Occurrence of semiclassical vacuum decay

Shao-Jiang Wang∗

Tufts Institute of Cosmology, Department of Physics and Astronomy,Tufts University, 574 Boston Avenue, Medford, Massachusetts 02155, USA

Recently, a novel phenomenon is observed for vacuum decay to proceed via classically alloweddynamical evolution from initial configurations of false vacuum fluctuations. With the help of someoccasionally developed large fluctuations in time derivative of a homogeneous scalar field that isinitially sitting at the false vacuum, the flyover vacuum decay is proposed if the initial Gaussianprofile of field velocity contains a spatial region where the field velocity is large enough to flyoverthe potential barrier. In this paper, we point out that, even if the initial profile of field velocityis nowhere to be large enough to classically overcome the barrier, the semiclassical vacuum decaycould still be possible if the initial profile of field velocity extends over large enough region.

I. INTRODUCTION

Quantum tunnelling in nonrelativistic quantum me-chanics of a single particle is a distinguishing featurefrom the classical mechanics where surmounting a po-tential barrier requires large enough energy instead ofquantum mechanically penetrating the potential barrierwith lower energy. This feature persists when generalizedinto the case of relativistic field theory with multiple po-tential minimums, one of which is absolute minimum inboth classical and (perturbative) quantum sense, whilethe rest of which is still classically stable but metastableby barrier penetration in quantum field theory. Althoughthe field version of barrier penetration consists of an in-finite number of degrees of freedom, the decay rate ofmetastable minimum could be expressed in terms of theEuclidean instanton solution [1, 2] as an analog to theWKB approximation in nonrelativistic quantum mechan-ics.

Although the instanton approach borrows the wisdomof classically forbidden quantum penetration under thebarrier, the physical picture behind the vacuum decayin field theory resembles the bubble nucleation in statis-tical physics [3, 4] with quantum fluctuations replacedby thermodynamic fluctuations. Indeed, there is a coldatom analog of vacuum decay in field theory from ex-perimental [5, 6], theoretical [7], and simulational [8, 9]perspectives. This inspires the request for vacuum decayin field theory via classically allowed channel, which is re-cently realized in lattice simulation [10] and formulatedin real-time formalism [11].

The semiclassical picture of vacuum decay introducedabove could be traced back to the stochastic approach tovacuum decay in [12, 13], where the initial conditionsfor the subsequent classical evolution could be drawnfrom realizations of random Gaussian fields that inheritthe same statistics as the quantum fluctuations [10].The simplest realizations for such initial conditions ofsemiclassical vacuum decay is the flyover vacuum de-cay [14, 15], where field velocity occasionally develops

∗ schwang@cosmos.phy.tufts.edu

a Gaussian profile large enough to classically overcomethe barrier directly from a homogeneous field at falsevacuum. This flyover vacuum decay can be also realizedwith the presence of gravity for both downward and up-ward transitions [15], which has far-reaching ramification[16] on the swampland criteria [17, 18] (see also [19, 20]):the eternal inflating regions are unavoidable even in theswampy landscapes [21].

To have a successful flyover vacuum decay, it is re-quired that there should be at least a spatial region ofa size larger than the minimal expansion radius, wherethe field velocity should also be larger than a critical ve-locity to classically flyover the barrier. We point out inthis paper that, even if the initial profile of field velocityis below the critical velocity everywhere in the region ofsurvey, the semiclassical vacuum decay could still occurprovided that the initial profile of field velocity coversspatial region large enough. This is a surprising resultsince, each local degree of freedom does not have enoughenergy to classically overcome the barrier, the whole re-gion could still be converted into the true vacuum alongclassical evolutions.

The outline of this paper is as follows: in Sec.II, wereview the flyover vacuum decay in order to fix the con-vention and setup we used in the following sections; inSec. III, we scan over all possible configurations of ini-tial profile of field velocity according to their width andheight. The last section is devoted to conclusion anddiscussions.

