Thin-Wall Approximation in Vacuum Decay: a Lemma

PHYSICAL REVIEW D 97, 105002 (2018)

Thin-wall approximation in vacuum decay: A lemma

Adam R. Brown* Physics Department, Stanford University, Stanford, California 94305, USA

(Received 16 February 2018; published 1 May 2018)

The “thin-wall approximation” gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the approximation is uncontrolled. In this paper I show that there are actually two different thin-wall approximations and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the thin-wall limit, the two approximations converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for non- perturbative vacuum instability.

DOI: 10.1103/PhysRevD.97.105002

Z ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Metastable states may decay by quantum tunneling. In σ ≡ f dϕ 2ðV½ϕ − V½ϕ Þ; the semiclassical (small ℏ) limit, the most important min f Zϕ contribution to the decay rate ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ ≡ f dϕ 2ðV½ϕ − V½ϕ Þ: ð Þ max t 3 ϕ rate ∼ A exp½−B=ℏ; ð1Þ t σ Coleman used only min and left Eq. (2) as an uncontrolled is the tunneling exponent B. In a famous paper Coleman approximation, in the sense that we are provided neither explained how to calculate B for a scalar field [1].He with an estimate of its accuracy nor with a bound on its showed that B is given by the Euclidean action of an error. We can do better. The main result of this paper is that, instanton that interpolates from the metastable (or “false”) for any potential VðϕÞ, vacuum toward the target (or “true”) vacuum. For a given field potential VðϕÞ, the instanton can be derived by 1 ð Þ∶ B¯ ½σ ≤ B ≤ B¯ ½σ : ð Þ Lemma no gravity tw min tw max 4 numerically integrating the second-order Euclidean equations of motion. The first inequality is proved in Appendix A1; the second For those who lack either the computing power or the V − V → 0 σ − in Appendix A2.As false true so too max patience to solve the equations of motion, or who seek an σ → 0 min and the two thin-wall approximations converge. intuitive understanding of the parametric dependence of the GV ≤ 0 GV > 0 decay rate, Coleman also showed that we can often use the false false “thin-wall” approximation. While the instanton is a bubble ¯ G always B ≤ B ½σmax? that smoothly interpolates from the false vacuum to near tw B ≥ B¯ G ½σ ? the true vacuum, the thin-wall approximation treats this always tw max transition as abrupt [1]; the decay exponent is then We can partially generalize this to include gravity. As approximated by first calculated by Coleman and de Luccia [2], gravitational backreaction changes B: the decay exponent now depends 27π2 σ4 − B ∼ B¯ ≡ : ð Þ not just on the difference Vfalse Vtrue, but also on Vfalse tw 2 ðV − V Þ3 2 false true and Vtrue separately, since zero-point energy curves spacetime. The gravitational generalization of the thin-wall B¯ G ½σ Consider these two expressions for the tension of the approximation to the decay exponent, tw , is given by bubble wall σ, Eq. (B5), and the partial generalization of Lemma 1 is that

Lemma2 *[email protected] ðwhenGV ≤ 0Þ∶ B ≤ B¯ G ½σ : ð5Þ Published by the American Physical Society under the terms of false tw max the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to This inequality is proved in Appendix B. The addition of B ≥ B¯ G ½σ the author(s) and the published article’s title, journal citation, gravity means it is no longer always true that tw min : and DOI. Funded by SCOAP3. GV > 0 a de Sitter false vacuum ( false ) with a short but very

2470-0010=2018=97(10)=105002(7) 105002-1 Published by the American Physical Society ADAM R. BROWN PHYS. REV. D 97, 105002 (2018)

FIG. 1. The field starts in the false vacuum ϕf and tunnels towards the true vacuum ϕt. The definition of ϕ is such that V½ϕ¼V½ϕf . Equation (3) gives two different definitions of the tension, with σmax > σmin.

