp , so, there may not be a ¯ t < t tp that satisfies Eq.(32). In this case, we find thepoint of closest approach 3 − φ (¯ t ) ¨ φ (¯ t )˙ φ (¯ t ) = 0 . (35)If multiple ¯ t s exist, we take the largest one. Then, the bounce should satisfy Eq.(34) forfinite t , and Eq.(32) for t → + ∞ .As trajectories near the bounce at large t satisfy Eq.(23), C is determined by the equation − C (¯ t + a ) = − C ¯ t (36)thus giving C = C (cid:18) − a ¯ t (cid:19) + h.o. (37)This equation also shows that the bounce corresponds to the limit ¯ t → + ∞ . Here h.o.indicates both terms of the power series expansion and higher order effects. Among these,12e have that a = a (¯ t ) when we match at finite ¯ t , as Eq.(25) defines a function of ¯ t (and thuson C by Eq.(36)) a (¯ t ) = a + O (cid:18) t (cid:19) . (38)Moreover, we expect a deviation from Eq.(23) at finite ¯ t sufficiently far from the bounce.These effects are important in the determination of C in the case a = 0 only, as we expectthat, for a (cid:54) = 0, we can always take ¯ t sufficiently large to make them negligible with respectto the a -dependent terms.We now use this result to prove that: • the action S EC has a stationary point for C = C ; • such point is a minimum for S EC on the bounce.Let us minimize the functional (26) with respect to C , and evaluate it in the vicinity of thebounce. We find dS EC dC = − π R ¯ t d ¯ tdC +2 π ¯ t (cid:18) V ( φ (¯ t )) − V (cid:18) C t (cid:19)(cid:19) + π C ¯ t + π (cid:90) + ∞ ¯ t dVdφ (cid:18) C t (cid:19) t dt + B φ + B g . (39)The second term vanishes for overshoot (undershoot) trajectories at finite (finite or infinite)¯ t . Here B φ , B g are boundary terms for the scalar field and for gravity, that appeared as weused the equations of motion in the variation of the first term of Eq.(26). There is one forthe scalar field B φ = ¯ t ˙ φ (¯ t ) δφδC (¯ t ) = − π C ¯ t , (40)while the gravitational one can be computed from the Hawking-Gibbons-York boundaryterm [38, 39] evaluated at t = ¯ tδS GHY = (cid:73) ∂V d x(cid:15) (cid:112) | h | n α V α (41)with V α = g µν δ Γ αµν + g αµ δ Γ νµν (42) δ Γ αβγ = 12 g αµ ( ∂ β δg γµ + ∂ γ δg βµ − ∂ µ δg γβ ) + 12 δg αµ ( ∂ β g γµ + ∂ γ g βµ − ∂ µ g γβ )and δg αβ is the variation of the metric, that has an inverse δg αβ = − g µα g νβ δg µν . Moreover, n α is the unit normal to ∂V and h is the determinant of h αβ , the induced metric on the13oundary. (cid:15) is +1 if ∂V is timelike, − n α = (1 , , , θ, ϕ, ψ ) coordinates is h αβ = (¯ t + a (¯ t )) θ )
00 0 sin( θ ) sin( ψ ) . (43)The variation of g αβ is δg αβ = t + a ) δaδC t + a ) δaδC sin( θ )
00 0 0 2( t + a ) δaδC sin( θ ) sin( ψ ) . (44)We find B g = π
16 ¯ t (cid:0) g αβ δ ˙ g αβ + δg αβ ˙ g αβ (cid:1) = 0 (45)The dominant term is the first one of Eq.(39) and gives dS EC dC ≈ − π R (¯ t ) ¯ t d ¯ tdC ≈ − C π a ¯ t = − C ( C − C ) π a (46)and thus it vanishes in the limit ¯ t → + ∞ . The second derivative instead is constant andnon-vanishing for ¯ t → + ∞ d S EC dC ≈ π a (47)Thus S EC has a minimum on the bounce. B. Light scalar field
As we stated above, Eq.(22) is not a good approximation in the case of a quadraticpotential for t ≈ t tp and large t tp , as the second derivative of the potential has a non-vanishing, constant value for φ tp →
0. However, as explained in Sect.II, Eq.(23) is valid atlarge times, up to some t ∗ such that t ∗ (cid:28) m . Thus, for the numerical method describedabove, we can choose φ C ( t ) = ¯ φ ( t ) 0 < t < ¯ tC t ¯ t < t < t ∗ t ∗ < t < + ∞ (48)14 | ∆ S E | S Esm S E C ¯ t Higgs 0 .