II. FLYOVER VACUUM DECAY

To have a semiclassical vacuum decay, we first choosea test potential as

V (φ) = φ2 − 2(1 + 2ω)φ3 + (1 + 3ω)φ4, (1)

where the difference between the false vacuum V (φ+) ≡V+ = 0 at φ+ ≡ 0 and the true vacuum V (φ−) ≡ V− =−ω at φ− ≡ 1 is ε ≡ V+−V− = ω, while the height of po-tential barrier Vb ≡ Vm−V+ is the difference between thefalse vacuum V+ and the top of potential barrier Vm =116 (1+4ω)/(1+3ω)3 at φm = 1/(2+6ω). As shown in the

arX

iv:1

909.

1119

6v2

[gr

-qc]

26

Nov

201

9

2

ω=0.01

ω=0.10

ω=0.30

-0.5 0.0 0.5 1.0 1.5

-0.2

0.0

0.2

0.4

0.6

ϕ

V(ϕ)

V(ϕ)=ϕ2-2(1+2ω)ϕ3+(1+3ω)ϕ4 A/A0=2/1 , R/R0=1/2

A/A0=1/1 , R/R0=1/1

A/A0=1/2 , R/R0=2/1

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

r / R0

ϕ (t=0,r)/ϕ c

ϕ(t=0,r)=A e-

r2

2 R2

A0= eϕc , R0= 2 σ

V+-V-

ϕc= 2 (Vm - V+)

σ=∫ϕ+

ϕ- 2 (V - V-) dϕ

ϕ ( t , r )

-0.5

0

0.5

1.0

1.5

ϕ ( t , r )

0

0.25

0.50

0.75

1.00

FIG. 1. Semiclassical picture of vacuum decay from an illustrative potential (upper left) with an initially vanished profile forthe field value and Gaussian profile for the field velocity (upper right), whose classical evolution gives rise to the convertedregions in true vacuum for both cases of thin wall (bottom left) and thick wall (bottom right).

upper left panel of Fig. 1, the thin-wall limit Vb � ε cor-responds to ω � 0.05, while the thick-wall limit Vb � εcorresponds to ω � 0.05. We will take ω = 0.01, 0.1, 0.3as illustrative examples for the cases of thin, intermedi-ate, and thick walls. The wall thickness is conventionallyestimated as δ ≡ φm/

√Vb = 2

√(1 + 3ω)/(1 + 4ω) to

minimize the combination [(φm/δ)2 + Vb]δ.

The initial condition for flyover vacuum decay is as-sumed as

φ(t = 0, r) = 0, φ̇(t = 0, r) = A exp

(− r2

2R2

), (2)

where the scalar field is initially sitting at the false vac-uum homogeneously, and the field velocity (time deriva-tive) occasionally develops from vacuum fluctuations aGaussian profile with amplitude A >

√2Vb large enough

to flyover the barrier directly due to 12 φ̇(t = 0, r)2 > Vb.

For a bubble to expand, the width of the Gaussian pro-file R > R0 is required to be larger than the minimal

expansion radius R0 = 2σ/ε with wall tension given by

σ =

∫ φ−

φ+

√2(V (φ)− V−)dφ. (3)

To uniformly specify the Gaussian profile of the ini-tial field velocity, one could normalize it as shown in theupper right panel of Fig. 1 by

φ̇(t = 0, r) =

(A

A0

)A0 exp

[− (r/R0)2

2(R/R0)2

], (4)

where A0 =√

2eVb is defined in such a way that thecritical velocity φ̇c ≡

√2Vb is realized exactly at r = R for

A = A0, as shown with the orange line in the upper rightpanel of Fig. 1. Therefore, for a general A 6= A0, onecould define a critical radius rc, if at all, by the maximalr within which the initial field velocity is larger than thecritical velocity, namely, φ̇(t = 0, r ≤ rc) ≥ φ̇c. It is easyto find that rc is given by

rc = R0

√1 + 2 ln

A

A0. (5)

3

Hence, the initial Gaussian profile for field velocity couldbe nowhere larger than the critical velocity if A/A0 <1/√e, as shown with the green line in the upper right

panel of Fig. 1. The blue line indicates a sub-R0 regionwith supercritical velocity.

