Table Of ContentNon-minimally coupled gravity and vacuum stability
Olga Czerwin´ska∗
InstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,Poland
E-mail: [email protected]
7
1
Zygmunt Lalak
0
2 InstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,Poland
n E-mail: [email protected]
a
J Marek Lewicki
0 InstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,Poland
2
ARCCentreofExcellenceforParticlePhysicsattheTerascale(CoEPP)&CSSM,
DepartmentofPhysics,UniversityofAdelaide,SouthAustralia5005,Australia
]
h E-mail: [email protected]
p
-
p Paweł Olszewski
e InstituteofTheoreticalPhysics,FacultyofPhysics,UniversityofWarsaw,Poland
h
E-mail: [email protected]
[
1
v We investigate the properties of vacuum decay taking into account a non-minimal coupling to
1 gravity.Weextendthestandardthin-wallsolutiontoincludethenon-minimalcouplingandverify
3
7 its validity by comparison with a full numerical study. We also investigate the implications of
5 a large cosmological constant whose influence on the geometry boosts the tunneling rate. Our
0
. analysis shows that the influence of the non-minimal coupling differs significantly between the
1
casesofMinkowskianddeSitterbackgrounds.
0
7 ADP-17-2/T1008
1
:
v
i
X
r
a
CorfuSummerInstitute2016"SchoolandWorkshopsonElementaryParticlePhysicsandGravity"
31August-23September,2016
Corfu,Greece
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
1. Introduction
Recent discovery of the Higgs boson initiated the thorough investigation of vacuum stability
in the Standard Model. Observed data indicate that the electroweak minimum in the SM effec-
tivepotentialismetastable,sothepotentialcanhaveasecondminimumtowhichtheelectroweak
vacuummaydecay[1,2]. Numerousmodificationsofthepotentialandtheirfeatureshavebeenin-
vestigated: additionalhigher-dimensionalinteractions[3],approximatescale-invariance[4],gauge
dependence[5,6],relationtoprimordialblackholes[7],tonameafew.
Alsothestudyofthegravitationalimpactonthemetastabilityhasbeenstudied,mostlyfollow-
ing the classic work of Coleman and De Luccia [8]. Using thin-wall approximation they showed
that the decay of Minkowski vacuum into anti-deSitter one in gravitational background is heavily
suppressed. Recent works have been devoted to describe the influence of gravity in more detail,
mostlyinsomewell-definedcosmologicalcontext[9–11].
Central point of this note is the question of the influence of non-minimal coupling between
the scalar field and scalar curvature, ξ, on vacuum decay. This coupling is required for the renor-
malizabilityofthescalarfieldincurvedspacetimeanditisacrucialfeatureoftheHiggsinflation
modelthatisstillallowedbytheexperimentaldata[12,13]. Sofaritsimpactonthevacuumdecay
has been investigated in case of the inflationary background [14–16] and in the Standard Model
case [17,18]. In this paper we explore a theory with a single scalar field and a renormalizable
potential which may seem simplified but it is dictated by the need to accommodate in a readable
mannerawidespectrumofparametersallofwhicharecontrollingtheinfluenceofgravity. Tunnel-
ingbothcloseandfarfromthethin-wallregimeisdiscussedandweconsiderdifferentgeometries
ofthetrueandfalsevacuum. Ourdiscussionaimstobeuniversalandapplicableintheplethoraof
contextsevokingthequantumtunnelinginthepresenceofgravitylikeinflation[19,20]orbaryo-
genesis[21]tonameafew.
The outline of the paper is as follows. In Section 2 we present our Lagrangian describing
scalarfieldnon-minimallycoupledtogravity,analysethenon-minimalcoupling’sinfluenceonthe
thin-wallapproximationandalsonumericallycalculatetheactionforthebouncesolutiontoverify
ouranalyticalapproximation. Wealsoinvestigatetheinfluenceofalargecosmologicalconstanton
thedecayofthefalsevacuumandtheconnectionbetweentunnelingviabubblenucleationandthe
Hawking-Moss solution. In Section 3 we present our conclusions and summarize the paper. For
moredetailssee[22].