071 10 − . . × Polynomial α = 10 − × − − .
59 1640 10 Polynomial α = 10 − × − − .
63 519 10 Polynomial α = 10 − × − − .
74 167 10 Polynomial α = 10 − × − × − .
16 57 . Polynomial α = 10 − × − − .
02 15 10 TABLE I. We compare here our results with the shooting method ( sm in the Table) ones. Inparticular, we compare the value of the bounce action S E . We also report the order of magnitudeof ¯ t , and C .We found the same initial condition φ in both numerical methods. instead of (28). If t ∗ is sufficiently large, S EC and its derivatives are not affected much. Thus,we can repeat the procedure in Sect.(III A) and find similar results. We may also consider atime-dependent mass, that, instead, may satisfy Eq.(22) if m → t tp → ∞ , as it wouldhappen by adding the non-minimal coupling φ R . This will be considered in future work. IV. APPLICATIONS
In this section, we show the validity of the numerical method with some examples, andwe compare it with shooting method results. In particular, we consider two potentials: • the Higgs potential V ( φ ) = λ ( φ )4 φ (49)where λ ( φ ) = λ ∗ + α ln ( φ ) + β ln( φ ) (50)and λ ∗ = − . , α = 1 . × − , β = 6 . × − . • a polynomial potential with a vanishing quadratic term V ( φ ) = α φ + α φ + α φ (51)where we choose α = 1, α = −
1, and we change α from 10 − to 10 − .15 .01 0.03 0.05 0.07 0.09220024002600 ϕ in S E C ϕ in S E C FIG. 2. Minimum of the action as a function of the initial condition φ for the potential (49) and(51) ( α = 10 − ). First of all, we would like to see whether Eq.s (15) are verified for φ tp →
0. First, let’sconsider the polynomial potential. In this case, we need only to verify that Eq.s (17),(18)hold. We should have (we retain the dominant term only)6 α t tp φ tp (cid:28) , α C (cid:28) . (52). Thus, we need φ tp ≈ (cid:15)t tp + h.o. , C (cid:28) α (53)with (cid:15) (cid:28)
1. Our prediction, the second equation of Eq.(22), gives φ tp = C α t tp (54)thus if 2 √ α C (cid:28) φ tp →
0. Eq.(22) gives φ tp ln( φ tp ) β = ¨ φ tp = C t tp . (55)Moreover, d j Vdφ j ≈ φ − j ln( φ ) j = 2 , , d j Vdφ j ≈ φ − j ln( φ ) j > . (56)Using these relations, we verified that Eq.(15) holds for every j >
1. Thus Eq.(13) holdsasymptotically for any value of the coupling.We compute the action S EC as described in Sect.(II) and find its minimum. Our results arereported in Table I, where we compare our findings with the ones obtained by the shooting16 ϕ tp - t t p ϕ V d ϕ ϕ tp - t t p ϕ V d ϕ FIG. 3. Top: t tp as a function of φ tp for the potentials in Eq.(49) (on the left) and Eq.(51)( α = 10 − , on the right), in the vicinity of the bounce ( φ tp → dVdφ as a function of φ . method ( sm in the Table). In particular, we report an order of magnitude estimate of thedeviation of our result for the on-shell action S E from the shooting method one S Esm as | ∆ S E | S Esm with | ∆ S E | = S Esm − S E . We did not report the difference among the initial condition φ ofour method and the shooting method one, as we found agreement up to the last digit. Wereport also the bounce action, S E , C and the order of magnitude of ¯ t on the minimum. Wecan consider ¯ t as an estimate of the time for which we need to solve numerically the equationsof motion to determine the bounce action with this precision. Such time is typically severalorders of magnitude