σ broad barrier has arbitrarily large min but still decays APPENDIX A: PROVING relatively promptly via a Hawking-Moss instanton [3],so NONGRAVITATIONAL RESULTS B ≪ B¯ G ½σ < B¯ G ½σ that tw min tw max ; and a Minkowski false GV ¼ 0 B 0 be greater than tw max for false ; I suspect it assume that there are no intervening minima between the may not.) false and true vacua, but otherwise leave the potential One application of these lemmas is diagnosing instability. completely general. I will now review how to calculate the GV ≤ 0 Gravity can stabilize superficially metastable false vacuum decay rate; this is all explained with great clarity in vacua [2], and there is great interest in determining which [1].In[1] it is shown that the tunneling exponent is given vacua decay and which endure (see e.g. [5]). Lemma 2 by the Euclidean action of an instanton. The instanton provides a sufficient condition for instability, which is that ϕ¯ ðτ; x⃗Þ R4 B¯ G ½σ < ∞ lives on and extremizes the Euclidean action tw max , or equivalently Z 1 sufficient condition for instability∶ S ¼ d4x ð∂ ϕÞ2 þ VðϕÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2 μ −V½ϕ − −V½ϕ Z σ < truepffiffiffiffiffiffiffiffiffi false : ð6Þ 1 max 6πG ¼ 2π2 dρρ3 ϕ_ 2 þ VðϕÞ ; ð Þ 2 A1 GV > 0 No such condition is required for false , since all de 2 2 2 2 2 2 Sitter false vacua are unstable [2]. where we have written ds ¼ dτ þ dx⃗ ¼ dρ þ ρ dΩ3 ϕ We can generalize Lemma 2 by replacing t with any and used that the instanton can be shown [6] to have O(4) ϕ ≤ ϕ < ϕ ¯ ¯ value in the range t mid , as shown in Fig. 2. Thus spherical symmetry ϕ ¼ ϕðρÞ. One boundary condition is the false vacuum is unstable if there exists any ϕ such ϕ¯ → ϕ ρ → ∞ mid that the field returns to the false vacuum f as ; that the other boundary condition is that the field immediately Z ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi after nucleation, given by σ ≡ f dϕ 2ðV½ϕ − V½ϕ Þ mid mid ϕ ∶ϕðt ¼ 0; x⃗Þ¼ϕ¯ ðτ ¼ 0; x⃗Þ; ð Þ midpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field after nucleation A2 −V½ϕ − −V½ϕ < midpffiffiffiffiffiffiffiffiffi false : ð7Þ has the same energy as the false vacuum (ΔE ¼ 0)and 6πG classically evolves toward the true vacuum. (In fact, the symmetry tells us that it will give rise to a bubble of When there are multiple fields, there are many possible approximately true vacuum that expands out at approaching routes over the barrier, and this condition applies to all of the speed of light.) The instanton gives a path through the them: if we can find any route for the σ integral such that mid space of ϕðx⃗Þs that connects the before-tunneling configu- Eq. (7) holds, the vacuum must be unstable, the field must ϕðτ ¼ −∞; x⃗Þ¼ϕ eventually decay, and spacetime is doomed [2]. ration f to the after-tunneling configuration ϕðτ ¼ 0; x⃗Þ. Indeed, the instanton is defined as the solution of minimum Euclidean action that satisfies these ACKNOWLEDGMENTS boundary conditions—in the language of [7],itisthemost 1 Thank you to Sonia Paban and Erick Weinberg probable decay path. for reading a draft of this paper, and to the organizers of TASI ’09 for their hospitality while these lemmas were 1Technically we insist that the decay path ends at its first established. intersection with the ΔE ¼ 0 surface.