2. Model
Wewanttodiscusstheimpactofthegravityonthevacuumdecayprocessconsideringavery
generaltoymodeldescribingasingleneutralscalarfield. ItsLagrangianissuplementedbythetwo
gravitational terms - usual Einstein-Hilbert action and non-minimal coupling of the scalar field to
theRicciscalar:
L = 1(∂φ)2−V+1R(cid:0)1−ξκφ2(cid:1) (2.1)
2 2κ
withtheverysimplebutinformativepotential
1 1 1
V =− a2(3b−1)φ2+ a(b−1)φ3+ φ4+a4c. (2.2)
4 2 4
1
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
It is intentionally chosen this way as it possesses all features we require to discuss tunneling, the
mostimportantis-ithastwominima: atφ =0andφ =a,seeFigure1. Weassumethatthefieldis
initiallyinahomogeneousconfigurationinthemoreshallowminimum(orfalsevacuum)atφ =0
whichwewilldenotebyφ andittunnellstothedeeperminimum(ortruevacuum)atφ =a.
f t
The only dimensionful parameter is a and the usual choice a = 1 means that true vacuum
is positioned at the Planck scale as we use natural units (M =1). Decreasing a corresponds to
p
attenuatingthegravitationaleffectsandbringingourresultsclosertoflatspacetimecase. Constant
cisresponsibleforthecharacterofourinitialfalsevacuum(itcorrespondstothevacuumenergy)
and we focus on a de Sitter false vacuum with c > 0 and Minkowski false vacuum with c = 0.
Parameter b controls the degeneration of the vacua. Figure 1 depicts our potential in the range of
parametersusedthroughoutthepaper,wefixa=1andc=0.
0.01
0.00
-0.01
V -0.02
b=1/25
b=1/20
-0.03
b=1/15
b=1/10
-0.04
b=1/5
-0.05
0.0 0.2 0.4 0.6 0.8 1.0
ϕ
Figure1: Toymodelpotentialfordifferentvaluesofbparameterandvanishingvacuumenergyc=0for
a=1.
2.1 Tunneling
Our discussion is based on the standard formalism of Coleman and De Luccia (CDL) [8],
whichassumesthatvacuumdecayproceedsthroughnucleationoftruevacuumbubbleswithinour
falsevacuum. Thedecayprobabilityisgivenby[23,24]
Γ=Ae−S, (2.3)
whereAisaprefactorcomingfromquantumcorrectionsthatisnotdiscussedinthepresentpaper
andSisingeneralthedifferenceoftheactionintegralbetweenfinalandinitialfieldconfigurations.
For the result including all the gravitational effects these are respectively the Coleman-DeLuccia
bounceφ andφ (andwedenoteSbyS )
CDL f CDL
S =S[φ ]−S[φ ]. (2.4)
CDL CDL f
Bubble (bounce) is a spherically symmetric scalar field configuration, φ = φ(τ), fulfilling
euclideanEOMwithappropriateboundaryconditionsimposedsothattheactiondifferenceisfinite
2
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
and coordinate singularities are prevented. Action of such a bubble in presence of non-minimal
couplingtogravityreads
S =2π2(cid:82) dτρ3(cid:0)1φ˙2+V−1R(cid:0)1−κξφ2(cid:1)(cid:1) (2.5)
E 2 2κ
=2π2(cid:82) dτ(cid:2)ρ3(cid:0)1φ˙2+V(cid:1)− 3(cid:0)1−ξκφ2(cid:1)ρ(cid:0)ρ˙2+1(cid:1)+6ξφ˙φρ˙ρ2(cid:3)+ 6π (cid:0)1−κξφ2(cid:1)ρ2ρ˙(cid:12)(cid:12)τmax
2 κ κ 0
withmetricds2 =dτ2+ρ(τ)2(dΩ)2 correspondingtotheFRW metricwiththecurvatureparam-
eterk=+1. HeredΩdenotesaninfinitesimalelementofthe3Dsphere,ρ(τ)istheradiusofthat
(cid:16) (cid:17)
sphere,φ˙ = dφ andR=−6 ρ¨ +ρ˙2 − 1 . IncaseofdSfalsevacuumtheboundarytermalways
dτ ρ ρ2 ρ2
vanishes.
