Particle creation in a Robertson-Walker Universe revisited

0.08 MB·English

by  F. Pascoal

Particlecreation inaRobertson-Walker Universerevisited
F. Pascoal∗
Departamento de F´ısica, Universidade Federal de Sa˜o Carlos
Via Washington Luis, km 235. Sa˜o Carlos, 13565-905, SP, Brazil
C. Farina†
Instituto de Fsica, Universidade Federal do Rio de Janeiro
Caixa Postal 68528, 21941-972, Rio de Janeiro, RJ, Brazil
(Dated:February2,2008)
Wereanalyze theproblemof particlecreationina3+1 spatiallyclosed Robertson-Walker space-time. We
computethetotalnumberofparticlesproducedbythisnon-stationarygravitationalbackgroundaswellasthe
correspondingtotalenergyandfindaslightdiscrepancybetweenourresultsandthoserecentlyobtainedinthe
literature[1].
7
PACSnumbers:11.10.-z
0
0
2
I. INTRODUCTION was investigated in [2, 5]. The 3 + 1 version of this prob-

n lemhasbeeninvestigatedinarecentwork[1][9]. However,
a In a curvedspace-time the conceptof particles is a subtle whengeneralizingthe1+1solutiontothe3+1one,thisauthor
J
oneandthemeaningofwhatisa particleorwhatisa detec- missed a degeneracyfactor leading to an incomplete answer
4 torbecomesmuchmoredifficultthaninaflatspace-time[2] forthetotalnumberofparticlescreatedaswellasthecorre-