By numerically evolving the initial condition (2) withA = A0 and R = R0 via the classical equation of motion(EOM)

φ̈ = φ′′ +2

rφ′ − dV

dφ(6)

with boundary condition φ′(t, r = 0) = 0, one obtains astack of time slices of the field value as shown in the bot-tom panels of Fig. 1 for the cases of thin (left) and thick(right) walls. For the thin-wall case, the bubble appearsaround t ≈ 0.05R0 and then starts to expand aroundt ≈ 0.1R0 with wall radius around R0. The wall expandsboth inward and outward as observed in [15]. The innertransition region oscillates around the two minimums,whose radius eventually shrinks to zero around t ≈ R0.This picture is fairly different from the corresponding in-stanton case. However, for the thick-wall case, the pic-ture is similar to the corresponding instanton case. Thebubble appears around t ≈ 0.6R0 and then starts to ex-pand around t ≈ R0 with velocity approaching the speedof light. Remarkably, it is found in [15] that the probabil-ity to have an initial velocity profile larger than the crit-ical velocity resembles the instanton estimation up to anumerical factor (although the numerical factor is largerin the thick-wall case than in the thin-wall case). How-ever, in the following sections, we will take the thick-wallcase as an example to present our finding, which shouldalso apply to the thin-wall case, however, with more sim-ulation cost.

III. SEMICLASSICAL VACUUM DECAY

The flyover vacuum decay reviewed above is not lim-ited to the critical case with A = A0 and R = R0 for theinitial Gaussian profile of field velocity. It is suggested in[15] that the flyover vacuum decay occurs as long as thecritical radius rc, within which the initial field velocityis supercritical, is larger than the minimal expansion ra-dius R0. We will see that this only serves as a sufficientcondition but not necessary.

To qualify the necessary condition for the aforemen-tioned realization of semiclassical vacuum decay, wepresent in the left panel of Fig. 2 the full-time evolu-tions of the total kinetic energy EK , gradient energy EG,and potential energy EV over the whole simulation region

of size L,

EK = 4π

∫ L

0

1

2φ̇2r2dr, (7)

EG = 4π

∫ L

0

1

2φ′2r2dr, (8)

EV = 4π

∫ L

0

V (φ)r2dr, (9)

where the conservative total energy ET = EK +EG+EVis also computed as a consistency check for the numeri-cal stability. It is easy to see that ET is initially equalto EK as it should be (apart from a compensation witha renormalized vacuum energy; see [11] for details), andEK is eventually converged with EG as EV becomes neg-ative (dashed part of green curve). Since the transitionregion first appears around t ≈ 0.6R0, as shown in thebottom right panel of Fig. 1, we plot a correspondingvertical dashed line, which intersects with EG at somevalue specified by a horizontal dashed line. We will de-note this critical energy threshold as Ec, whose value willbe estimated at the end of this section. The appearanceof true vacuum bubble will be defined as the momentwhen EG eventually surpasses Ec.

A simple conjecture that could be imposed as the nec-essary condition expected for the semiclassical vacuumdecay is that the initial total kinetic energy should belarger than some critical energy threshold Ec,

E0K = 4π

∫ ∞0

1

2φ̇(t = 0, r)2r2dr =

π32

2A2R3 ≥ Ec, (10)

even if the kinetic energy density for each local degreeof freedom is insufficient to classically flyover the barrier.The key factor is the nonlinear evolution of EOM (6)that converts some of the kinetic energy into the gradientenergy to eventually form a bubble. The central task isto determine the precise form/value of Ec.

One of the approaches to determine the existence of Ecis to scan over the whole parameter space of the initialGaussian profile of field velocity parametrized by R/R0

and A/A0, as shown in the right panel of Fig. 2. A van-ishing rc is specified by the white dashed horizontal linewith A/A0 = 1/

√e, below which the whole initial pro-

file of field velocity is smaller than the critical velocity.A comparable rc to the minimal expansion radius R0 isspecified by the white dashed contour line, above which isthe originally proposed flyover vacuum decay. However,as specifically checked by numerical simulations with alldifferent combination values of R/R0 and A/A0, the blueregion is successful to implement the semiclassical vac-uum decay, which is obviously larger than the originalregion suggested by rc ≥ R0. The semiclassical vacuumdecay is possible for rc smaller than the minimal expan-sion radius R0 and still possible even when rc does notexist. The orange region does not possess a semiclassicalvacuum decay, which is separated from the blue region bya fuzzy zone. However, the fuzzy zone is manifest itself