Fromtheaboveaction(2.5)weobtaintheequationofmotionofthescalarfield,
ρ˙ ∂V
φ¨+3 φ˙−ξφR= , (2.6)
ρ ∂φ
thefirst
κρ2 (cid:18)1 ρ˙(cid:19)
ρ˙2=1+ φ˙2−V+6ξφ˙φ (2.7)
3(1−κξφ2) 2 ρ
andthesecondFriedmanequation
(cid:18) (cid:18) (cid:19)(cid:19)
κρ ρ˙
ρ¨ = −φ˙2−V+3ξ φ˙2+φ¨φ+φ˙φ . (2.8)
3(1−κξφ2) ρ
UsingthefirstFriedmanequationwecanalsofurthersimplifytheaction(2.5)
SE =4π2(cid:90) dτ(cid:20)ρ3V−3ρ (cid:0)1−ξκφ2(cid:1)(cid:21)+ 6π (cid:0)1−κξφ2(cid:1)ρ2ρ˙(cid:12)(cid:12)(cid:12)τmax. (2.9)
κ κ (cid:12)
0
Scalefactorρ crosseszeroatleastonce[25]andwithoutlossofgeneralitywecanchosevalue
of τ of the first zero to be τ =0 and the other at τ . Then the appropriate boundary conditions
max
read
φ˙(0)=φ˙(τ )=0,
max
ρ(0)=0,
ρ(τ )=0 (fordSfalsevacuum),
max
ρ(τ )=ρ (cid:54)=0 (forMinkowskifalsevacuum).
max max
InourdefinitionofRsecondpowerofρ appearsinthedenominatorsothereisapossibilityof
divergence here. Therefore, it is much more convenient for numerical calculations to express the
scalarcurvatureusingtheFriedmanequationsas
(cid:18)ρ¨ρ+ρ˙2−1(cid:19) κ (cid:18) (cid:18) ρ˙(cid:19)(cid:19)
R=−6 = φ˙2+4V−6ξ φ˙2+φφ¨+3φ˙φ . (2.10)
ρ2 (1−κξφ2) ρ
This way R contains only the Hubble parameter that is already present in the scalar field’s EOM
andthushastobenumericallystable.
3
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
For gravitational background we assume a constant field configuration, which results in the
simplifiedfirstFriedmannequation(2.7)
(cid:115)
dρ κρ2V
= 1− , (2.11)
dτ 3(1−κξφ2)
whereV =V(φ)andφ isthechosenconstantfieldvalue. Equation(2.9)correspondsthento
24π2(1−κξφ2)2
S[φ ]=− f (fordS), (2.12)
f κ2Vf
S[φ ]=0 (forMinkowski). (2.13)
f
Inourpotentialfalsevacuumisalwayspositionedatφ =0sothereisnomodificationofthefalse
f
vacuumenergy. However,thesamereasoningappliestothetruevacuumenergywhichismodified
by the non-minimal coupling - energy of true vacuum can be increased beyond that of the false
vacuum (in the case whenV(φ)>0) making false vacuum stable. This is especially visible for
t
large vacuum energies where the true vacuum can disappear altogether as shown in Figure 2. We
alwaysneglecttunnelinginsuchcasesbecauseeventhoughthebubbleprofilecansometimesstill
becalculated,suchbubbleisnotenergeticallyfavourableandwouldnotgrowafternucleation.
0.08
0.16
0.00
b=1/25
0.06 b=1/20
0.14 b=1/15
-0.02 b=1/10
221-κξϕ)-0.04 0.04 0.12 b=1/5
V/( bb==11//2250 b=1/25
-0.06 bb==11//1150 0.02 bbb===111///211050 0.10
b=1/5 b=1/5
-0.08 0.00 0.08
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
ϕ ϕ ϕ
Figure 2: Modified potential V/(1−κξφ2)2 for different choices of the vacuum energy c and with the
non-minimalcouplingsettoξ =0.2. Valueofthecconstantwassettoc=(0;0:05;0:1)fromlefttoright.
2.2 TWapproximation
Thin-wall (TW) approximation originating from [8,23], assumes the true vacuum bubble
stretches to some ρ¯ having a constant value V inside and on the outside of the bubble our solu-
t
tionisidenticaltothefalsevacuumV. Theoverallaction
f
S =S +S[φ ]−S[φ ] (2.14)
TW wall tv fv
acquiresthenewtermS describingthewall. Oneoftheadvantagesofthisapproximationisthe
wall
factthatinsomecasesitcanprovidetheanalyticalexpressionforthebouncesolution.