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FIG. 2. If we can find any path across the barrier that satisfies Eq. (7), the vacuum must be unstable. Left: the σ integral need not be taken all the way to ϕtrue, and can instead stop anywhere in the range ϕtrue ≤ ϕmid < ϕ. Right: when there are multiple fields, there are many escape routes, and the decay path may take any of them. Z ∞ 1 τ ¼ 0 B ¼ S ðϕ¯ Þ−S ðϕ¯ Þ¼2π2 dρρ3 ϕ_ 2 þVðϕÞ−V : (With no constraint on the energy of the configu- E E f false ration, the instanton has exactly one negative mode [8]; this 0 2 negative mode is associated with changing the energy of the ðA6Þ nucleated bubble away from ΔE ¼ 0. In this paper we will freeze this negative mode by only considering paths that Since tunneling conserves energy, immediately after nucle- end with the same energy as the false vacuum ΔE ¼ 0; ation the bubble must have the same energy as the false amongst this set of paths the instanton is a minimum of the vacuum, Euclidean action [9].) Z ∞ 1 To minimize the action, the instanton must satisfy the ΔE ≡ 4π dρρ2 ϕ¯_ 2 þ Vðϕ¯ Þ − V ¼ 0: ð Þ false A7 Euler-Lagrange equation, 0 2

But the energy density is not zero everywhere. Instead, the d 1 _ 2 3 _ 2 ϕ¯ − Vðϕ¯ Þ ¼ − ϕ¯ : ð Þ energy density is positive in the “wall” of the bubble where dρ 2 ρ A3 the field traverses the barrier, and then negative inside the bubble. Fig. 3 shows a cross section of a typical bubble. 1 ¯_ 2 ¯ This equation implies that 2 ϕ − VðϕÞ monotonically “Outside” the bubble (ϕ > ϕ) the energy density is bigger ρ “ ” V “ ” ϕ < ϕ decreases as increases (in [1] this is called friction ), than false; inside the bubble ( ) we know only that V which in turn implies that the energy density is bigger than true. Consider the field value at the very center of the bubble, 1 _ 2 ϕðρ ¼ 0Þ. It follows from the conservation of energy that −V < ϕ¯ − VðϕÞ < −V ðA4Þ false 2 true ϕð0Þ ≤ ϕ; conversely it follows from Eq. (A5) that ϕð0Þ ≥ ϕ ϕ ≤ ϕð0Þ ≤ ϕ t. Indeed, the fact that t is what σ σ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi originally motivated the definitions of min and max in → 2ðV − V Þ < ϕ¯_ < 2ðV − V Þ: ð Þ false true A5 Eq. (3): the two lower limits of integration capture the full range of possible values of ϕð0Þ. The tunneling exponent is then given by the difference in For the remainder of Appendix A we will add a constant V ¼ 0 Euclidean action between the instanton and the false vacuum to the potential to set false .

FIG. 3. A cross-section through a typical bubble at the moment of nucleation. The center of the bubble may have an energy density as V ρ > ρ 1 ϕ_ 2 þ VðϕÞ ≥ VðϕÞ ≥ V low as true; but for the energy density necessarily exceeds that of the false vacuum 2 false. The total integrated energy is equal to that in the false vacuum, ΔE ¼ 0.

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ˆ FIG. 4. V½ϕ is constructed by deleting the part of VðϕÞ that lies between ϕt and ϕ. The corresponding bubble instanton has pure true ˆ vacuum inside some radius that we will call ρˆ. The bubble has energy ΔE ¼ 0.