4

EK

EG

EV

ET

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.01

0.10

1

10

100

1000

t / R0

ω = 0.3 , A / A0 = 1 / 1 , R / R0 = 1 / 1

FIG. 2. Left : the time evolutions of the integrated kinetic energy EK , gradient energy EG, potential energy EV (shown asdashed part of green curve when turning to negative) and total energy ET for the thick-wall case with ω = 0.3 and initialGaussian profile of field velocity with A = A0 and R = R0. The vertical dashed line indicates the appearance time t ≈ 0.6R0 ofthe transition region as shown in the bottom right panel of Fig. 1. The vertical dashed line intersects with EG at some criticalvalue Ec specified by the horizontal dashed line. Right : the parameter space supported by the width R/R0 and height A/A0 ofthe initial Gaussian profile of field velocity. The white dashed horizontal line sets the boundary of vanishing critical radius rc,below which there is nowhere in the initial velocity profile to be larger than the critical velocity required to classically overcomethe barrier. The white dashed contour line signifies a critical radius rc = R0, above which is the originally proposed flyovervacuum decay. The red dashed contour line gives rise to a relaxed condition for the occurrence of semiclassical vacuum decaywith the total kinetic energy larger than a critical energy, above which a pop-up vacuum decay could be possible even if it isbelow the white dashed horizontal line. We scan over the whole parameter space, where the orange region does not possessa semiclassical vacuum decay, while the blue region could have a transition region expanding outward. The fuzzy region ismanifest due to a delayed time for bubble to appear that is exceeding the maximal simulation time limited by the numericalinstability.

only for a practical reason that the time for the bubbleto appear, if at all, has exceeded the maximal simulationtime limited by the numerical instability. Nevertheless,since there is either an expanding bubble or not, thisfuzzy zone should be theoretically a sharp transition lineas shown with the red dashed contour line by E0

K = Ec.

A. Flyover vacuum decay

The region enclosed by rc ≥ 0 and E0K ≥ Ec could

be regarded as a general type of flyover vacuum decay,since there is a region of size rc where field velocity issupercritical, which is large enough for any local degreeof freedom inside rc to classically flyover the potentialbarrier directly. Three examples of such general type offlyover vacuum decay are given in Fig. 3, the first twoof which are the original type of flyover vacuum decay,while the third one has rc below R0. The difference lies inthe bubble radius just before the expansion at the speedof light, which is larger than R0 in the first two exam-ples while smaller than R0 in the third example. Asidefrom the time evolutions of field value in the left column,we also present in the right column the time evolutionsof the integrated kinetic energy, gradient energy, poten-tial energy, and total energy. The dashed part of green

curve denotes the absolute value of a negative potentialenergy when the bubble expands after its appearance.The kinetic energy is eventually converged with the gra-dient energy after long-time expansion. The horizontaldashed line is the same critical energy threshold Ec, asshown in Fig. 2, which always intersects EG exactly atthe time when the bubble appears. This will be a pat-tern that keeps showing up in the numerical simulations,which also reinforces the very existence of such a criticalenergy threshold Ec.

B. Pop-up vacuum decay

For the region enclosed by E0K ≥ Ec but below rc = 0,

we also find the appearance of bubble with subsequentexpansion, even if the initial profile of field velocity iseverywhere insufficient to classically overcome the po-tential barrier directly. We also present three examplesof such semiclassical vacuum decay in Fig. 4, where thetime evolutions of field value (left column) and the in-tegrated energies (right column) are presented with thesame convention as in the previous sections. In all threeexamples, compared to the flyover cases, the appearanceof bubble is significantly delayed, before which the totalkinetic energy and potential energy are oscillating with

5

ϕ ( t , r )

0

0.25

0.50

0.75

1.00

1.25

EK

EG

EV

ET

0.0 0.5 1.0 1.5 2.0 2.5

0.01

0.10

1

10

100

1000

104

t / R0

ω = 0.3 , A / A0 = 3 / 2 , R / R0 = 1 / 1

ϕ ( t , r )

0

0.25

0.50

0.75

1.00

1.25

EK

EG

EV

ET

0.0 0.5 1.0 1.5 2.0 2.5

0.01

0.10

1

10

100

1000

104

t / R0

ω = 0.3 , A / A0 = 1 / 1 , R / R0 = 3 / 2

ϕ ( t , r )