Wedeveloptheoriginalsetupbyincludingthenon-minimalcouplingtogravity. ThefullEOM
readsthen
3 ∂V
φ¨+ φ˙−ξφR= , (2.15)
τ ∂φ
4
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
where according to TW approximation we neglect the term proportional to φ˙ for the wall and
ρ =ρ¯ =constsothatcurvaturecanbeapproximatedbyR≈ 6 . Integrating(2.15)onceweobtain
ρ¯2
dφ (cid:113) (cid:0) (cid:1)
=− 2(V−V)+ξR φ2−φ2 , (2.16)
dτ t t
sotheactionofthebubblewallreads
B =2π2ρ¯3(cid:82)τmax(cid:2)2(V−V)+ξR(φ2−φ2)(cid:3)dτ (2.17)
wall 0 t t
(cid:18) (cid:19)
≈2π2 ρ¯3(cid:82)φt(cid:112)2(V−V)dφ+ξρ¯(cid:82)φt √3(φ2−φt2)dφ (2.18)
φf t φf 2(V−Vt)
=2π2(cid:0)ρ¯3S +ξρ¯S (cid:1), (2.19)
0 1
whereweexpandedtothefirstorderinξ. S istheusualresultwewouldobtainneglectinggravity:
0
S =ρ (V −V)/3,whereρ isthesizeofthebounceobtainednumericallyneglectinggravity,and
0 0 f t 0
S isthelinearcorrectionduetothenon-minimalcoupling.
1
Expansion to the second order in ξ slightly increases the action but it does not improve sig-
nificantly the accuracy of our results. We prove in Section 2.4 that both approximations we use
overestimate the correct result. Interpretation of this error presumes that it comes from our as-
sumption on the shape of the bounce rather than from the expansion in the non-minimal coupling
ξ.
Gravitational part of the action corresponds again to the constant field configuration in (2.9)
withsomeupperlimitoftheintegratiom,namelytheradiusofthebubbleρ¯,whichresultsin
2(cid:0)1−ρ¯2ΛV(cid:1)3/2−1
S =2π2 , (2.20)
grav 3 Λ2V
whereΛ=κ/(1−κξφ2)andφ istheconstantfieldvalue.
Final expression for the action for TW approximation including gravitational contributions
reads
(cid:32) (cid:33)
2(1−ρ¯2ΛV)3/2−1 2(1−ρ¯2ΛV)3/2−1
S =2π2 ρ¯3S +ξρ¯S − f f + t t , (2.21)
TW 0 1 3 Λ2V 3 Λ2V
f f t t
where Λ = κ/(1−κξφ2) and Λ = 1/(1−κξφ2) are constant field values. In the flat case -
f f t t
Minkowski background withV =0, false vacuum gravity action should be replaced with the ap-
f
propriatelimitS →V →0−2ρ¯2/Λwhichgivestheoverallactionoftheform
grav
(cid:32) (cid:33)
ρ¯2 2(1−ρ¯2ΛV)3/2−1
S =2π2 ρ¯3S +ξρ¯S + + t t . (2.22)
TW 0 1 Λ 3 Λ2V
f t t
Bi-quadraticequationforthesizeofthebubbleρ¯ usedin(2.21)or(2.22)canbeobtainedby
differentiatingtheactionwithrespecttoρ¯ andexpandingittothelinearorderinξ reads
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:16) (cid:17)2 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)2
1 − 1 −3ξS S 1 + 1 +ρ¯4 9S2 Vf + Vt + Vt − Vf +81S2 + (2.23)
Λ2f Λ2t 0 1 Λ2f Λ2t 2 0 Λf Λt Λt Λf 16 0
(cid:104) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
+ρ¯2 −2 Vf + Vt −9S2 1 + 1 + 2 Vf + Vt +3ξS S Vf + Vt +9S2 =0.
Λ3f Λ3t 2 0 Λ2f Λ2t ΛfΛt Λt Λf 0 1 Λf Λt 4 0
5
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
Intheflatcaseourequationofmotionforthescalarfieldsimplifiesto
3 ∂V
φ¨+ φ˙ = (2.24)
τ ∂φ
withnewboundaryconditions
φ˙(0)=φ˙(τ )=0,
max
limφ =V .
f
τ→∞
2.3 HMsolution
Hawking-Moss(HM)instantonsdescribetheprobabilityforawholehorizonvolumetotransi-
tiontothetopofthebarrierandcontinuebyaclassicalroll-down[26],seeFigure3forcomparison
withCDLinstanton. ActionofHMinstantonisthedifferencebetweenactionofourfalsevacuum
and the energy of a homogenous solution on top of the potential barrier. Non minimal coupling
modifiestheseenergiesasdescribedin(2.12)andweget
24π2(1−κξφ2 )2 24π2(1−κξφ2)2
S =− max + f , (2.25)
HM κ2V κ2V
max f
whereφ andV correspondtothepotentialandfieldvaluesatthetopofthebarrier.
max max
V
HM rol l -cldaossical ϕ
wn
CDL tunneling
Figure3: ComparisonbetweenColeman-deLuciaandHawking-Mosssolutions.