less than the one needed for the shooting method. In the Higgs case,the typical time needed for the shooting method is ¯ t ≈ , in the polynomial case it isroughly one order of magnitude more than ours, for all the values of α considered, but for α = 10 − . In this case, the typical time needed for the shooting method matches ours. InFig.2 we reported the functional S EC in both cases, as a function of the initial condition φ in .Notice that neglecting the Higgs mass affects the bounce at t ≥ , but, as we can see17rom the results in Table I, ¯ t ≤ is sufficient to determine the on-shell action with goodprecision. Moreover, our value of C for the Higgs potential roughly corresponds to the onederived from minimization of the on-shell action, with a small backreaction [14] C = 2 lim t → + ∞ h ( t ) t = 4 √ | λ | R = 17 . × with R = 350 . In both cases, we compare t tp as a function of φ tp as given by our theoretical prediction(22), with a numerical evaluation. We expect them to match for sufficiently large t tp , andthat t tp → + ∞ for φ tp →
0. We reporte our results in Fig.3. On the top, we have t tp as a function of φ tp for the potentials in Eq.(49) (on the left) and Eq.(51) ( α = 10 − , onthe right), in the vicinity of the bounce. The continuous line is the numerical prediction:we solved numerically the equations of motion for undershoot trajectories in the vicinityof the bounce and reported ( t tp , φ tp ) for each. The red line highlights the position of theminimum of such function. The dashed line represents t tp as a function of φ tp , as determinedby Eq.(22). We took C s to be the ones reported in Table I. In the Higgs case, the dashedline is superimposed with the continuous one, and thus we omitted it. We can see that theminimum of the function lies where the derivative of the potential has its maximum. In theother one, the two functions have the same asymptotic behaviour for φ tp → V. CONCLUSIONS AND FUTURE PROSPECTS
The study of bounce solutions for single scalar field theories is usually addressed withnumerical techniques, despite some analytical solutions have been found. Even when thelatter is possible, including dynamical gravity in the picture usually results in the necessityof using numerical methods, too. Knowing the asymptotic behaviour of the bounce in ananalytical form allows to control the error in such numerical techniques, and also to introducenew ones. In this paper, we considered single scalar field theories with and without dynamicalEinstein-Hilbert gravity and found that, on the bounce, not only the asymptotic behaviourof the scalar field is independent on the potential but it may also be determined in closedform. The most stringent restriction that we need to impose is that the scalar field shouldbe massless. If the scalar field is very light (as measure with respect to the Planck mass)the same holds up to a cut-off that corresponds to the inverse of the scalar field mass. Withthis result at hand, we are able to determine the bounce by finding stationary points of the18ction. As such stationary point is a saddle point, we modified the action functional in orderto turn it into a minimum, at least in the case in which gravity is dynamical. The resultingprocedure is very simple and allows some control on our error in the determination of