¯ Z Z 1. Proving B ≥ Btw½σmin ρˆ ∞ 1 ≥ 4π dρρ2V þ 4πρˆ2 dρ ϕ¯_ 2 þ V½ϕ¯ true 2 Proof strategy: The instanton is a bubble configuration 0 ρˆ with zero energy, ΔE ¼ 0. In order to have ΔE ¼ 0,any 4π ≥ ρˆ3V þ 4πρˆ2σ : ð Þ bubble must be big, and if it is big enough it must 3 true min A11 have B ≥ B¯ ½σ . tw min ˆ First let’s construct a new potential Vˆ ½ϕ that decays Tunneling conserves energy, ΔE ¼ 0, so the bubble must faster than V½ϕ, be large 3σ min V½ϕ for ϕ > ϕ ρˆ ≥ : ðA12Þ Vˆ ½ϕ ≡ ð−V Þ V½ϕ − ϕ þ ϕ ϕ < ϕ : true t for ˆ ˆ ˆ ˆ πρˆΔE Since ΔE ¼ 0 implies B ¼ B − 2 , we can use Eqs. (A6) V½ϕ ϕ ϕ The region of between t and has been excised, and (A10) to prove our result with the ϕ > ϕ part glued straight onto the ϕ < ϕ part. Z t ∞ 1 This does not affect σ or V or V , but since we have B ≥ Bˆ ¼ 2π2 dρρ2ðρ − ρˆ Þ ϕ¯_ 2 þ Vˆ ½ϕ¯ ð Þ min false true 2 A13 removed part of the barrier the decay rate is faster Z 0 ρˆ 1 ≥ 2π2 dρρ2ðρ − ρˆ Þ ϕ¯_ 2 þ Vˆ ½ϕ¯ ð Þ B½VðϕÞ ≥ Bˆ ≡ B½Vˆ ðϕÞ: ð Þ 2 A14 A8 Z0 ρˆ ≥ 2π2 dρρ2ðρ − ρˆ ÞV ð Þ The shape of the bubble is plotted in Fig. 4. The bubble true A15 ρ ¼ ρˆ 0 stays uniformly in the true vacuum until , and then ρˆ4ð−V Þ proceeds towards the false vacuum with a profile given ≥ 2π2 true ð Þ 12 A16 by Eq. (A3). Bˆ 27π2 σ4 Let us calculate . First notice that by changing variables ≥ min ¼ B¯ ½σ : ð Þ ρ ϕ 2 ð−V Þ3 tw min A17 from to , true Z ∞ 1 ¯_ 2 ¯ ¯ dρ ϕ þ V½ϕ 2. Proving B ≤ Btw½σmax ρˆ 2 Z Proof strategy: The instanton is the path of minimum ϕ dϕ 1 ¼ f ϕ¯_ 2 þ V½ϕ¯ action that interpolates from the false vacuum to a ΔE ¼ 0 ¯_ 2 ϕ ϕ state on the true vacuum side of the barrier. I will explicitly pffiffiffiffiffiffi pffiffiffiffiffiffi B¯ ½σ Z 1 ¯_ 2 ¯_ construct an interpolating path with action tw max . ϕf ðϕ − 2VÞ þ 2Vϕ 2 ¼ dϕ 2 ≥ σ : ðA9Þ Consider the one-parameter family of field-profiles ¯_ min ¯ ϕ ϕ ϕR¯ ðρÞ parameterized by R and defined by

The total energy of the bubble, relative to the false 2This field-profile has a discontinuous first derivative at both vacuum, is ρ ¼ R¯ and ρ ¼ 0. This indicates the field-profile is not a Z minimum of the Euclidean action, but does not prevent the ρˆ 1 field-profile from contributing to the path integral—to contribute ˆ 2 ¯_ 2 ¯ ΔE ¼ 4π dρρ ϕ þ V½ϕ to the path integral, a field-profile only needs to be continuous, 0 2 Z not differentiable. If desired, my proof can be reformulated ∞ 1 _ 2 entirely in terms of differentiable field-profiles: smoothing the þ 4π dρρ2 ϕ¯ þ V½ϕ¯ ð Þ ρ ¼ 0 ρ ¼ R 2 A10 field at and will only make a tiny (second-order) ρˆ change to the action, so the inequality would still follow.