2.4 Numericalresults
In a flat case we solve equation of motion for the scalar field (2.24) numerically using the
shootingmethodsimilarto[27]andwefindthebubblesizeρ =τ(cid:0)φ = Vt+Vf(cid:1)laterusedin(2.21).
0 2
This method of obtaining the bubble size is more accurate than the simpler flat spacetimete thin-
wall result so the initial flat spacetime error does not influence our thin-wall approximation con-
cerniggravity.
In numerical calculations containing ξ we solve the field EOM (2.6) with Ricci scalar ex-
pressedbythescalarfield(2.10)andthesecondFriedmanequation(2.8)withboundaryconditions
(2.10), approximating ρ(0) as proportional to initial τ =ε and ρ˙ =1. Our EOM is the equation
of a particle in potential −V(φ) with a time-dependent friction 3ρ˙ and we expand it around both
ρ
vacua in ε that can be arbitrarily small. We neglect higher orders of both expansions and we find
iterativelytheirleadingordersthatfulfilltheboundarycondition. Lastinitialcondition-fieldvalue
6
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
1.0
ξ=0. ξ=0.
1.0
SCDL=282.9 SCDL=2819.
0.8
ξ=0.1 0.8 ξ=0.1
0.6 SCDL=387.1 SCDL=2267.
ξ=0.2 0.6 ξ=0.2
ϕ0.4 SCDL=549.3 ϕ SCDL=1938.
0.4
0.2 0.2
0.0 0.0
0 5 10 15 20 0 5 10 15 20 25 30 35
τ τ
Figure4:CDLbubbleprofilesfortunnelingfromdSfalsevacuumc=0.05(left)andfromMinkowskifalse
vacuum(right)forseveralvaluesofthenon-minimalcouplingξ forb=1/10.
φ corresponding to CDL, can be found by a undershoot/overshoot method known from the flat
0
setup,seee.g.[3]. ResultingbubbleprofilesarepresentedinFigure4.
Now also the influence of the boundary term in the action (2.9) when the false vacuum has a
vanishingenergyisstraightforwardlyvisible. WecanseeinthetheFigure5thatρ asymptotesto
a linear function instead of crossing zero again at τ and the effect of the boundary term is not
max
negligible.
8
40
6
30
ρ4 ρ20
ξ=0. ξ=0.
2 ξ=0.1 10 ξ=0.1
ξ=0.2 ξ=0.2
0 0
0 5 10 15 20 0 5 10 15 20 25 30 35
τ τ
Figure5:BubbleradiusfortunnelingfromdSfalsevacuumc=0.05(leftpanel)andfromMinkowskifalse
vacuum(rightPanel)forseveralvaluesofthenon-minimalcouplingξ forb=1/10.
Tocalculatetheactionofoursolutionandconsequentlythetunnelingprobabilitywenumeri-
callyperformtheactionintegral(2.9),knowingtheCDLsolutionforφ(τ)andρ(τ),andputitin
(2.4)togetherwiththebackgroundaction(2.12). Figures6and7showtheresultingactionforall
four methods discussed in this paper - S is the numerically obtained result including Einstein-
CDL
Hilberttermandnon-minimalcouplingtogravity,S istheresultofitsthin-wallapproximation,
TW
S comesfromHawking-MosssolutionandS isthenumericallyobtainedflatspacetimeresult
HM flat
completelyneglectinggravity.
Figures6and7showthatbothapproximations, TWandHM,alwaysoverestimatetheaction
which includes the influence of gravity. But the validity of them is different - for relatively large
7
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
a=1,b=0.05,c=0.
107 a=1,b=0.1,c=0.