thebounce. Another possibility is to use the shooting method and compare trajectories nearthe bounce with the asymptotic behaviour that we found, in order to fix an optimal cut-off.Despite we largely focussed on the numerical method and on Einstein-Hilbert gravity,our findings allow for a wider range of applications. As we mentioned, in more genericscenarios, knowing the asymptotic behaviour of the bounce has important consequences inthe determination of the decay rate of the false vacuum. In particular, it might be thecase that the system indeed reaches the false vacuum for arbitrary large t , but the integralof the Lagrangian, the on-shell action, does not converge. Another possibility is that theasymptotic behaviour of scalar fields is such that the appropriate boundary conditions forgravity are not met. Moreover, our procedure may be generalized to include modified gravityor multiple scalar fields. In the future, we would like to extend our work to include suchscenarios, with particular attention to the case of Higgs decay. 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
Here, we compute the coefficients f n of Eq.(14). We denote time derivatives of arbitraryorder with the index ( n ), while time derivatives of first and second order are denoted by onedot or two dots respectively. Derivatives of the potential with respect to the scalar field oforder i are indicated as d i Vdφ i . Using Eq.(3) we can write d i Vdφ i ( n +1) = (cid:18) d i +1 Vdφ i +1 ˙ φ (cid:19) ( n ) , φ ( n ) = (cid:18) d Vdφ ˙ φ (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.19sing the first equation in (A1), we can write the (n+1)-th derivative of dVdφ as dVdφ ( n +1) = d Vdφ φ ( n +1) + · · · + (cid:18) d Vdφ (cid:19) ( n − ¨ φ = ¨ φ (cid:32)(cid:18) d Vdφ (cid:19) ( n − ¨ φ + · · · + d Vdφ φ ( n − (cid:33) ++ φ (3) (cid:32)(cid:18) d Vdφ (cid:19) ( n − ¨ φ + · · · + d Vdφ φ ( n − (cid:33) + · · · + d Vdφ φ ( n +1) , (A2)which can be further expanded using again Eq. (A1). We set ˙ φ = 0, as f n s in Eq.(14) areevaluated at the turning point time t tp . We obtain dVdφ ( n +1) = d Vdφ φ ( n +1) + d Vdφ ( ¨ φ φ ( n − + φ (3) φ ( n − + · · · + φ ( n +1) / φ ( n +1) / )++ d Vdφ ( ¨ φ φ ( n − + φ (3) ¨ φφ ( n − + · · · + φ ( n +1) / φ ( n +1) / φ ( n +1) / ) + . . . . (A3)Each term d i Vdφ i in Eq. (A3) is multiplied by i − φ . Suchderivatives are of order n + 5 − i or lower, thus these terms are non-vanishing only if n + 5 − i >
1. So, the highest-order derivative d ¯ ı Vdφ ¯ ı that appears in (A3) is the onesatisfying n + 5 − ı = 3 for even n and n + 5 − ı = 2 for odd n . For example, thesixth-order time derivative dVdφ (6) ( n = 5) is expanded in terms of d Vdφ , d Vdφ and d Vdφ as: dVdφ (6) = d Vdφ φ (6) + d Vdφ ( ¨ φ φ (4) + φ (3) φ (3) ) + d Vdφ ¨ φ . (A4)We expand time derivatives of ¨ φ in Eq.(A3) using Eq.s (A1). We find φ ( n +1) = d Vdφ φ ( n − + d Vdφ ( φ ( n − ¨ φ + φ ( n − φ (3) + . . . ) + n (cid:88) i =2 B i φ ( i ) ( t + a ) n − i (A5)As a result, we can express dVdφ ( n +1) ( t tp ) in terms of derivatives of the potential with respectto the scalar field, ¨ φ ( t tp ) = dV ( φ tp ) dφ and t tp only. We order such terms according to the orderof the derivative of the potential in the scalar field. In particular, we have from Eq.(A3)that the highest-order derivative (the ¯ ı -th term) is multiplied only by time derivatives of thescalar field of order 2 or 3 and thus it contributes as d ¯ ı Vdφ ¯ ı ¨ φ ¯ ı − ( t tp + a ) even n , d ¯ i Vdφ ¯ ı ¨ φ ¯ ı − odd n (A6)20o f n .The second-highest derivative ¯ ı − , , ,