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ϕ ðρÞ¼ϕ ρ > R¯ ð Þ R¯ f for A18 APPENDIX B: PROVING GRAVITATIONAL RESULT pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ_ ðρÞ¼ 2ðV½ϕ ðρÞ − V Þ ρ < R:¯ ð Þ R¯ R¯ true for A19 When gravity is included, the decaying field curves spacetime. Despite this complication, the same general ϕ < ϕ ðρÞ ≤ ϕ From the definition it follows that t R¯ f. The proof strategy will apply as for the nongravitational case of Euclidean action of ϕR¯ ðρÞ is Sec. A2. Z The formalism that governs the gravitational decay of the R¯ 1 ¯ 2 3 _ 2 false vacuum was described with great clarity in [2]. With SE½R¼2π dρρ ϕR¯ þ V½ϕR¯ ðA20Þ 0 2 gravity included, it has not been proved that the dominant Z Z instanton is still Oð4Þ-symmetric; following conventional R¯ R¯ ¼ 2π2 dρρ3ð2ðV½ϕ − V ÞÞ þ 2π2 dρρ3ðV Þ wisdom, we will assume that it is. The metric may then be true true 0 0 written

ðA21Þ 2 2 2 2 ds ¼ dξ þ ρðξÞ dΩ3: ðB1Þ Z ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2π2 ≤ 2π2R¯ 3 f dϕ 2ðV½ϕ −V Þ þ V R¯ 4 Matter tells space how to curve true 4 true ϕR¯ ½0 8πG 1 ðA22Þ ρ_2 ¼ 1 þ ρ2 ϕ_ 2 − VðϕÞ ; ð Þ 3 2 B2 π2 ≤ 2π2R¯ 3σ − ð−V ÞR¯ 4 ð Þ max 2 true A23 and space tells matter how to move 27π2 σ4 d 1 _ 2 ρ_ðξÞ _ 2 ≤ max ¼ B¯ ½σ : ð Þ ϕ¯ − Vðϕ¯ Þ ¼ −3 ϕ¯ : ðB3Þ 2 ð−V Þ3 tw max A24 dξ 2 ρðξÞ true

Our family of field-profiles ϕR¯ ðρÞ must contain as “escape When the gravitational constraint Eq. (B2) is satisfied, the path,” in the language of [7]: while small values of R¯ gives action is given by Eq. (3.9) of [2] as ¯ positive energy ΔE>0, arbitrarily large values of R give Z Z 3 ρ3V − 3 ρ arbitrarily negative energies, so there must be an intervening S ¼ 4π2 dξ ρ3V − ρ ¼ 4π2 dρ 8πG : ¯ E value RE¼0 such that ΔE½ϕR¯ ðt ¼ 0; x⃗Þ ¼ 0. Therefore the 8πG ρ_ B ≤ S ½R¯ ≤ B¯ ½σ most probable escape path has E E¼0 tw max . ðB4Þ This proof would still have gone through had we V V ≡ V½ϕ 3 replaced true by mid mid in Eq. (A19), for any The thin-wall approximation to the tunneling exponent ϕ ϕ ≤ ϕ < ϕ mid in the range t mid . Thus a strengthening of [2,14] is the lemma is that 8πG 2 3 3 ð1 − ρ¯ V Þ2 − 1 27π2 σ4 B¯ G ≡ 2π2ρ¯3σ þ 3 true B ≤ mid ; where tw 16 G2V 2 ðV − V Þ3 true false mid Z 8πG 2 3 ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð1 − ρ¯ V Þ2 − 1 σ ≡ f dϕ 2ðV½ϕ − V½ϕ Þ ð Þ − 3 false ; ðB5Þ mid mid A25 16 G2V ϕmid false ϕ for every possible mid, and (for multifield potentials) for where ρ¯ is the radius of the bubble wall that maximizes every possible route over the barrier. Eq. (B5), namely (for GVf ≤ 0)

3σ ρ¯ ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi : ðB6Þ ð −V − −V Þ2 − 6πGσ2 ð −V þ −V Þ2 − 6πGσ2 t f t f

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi σ ð −V − −V Þ= 6πG ρ¯ B¯ G As approaches t f both and tw diverge; for larger values of the tension, the thin-wall approximation predicts that the false vacuum is stable. 3For numerical investigations of the reliability of the thin-wall approximation in the nongravitational case, see [10]; see also To prove the gravitational lemma [Eq. (5)] we will follow [11]. For numerical investigations in the gravitational case, see Sec. A2in constructing a family of paths and showing that [4,12]; see also [13]. one member of the family is an escape path, and every