0

0.2

0.4

0.6

0.8

1.0

EK

EG

EV

ET

0.0 0.5 1.0 1.5 2.0 2.5

0.01

0.10

1

10

100

1000

t / R0

ω = 0.3 , A / A0 = 2 / 1 , R / R0 = 1 / 2

FIG. 3. Three illustrative examples of general flyover vacuum decay with rc ≥ 0 and E0K ≥ Ec, where the time evolutions of

field value (left column) and integrated energies (right column) are presented. The first two examples have rc > R0, while thethird one has rc < R0 by our parameter choices specified in the figures. In the right column, the dashed part of green curve isthe opposite value of potential energy, the vertical dashed line is the bubble appearance time, and the horizontal dashed lineis the critical energy threshold Ec.

6

ϕ ( t , r )

-0.25

0

0.25

0.50

0.75

1.00

1.25

EK

EG

EV

ET

0 1 2 3 4 5 6 70.001

0.010

0.100

1

10

100

1000

t / R0

ω = 0.3 , A / A0 = 3 / 5 , R / R0 = 3 / 2

ϕ ( t , r )

-0.25

0

0.25

0.50

0.75

1.00

EK

EG

EV

ET

0 2 4 6 8

0.001

0.010

0.100

1

10

100

1000

t / R0

ω = 0.3 , A / A0 = 2 / 5 , R / R0 = 2 / 1

ϕ ( t , r )

0

0.25

0.50

0.75

1.00

1.25

EK

EG

EV

ET

0 1 2 3 4 5 60.001

0.010

0.100

1

10

100

1000

t / R0

ω = 0.3 , A / A0 = 1 / 2 , R / R0 = 2 / 1

FIG. 4. Three illustrative examples of pop-up vacuum decay with E0K ≥ Ec but without rc (namely, the initial profile of field

velocity is nowhere larger than the critical velocity), where the time evolutions of field value (left column) and the integratedenergies (right column) are presented. In the right column, the dashed part of green curve is the opposite value of potentialenergy, the vertical dashed line is the bubble appearance time, and the horizontal dashed line is the critical energy thresholdEc.

7

|ϕ|( t , r )

0

0.5

1.0

1.5

|ϕ'|( t , r )

0

0.5

1.0

1.5

FIG. 5. The time evolutions of kinetic term |φ̇| (left) and gradient term |φ′| (right) of the pop-up vacuum decay shown in thethird example of Fig. 4.

decreasing amplitudes that are likely converted into thegradient energy. Finally, the bubble appears when thegradient energy exceeds the same critical energy thresh-old Ec presumed in the previous sections. Soon after theappearance of bubble, the total potential energy quicklyturns to zero before becoming negative as bubble expand-ing. An intriguing case is the first example, where thebubble appears and disappears around t ≈ 2R0 and thenreappears around t ≈ 4R0 with successful expansion fol-lowed. It seems that the expanding bubble finally ap-pears after a (or several) short pop-up(s) of collapsingbubble(s), hence the name of pop-up vacuum decay.

To see how such a bubble could appear even if the ini-tial kinetic energy density is everywhere insufficient toclassically overcome the barrier, the time evolutions ofthe to-be-integrated kinetic term |φ̇| and gradient term|φ′| at each time slice are presented in Fig. 5 for the lastexample of Fig. 4. Near the emergence of bubble wall,the kinetic term |φ̇| looks like some dark blue valleys sur-rounded by the white ridges, while the gradient term |φ′|is suddenly enlarged as a white spark around t ≈ 3.2R0

after gradually gathering itself around the would-be wallregion through the decreasing kinetic energy. Therefore,the initial total kinetic energy E0

K = 12π

3/2A2R3 shouldbe at least larger than the total gradient energy at theonset of wall appearance as the critical energy thresholdEc,

EG = 4π

∫1

2φ′2r2dr ∼

(φmδ

)2

R20δ ∼ VbR2

0δ ≡ Ec.