5. Using Eq. (A5), derivatives of order 4 and 5 may be expressed in terms of lowerderivatives. As can be seen from Eq.(A5), this results in an additional d Vdφ contribution(we omit numerical coefficients for simplicity) d ¯ ı − Vdφ ¯ ı − ¨ φ ¯ ı − ( t tp + a ) (cid:18) A d Vdφ ( t tp + a ) (cid:19) even n , (A7) d ¯ ı − Vdφ ¯ ı − ¨ φ ¯ ı − ( t tp + a ) (cid:18) A d Vdφ ( t tp + a ) (cid:19) odd n. The third-highest derivative ¯ ı − , , , , ,
7. Using Eq. (A5), to express derivatives of order 4 , , , d ¯ ı − Vdφ ¯ ı − ¨ φ ¯ ı − ( t tp + a ) (cid:32) A d Vdφ ( t tp + a ) + A (cid:18) d Vdφ ( t tp + a ) (cid:19) + A d Vdφ t ¨ φ (cid:33) even n ,(A8) d ¯ ı − Vdφ ¯ ı − ¨ φ ¯ ı − ( t tp + a ) (cid:32) A d Vdφ ( t tp + a ) + A (cid:18) d Vdφ ( t tp + a ) (cid:19) + A d Vdφ ( t tp + a ) ¨ φ (cid:33) odd n. In general, the ¯ ı − i th term has contributions from terms in Eq.(A3) that are multiplied witha time derivative of the scalar field of order n − i + 3 or higher. In this way, the dependenceof the ¯ ı − i th term on ¨ φ and d j Vdφ j can be fully determined. The dependence on t tp + a canbe fixed by dimensional consistency. In particular, in each ¯ ı − i -th term, these contributionsappear always in the combination d j Vdφ j ¨ φ j − ( t tp + a ) j − with j ≥ φ ( t tp ) = Ct ktp with k < −
4. If all derivatives of the potential in the scalar field arefinite for φ →
0, we have that d j Vdφ j ¨ φ j − ( t tp + a ) j − (cid:28) j > t tp . (A9)Thus, to a good approximation, if also d Vdφ t tp (cid:28)
1, we have dVdφ ( n +1) = i =¯ ı − (cid:88) i =0 ˜ A i dV ¯ ı − i dφ ¯ ı − i ¨ φ ¯ ı − i − ( t + a ) i +1 even n , (A10) dVdφ ( n +1) = i =¯ ı − (cid:88) i =0 ˜ A i dV ¯ ı − i dφ ¯ ı − i ¨ φ ¯ ı − i − ( t + a ) i odd n ,21nd the sum in Eq.(14) is negligible with respect to dV ( φ tp ) dφ for large t tp if (A9) holds.To clarify further our discussion, we report the expression of f n in terms of ¨ φ and d i Vdφ i for 1 ≤ n ≤ n = 1 dVdφ (2) = d Vdφ ¨ φn = 2 dVdφ (3) = − d Vdφ ¨ φt tp + an = 3 dVdφ (4) = 3 d Vdφ ¨ φ + d Vdφ ¨ φ ( t tp + a ) (cid:18)
15 + d Vdφ ( t tp + a ) (cid:19) n = 4 dVdφ (5) = − d Vdφ ¨ φ t tp + a − d Vdφ ¨ φ ( t tp + a ) (cid:18)
15 + d Vdφ ( t tp + a ) (cid:19) n = 5 dVdφ (6) = 15 d Vdφ ¨ φ + 15 d Vdφ ¨ φ ( t tp + a ) (cid:18)
21 + 65 d Vdφ ( t tp + a ) (cid:19) ++ 18 d Vdφ ¨ φ ( t tp + a ) (cid:32)
35 + 156 d Vdφ ( t tp + a ) + 118 (cid:18) d Vdφ (cid:19) ( t tp + a ) (cid:33) n = 6 dVdφ (7) = − d Vdφ ¨ φ t tp + a − d Vdφ ¨ φ ( t tp + a ) (cid:18)
11 + 67 d Vdφ ( t tp + a ) (cid:19) + − d Vdφ ¨ φ ( t tp + a ) (cid:32)
56 + 256 d Vdφ ( t tp + a ) + 110 (cid:18) d Vdφ (cid:19) ( t tp + a ) (cid:33) n = 7 dVdφ (8) = 5 d Vdφ ¨ φ + 5 d Vdφ ¨ φ ( t tp + a ) (cid:18)
54 + 45 d Vdφ ( t tp + a ) (cid:19) + 21 d Vdφ ¨ φ ( t tp + a ) ×× (cid:32) d Vdφ ( t tp + a ) + 277 (cid:18) d Vdφ (cid:19) ( t tp + a ) + 4 d Vdφ ( t tp + a ) ¨ φ (cid:33) + 3 d Vdφ ×× ¨ φ ( t tp + a ) (cid:32) d Vdφ ( t tp + a ) + 30 (cid:18) d Vdφ (cid:19) ( t tp + a ) + 13 (cid:18) d Vdφ (cid:19) ( t tp + a ) (cid:33)
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