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pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi σ < ð −V − −V Þ= 6πG ΔS ½R¯ B¯ G ½σ FIG. 5.p Left:ffiffiffiffiffiffiffiffi forpffiffiffiffiffiffiffiffiffimax pffiffiffiffiffiffiffiffiffi t f , the function E from Eq. (B10) never exceeds tw max . Right: for ¯ σmax ≥ ð −Vt − −Vf Þ= 6πG, the function ΔSE½R may (or may not) grow without bound. ΔS ≤ B¯ G ½σ member of the family has E tw max . The family of configurations contribute to the gravitational path integral. field-profiles will be parametrized by R¯, Notice also that ρ_ ≥ 1 > 0 so ρ monotonically increases 8πG with ξ and the topology is R4. The field and metric differ ϕ ðξÞ¼ϕ → ρ_2 ¼ 1 − ρ2V ρ > R¯ ð Þ ¯ R¯ f 3 false for B7 from the false vacuum only inside ρ < R, so the difference ΔS ¼ S ½ϕ ðρÞ − S ½ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in action E E R¯ E f is ϕ_ ¯ ðξÞ¼ 2ðV½ϕðξÞ − V Þ 0 1 R true Z 8πG R¯ B ρ3V − 3 ρ ρ3V − 3 ρ C → ρ_2 ¼ 1 − ρ2V ρ < R:¯ ð Þ ΔS ¼ 4π2 dρ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8πG − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifalse 8πG true for B8 E @ A 3 0 1 − 8πG ρ2V 1 − 8πG ρ2V 3 true 3 false Notice that while this family need not satisfy the equation of motion for the field, Eq. (B3), it is required to satisfy the ðB9Þ gravitational constraint, Eq. (B2), because only such 0 1 Z R¯ B ρ3ðV − V Þ ρ3V − 3 ρ ρ3V − 3 ρ C ¼ 4π2 dρ@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrue þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrue 8πG − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifalse 8πG A 0 1 − 8πG ρ2V 1 − 8πG ρ2V 1 − 8πG ρ2V 3 true 3 true 3 false

Z 3 8πG 2 3 8πG 2 3 ϕ dϕρ ðV − V Þ 3 ð1 − R¯ V Þ2 − 1 3 ð1 − R¯ V Þ2 − 1 ¼ 4π2 f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit þ 3 t − 3 f 2ðV − V Þ 16 G2V 16 G2V ϕR¯ ½0 t t f 8πG 2 3 8πG 2 3 3 ð1 − R¯ V Þ2 − 1 3 ð1 − R¯ V Þ2 − 1 ≤ 2π2R¯ 3σ þ 3 true − 3 false ð Þ max 16 G2V 16 G2V B10 true false

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ≤ B¯ G ½σ : ð Þ σ ≥ð −V − −V Þ= 6πG tw max B11 max t f the family may not contain B¯ G ½σ ¼∞ an escape path, but since tw max , Lemma 2 trivially There are two possiblepffiffiffiffiffiffiffiffi cases,pffiffiffiffiffiffiffiffiffi bothp shownffiffiffiffiffiffiffiffiffi in Fig. 5. First holds. Thus we have proved Lemma 2 for all values of σ . consider σ < ð −V − −V Þ= 6πG. In this case, max ¯max ¯t f As in the nongravitational case, this proof would still both ΔSE½R and ΔE½R become unboundedly negative ϕ ¯ have gone through had we replaced t with any value in the at large R, so the family contains an escape path; ϕ ≤ ϕ < ϕ range t mid , yielding the more powerful con- since no member of the family has an action that dition of Eq. (7). B¯ G ½σ exceeds tw max , Lemma 2 holds. By contrast, for the case

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