(11)

The criterion (10) now gives rise to a form of(A

A0

)2(R

R0

)3

&δ

R0(12)

up to some universal numerical factor. In particular, forthe thick-wall case, we have tested so far with compara-ble thickness to bubble radius δ ∼ R0, the total gradi-ent energy as the critical energy threshold Ec could berewritten as

EG ∼ VbR20δ ∼

4π

3VbR

30 ≡ Ec, (13)

which is the total potential energy with converted regionof minimal expansion size at the top of potential barrier.The criterion (12) is now replaced by(

A

A0

)2(R

R0

)3

&4

3e√π, (14)

which is shown as the red dashed curve in the right panelof Fig. 2. It is worth noting that, our criteria for the ap-pearance of an expanding bubble are expressed in termsof a combination ∼ A2R3, while the flyover transitionrate found in [15] goes as ∼ exp(−A2R3). This sug-gests that transitions with different A and R but thesame value of A2R3 may have nearly the same probabil-ity shown in the second panel of Fig.2.

C. No semiclassical vacuum decay

If the initial total kinetic energy is smaller than thecritical energy threshold specified above, the field value(left column) will simply oscillate around the false vac-uum minimum with nearly constant (first example), de-creasing (second example), or damping (third example)amplitudes for the time evolutions of total gradient en-ergy (right column), as shown with three illustrative ex-amples in Fig. 6. This is the case shown with orange

8

ϕ ( t , r )

-0.050

-0.025

0

0.025

0.050

0.075

EK

EG

EV

ET

0 2 4 6 810-5

10-4

0.001

0.010

0.100

1

10

t / R0

ω = 0.3 , A / A0 = 1 / 4 , R / R0 = 3 / 2

ϕ ( t , r )

-0.2

-0.1

0

0.1

0.2

0.3

EK

EG

EV

ET

0 1 2 3 4 50.001

0.010

0.100

1

10

t / R0

ω = 0.3 , A / A0 = 1 / 1 , R / R0 = 1 / 2

ϕ ( t , r )

-0.04

-0.02

0

0.02

0.04

EK

EG

EV

ET

0 1 2 3 4 510-5

10-4

0.001

0.010

0.100

1

10

t / R0

ω = 0.3 , A / A0 = 1 / 4 , R / R0 = 1 / 2

FIG. 6. Three illustrative examples of parameter choices without the appearance of expanding bubble. The time evolutions offield value and the integrated energies are presented in the left and right columns, respectively.

shading in the right panel of Fig. 2. As an aside, thegeneral feature of damping/decreasing amplitude for thetime evolution of total gradient energy indicates that thenecessary condition for such a semiclassical vacuum de-cay could also be found by identifying the growing modes

of total gradient energy.

9

IV. CONCLUSION AND DISCUSSIONS

The flyover vacuum decay is recently proposed as aspecific channel of semiclassical vacuum decay, which isrealized with an initial Gaussian profile in time deriva-tive of a homogeneous field that is initially sitting at falsevacuum. The flyover vacuum decay converts the falsevacuum region into the true one but purely along theclassical evolutions of the initial condition that is largeenough to classically flyover the potential barrier. Wegeneralize the flyover vacuum decay channel by includ-ing those initial Gaussian profiles of field velocity thatare nowhere larger than the critical velocity to overcomethe barrier directly for each local degree of freedom. Wehave shown by scanning over the whole parameter spaceof initial condition with numerical simulations that thedubbed pop-up vacuum decay could be possible as longas the initial kinetic energy of the total simulation regionis larger than some critical energy threshold, of whichan approximated form is estimated and evaluated for thethick-wall case. The thin-wall case should also have thesame form but with different numerical prefactor.

Several improvements could be made as follows: first, aprecise determination of such a critical energy thresholdshould be carried out analytically, possibly by studyingthe growing modes of the total gradient energy. Second,more general initial configurations with fluctuations inboth field value and field velocity should be explored withcorresponding necessary condition to be determined forsuch a semiclassical vacuum decay. Third, similar to theflyover vacuum decay, the gravity effect should also beincluded for the general semiclassical vacuum decay.

ACKNOWLEDGMENTS

The numerical simulation is implemented in this paperby Mathematica 11.3.0. I would like to thank HelingDeng for the help of numerical simulation carried outoriginally by C++. I am grateful to Jose J. Blanco-Pillado, Laurence Ford, Mark Hertzberg, Matthew John-son, Ken Olum, Alexander Vilenkin and Masaki Yamadafor the stimulating discussions during many of the lunchtime. S. -J. W. is supported by the postdoctoral schol-arship of Tufts University from NSF.