106
106 SSSTCflaWDtL 105 SSSTCflaWDtL
S 105 S
104
104
1000
1000
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
ξ ξ
Figure 6: Tunneling action for Minkowski false vacuum as a function of non-minimal coupling obtained
using four different methods for different values of b parameter. S is the numerically obtained result
CDL
fully including gravity, S is the result of our thin-wall approximation, S comes from Hawking-Moss
TW HM
solutionandS isthe,numericallyobtained,flatspacetimeresultcompletelyneglectinggravity.
flat
vacuumenergiesHMsolutiongivesactionsmallerthanthin-wallandisaverygoodapproximation
while for smaller vacuum energy thin-wall approximation becomes better and the suppression of
theactionduetogravitationaleffectslowers. However,bothapproximationsbecomelessaccurate
asthevacuumenergydecreases. Inparticular,intheMinkowskicasegravitationaleffectssuppress
vacuumdecaybyincreasingtheactionandtheHMsolutiondoesnotexist(S wouldbeinfinite).
HM
AtthesametimeTWseverelyoverestimatestheactionduetononzerocouplingξ.
Moreover, action quickly decreases as the false vacuum energy increases which can be inter-
preted as a result of a temperature effect coming from an effective temperature induced by our
compact spacetime [28]. Presence of non-minimal coupling weakens this effect as it makes the
potential more and more flat as the vacuum energy increases, thus also increasing the action. In
thiscasebouncesolutionsdonothavetoreachthefalsevacuumexactlybuttheyonlyhavetopass
the bubble wall. Figure 8 depicts the potentials with different values of the vacuum energy c and
partofthepotentialactuallyprobedbythetunnelingsolution.
For a fixed positive vacuum energy (given c) increasing ξ also results in more flat potential.
It means that the bounce probes only values closer to the top of the barrier which makes it more
similar to the HM solution. Furthermore, when value of ξ is too large potential becomes too flat
andCDLbouncesceasetoexist[29,30]. Thus,asthevacuumenergydecreaseslargervaluesofξ
allowtunneling.
3. Conclusionsandsummary
We analysed the vacuum decay process in presence of non minimal coupling to gravity for a
generalbutsimplemodelconsistingofasingleneutralscalarwiththegenericpotentialdescribed
in Section 2. We developed a thin-wall solution concerning ξ and provided explicite formulae
needed to compute the decay exponent in a generic model. Analytical results were verified by a
precisenumericalcalculation.
8
Non-minimallycoupledgravityandvacuumstability OlgaCzerwin´ska
a=1,b=0.05,c=0.1 a=1,b=0.1,c=0.1
5000
500
2000
S1000 SSSSfHTClaWMDtL S SSSSfHTClaWMDtL
500 200
200
100
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
ξ ξ
a=1,b=0.05,c=0.05 a=1,b=0.1,c=0.05
1000
5000 SSSSfHTClaWMDtL 680000
S2000 S
400
1000 SSSSfHTClaWMDtL
500
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.05 0.10 0.15 0.20 0.25 0.30
ξ ξ
Figure7: TunnelingactionfordeSitterfalsevacuumasafunctionofnon-minimalcouplingobtainedusing
fourdifferentmethods.ParameterchoiceandlabelsarethesameasinFigure6.Rowscorrespondtodifferent
falsevacuumenergydensitiesparametrisedbyc=(0.1, 0.05)fromtoptobottom.
ξ=0. ξ=0.1
0.10
0.08 0.10
c=0.1 c=0.1
0.06 SCDL=115.014 22ϕ) SCDL=185.648
V 00..0024 cS=C0D.0L=5282.948 V1/(-κξ0.05 cS=C0D.0L=5387.079
c=0. c=0.
0.00 SCDL=2818.61 0.00 SCDL=2267.27
-0.02
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
ϕ ϕ
Figure8: Potentialswithdifferentvaluesofthevacuumenergyc. Thepartofthepotentialactuallyprobed
bythetunnelingsolutionisdashed. Forthisexamplethenon-minimalcouplingwassettoξ =0(leftpanel)
andξ =0.05(rightpanel)whilethevacuasplittingparameterb=1/10.
Influence of non-minimal coupling to gravity is very different in cases of Minkowski and dS
false vacua. For dS vacuum decay probability quickly decreases as the coupling grows and the
vacuum can be made absolutely stable. In Minkowski case effect is much weaker and the decay
rate increases for small values of ξ. Also, for a flat spacetime TW approximation works worse
significantly overestimating tha action due to ξ term. Even though TW approximation may not
give a precise result in a specific model, at least the order of magnitude is right - especially in dS
9