[1] S. R. Coleman, “The Fate of the False Vacuum. 1.Semiclassical Theory,” Phys. Rev. D15 (1977)2929–2936. [Erratum: Phys. Rev.D16,1248(1977)].

[2] C. G. Callan, Jr. and S. R. Coleman, “The Fate of theFalse Vacuum. 2. First Quantum Corrections,” Phys.Rev. D16 (1977) 1762–1768.

[3] J. S. Langer, “Theory of the condensation point,”Annals Phys. 41 (1967) 108–157. [AnnalsPhys.281,941(2000)].

[4] J. S. Langer, “Statistical theory of the decay ofmetastable states,” Annals Phys. 54 (1969) 258–275.

[5] O. Fialko, B. Opanchuk, A. I. Sidorov, P. D.Drummond, and J. Brand, “Fate of the false vacuum:Towards realization with ultra-cold atoms,” EPL(Europhysics Letters) 110 no. 5, (Jun, 2015) 56001.https:

//doi.org/10.1209%2F0295-5075%2F110%2F56001.[6] O. Fialko, B. Opanchuk, A. I. Sidorov, P. D.

Drummond, and J. Brand, “The universe on a table top:engineering quantum decay of a relativistic scalar fieldfrom a metastable vacuum,” J. Phys. B50 no. 2, (2017)024003, arXiv:1607.01460 [cond-mat.quant-gas].

[7] J. Braden, M. C. Johnson, H. V. Peiris, andS. Weinfurtner, “Towards the cold atom analog falsevacuum,” JHEP 07 (2018) 014, arXiv:1712.02356[hep-th].

[8] T. P. Billam, R. Gregory, F. Michel, and I. G. Moss,“Simulating seeded vacuum decay in a cold atomsystem,” arXiv:1811.09169 [hep-th].

[9] J. Braden, M. C. Johnson, H. V. Peiris, A. Pontzen, andS. Weinfurtner, “Nonlinear Dynamics of the Cold AtomAnalog False Vacuum,” arXiv:1904.07873 [hep-th].

[10] J. Braden, M. C. Johnson, H. V. Peiris, A. Pontzen,and S. Weinfurtner, “New Semiclassical Picture of

Vacuum Decay,” Phys. Rev. Lett. 123 no. 3, (2019)031601, arXiv:1806.06069 [hep-th].

[11] M. P. Hertzberg and M. Yamada, “Vacuum Decay inReal Time and Imaginary Time Formalisms,” Phys.Rev. D100 no. 1, (2019) 016011, arXiv:1904.08565[hep-th].

[12] J. R. Ellis, A. D. Linde, and M. Sher, “Vacuum stability,wormholes, cosmic rays and the cosmological bounds onm(t) and m(H),” Phys. Lett. B252 (1990) 203–211.

[13] A. D. Linde, “Hard art of the universe creation(stochastic approach to tunneling and baby universeformation),” Nucl. Phys. B372 (1992) 421–442,arXiv:hep-th/9110037 [hep-th].

[14] H. Huang and L. H. Ford, “work in progress (2019),seminar given by H. Huang at Tufts University,,”.

[15] J. J. Blanco-Pillado, H. Deng, and A. Vilenkin,“Flyover vacuum decay,” arXiv:1906.09657 [hep-th].

[16] A. R. Brown and A. Dahlen, “Populating the WholeLandscape,” Phys. Rev. Lett. 107 (2011) 171301,arXiv:1108.0119 [hep-th].

[17] G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, “DeSitter Space and the Swampland,” arXiv:1806.08362

[hep-th].[18] H. Ooguri, E. Palti, G. Shiu, and C. Vafa, “Distance

and de Sitter Conjectures on the Swampland,”arXiv:1810.05506 [hep-th].

[19] U. H. Danielsson and T. Van Riet, “What if stringtheory has no de Sitter vacua?,” Int. J. Mod. Phys. D27no. 12, (2018) 1830007, arXiv:1804.01120 [hep-th].

[20] S. K. Garg and C. Krishnan, “Bounds on Slow Roll andthe de Sitter Swampland,” arXiv:1807.05193

[hep-th].[21] J. J. Blanco-Pillado, H. Deng, and A. Vilenkin,

“Eternal Inflation in Swampy Landscapes,”

10

arXiv:1909.00068 [gr-qc].