. Basically, this happensbecause in a generalcurvedspace- spondingtotalenergy.
1
v timewedonothaveLorentzsymmetryanymoreanditispre- Themainpurposeofthisshortpaperistocomputetheto-
3 cisely this symmetry that, in a flat space-time, allows us to talnumberofparticlescreatedandthetotalenergyrelatedto
3 identify the best vacuum state for the theory. Lorentz sym- them taking into account the correct degeneracy factor that
0 metryassuresthatthevacuumstateisthesameforallinertial ariseswhenweareina3+1dimensions. Withtheaidofap-
1 observers. propriatedgraphics,we also makea briefdiscussiononhow
0
Fortunately,inmanyproblemsofourinterest,therearere- theparametersthatappearinthemetricaffectthetotalnumber
7
0 gionswhereachoiceofthephysicalvacuumisquitenatural. ofparticlesproduced.
/ Thathappenswhenthetimedependenceofthefieldinthese
h
regions is harmonic (or, at least, almost harmonic). A basic
t
- conditiontohaveafieldwithaharmonictimedependenceis
p II. PARTICLECREATION
thatthebackgroundmetricandtheeventualboundariesdonot
e
h dependontime. Insomespecificcases,thebackgroundmet-
We consider the case of an expanding spatially closed
: ric respects this condition only in asymptotic times (distant
v Robertson-Walkeruniversewhoselineelementandscalarcur-
future and remote past). In these cases, the physical inter-
i vaturearegiven,respectively,by
X pretationof particlesis possible at those asymptoticaltimes,
r butnotbetweenthem. Asaconsequence,anon-staticcurved
a space-timebackgroundmayleadtothephenomenonofparti- ds2 = a2(η) dη2−dρ2− sen2ρ(dθ2+ sen2θdϕ) ,
clecreation.
(cid:16) (cid:17)
The first one to discuss the problem of particle creation R = 6a−3 ∂2a+a , (1)
η
due to a curved space-time background was Schro¨dinger in
(cid:0) (cid:1)
1939[3], butthefirstonethatcarefullyinvestigatedthisphe- where a(η) is the scale factor, η is the conformal time and
nomenon was Parker in the late 60s [4]. The particle cre- 0≤ ρ ≤ π. Inthiscasetheequationofamassivescalarfield
ationina1+1spatiallyclosedRobertson-Walkerspace-time conformallycoupledtothemetriciswrittenas
∂∂2ηu2k + a2∂∂ηa∂∂uηk + m2a2+1+ a1∂∂2ηa2 uk−∇anguk =0, (2)
(cid:18) (cid:19)
where ∇ is the angular part of the Laplacian operator on a 3-sphere. This operator has the hiper-spherical harmonics as
ang
eigenfunctions,thatsatisfythefollowingdifferentialequation[7]
∇ Y(l,m ,m ;ϕ,θ,ρ)=−l(l+2)Y(l,m ,m ;ϕ,θ,ρ), (3)
ang 1 2 1 2
where l = 0, 1, 2, ..., m = 0, ±1, ±2, ..., ±l and m = geststheansatz,
1 2
0, ±1, ±2, ..., ±m . Hence, the ∇ in equation (2) sug-
1 ang 1
uk(ϕ,θ,ρ,η)= a(η)Y(l,m1,m2;ϕ,θ,ρ)gl(η). (4)
2
Substituting(4)in(2),weobtainthedifferentialequationfor B
g (η), +
l
A
d2g (η)
l +ω2(η)g (η)=0, (5)
dη2 l l
)
η
( A
where 2
a
ω2(η)=(l+1)2+m2a2(η). (6)
l
B
Notethatthisequationissimilartothatofamechanicalhar-
−
monicoscillatorwithatime-dependentfrequency. A
Now,weconsideranexactlysolvablecaseduetoaconve-
−3η0 0 3η0
nientchoiceofa(η),namely η
FIG.1:Graphicofa2(η)versusηforaclosedexpandinguniverse.
η
a(η)= A+Btanh , (7)
s (cid:18)η0(cid:19)
where A, B and η are constantsandA > B. Since we are As we can see from Figure 1, there are two asymptotic
0
consideringan expandinguniverse, B > 0 and η > 0. An times where the backgroundmetric is almost static, namely,
0
inspectioninFigure1allowustointerpretη asthetimescale the remote pass, characterized by η ≪ −η and the distant
0 0
oftheexpansionoftheUniversewhileA−BandA+Bare, future, characterizedby η ≫ η . At those asymptotic times
0
respectively,thescalesofthe size oftheuniversebeforeand the corresponding vacuum states may be well defined if the
aftertheexpansion. solutionsof(5)havethefollowingasymptoticlimit
lim g(p,f)(η) = e−iωl(p,f)η ,
η→∓∞ l 2ω(p,f)Nk
l
ω(p,f) = q(l+1)2+m2(A∓B), (8)
l
p
wherethesuperscriptspandf standforpastandfuturesolutions,respectively. Thesolutionsof(5)thatrespecttheasymptotic
limits(8)aregiven,respectively,by
gp = ξ2iωlfη0(1−ξ)−2iωlpη0 F 1+iω−η ,iω−η ;1−iωpη ;1−ξ ,
l (2ωpNk)1/2 2 1 l 0 l 0 l 0
l
gf = ξ2iωlfη0(1−ξ)−2iωlpη0 F (cid:0)1+iω−η ,iω−η ;1+iωfη ;ξ , (cid:1) (9)
l 1/2 2 1 l 0 l 0 l 0
2ωlfNk (cid:16) (cid:17)
(cid:16) (cid:17)
where 2F1 isthehipergeometricfunction[8]andwehavedefinedωl(±) := 12 ωlf ±ωlp andξ := 1−e2η/η0 −1. Sinceupk
andufk arenotequal,thecorrespondingBogolubovcoefficientsmustbenon-v(cid:16)anishing. U(cid:17)singsome(cid:0)wellknown(cid:1)proprietiesof
thehypergeometricfunction,wecanwriteup intermsofuf anduf∗asfollows
k k k
upk = αkk′ufk′ +βkk′ufk′∗ , (10)
k′
X(cid:16) (cid:17)
withtheBogolugovcoefficientsgivenby
ωf Γ(1−iωpη )Γ(−iωfη )
αk′k = δkk′sωllp Γ(−iωl+ηl0)Γ0(1−iωll+η00);
ωf Γ(1−iωpη )Γ(iωfη )
βk′k = δkk′sωlp Γ(iω−η )lΓ(01+iωl−η0). (11)
l l 0 l 0
3
Intheremotepastalltheinertialparticledetectorsregisterthecompleteabsenceofparticlesinthestate|0pi(thevacuumstate
associatedtoup). However,inthedistantfuture,anyinertialparticledetectorwillregisteranumberofparticleswithquantum
k
numberskinthe|0pistategivenby
sinh2(πω(−)η )
h0p|Nkf|0pi= |βkk′|2 = sinh(πωpη )silnh(π0ωfη ). (12)
k′ l 0 l 0
X
Thetotalnumberofproducedparticlesandthetotalenergyassociatedtothemaregiven,respectively,by
∞ sinh2(πω(−)η )
h0p|Nf|0pi= |βkk′|2 = (l+1)2 l 0 , (13)
sinh(πωpη )sinh(πωfη )
k′ k l=0 l 0 l 0
XX X
∞ ωfsinh2(πω(−)η )
E = h0p|Nf|0piωf = (l+1)2 l l 0 . (14)
k l sinh(πωpη )sinh(πωfη )
k l=0 l 0 l 0
X X
Since we were not able to perform summation (13) analyti- onthoseparameters. We dothatbyplottingthegraphofthe
cally,letusmakeanumericalanalysisoftheprecedingresults totalnumberofcreatedparticlesasafunctionofthefieldmass
constructing,forinstance, the graphicof thetotalnumberof forseveralvaluesofA−B,A+Bandη (Figure3). Atfirst
0
particlescreatedN versusthemassofthefieldexcitationsm. we plot a control curve (in Figure 3, it is represented as a
Naively,wecouldexpectthatthetotalnumberofparticlescre- continuousline), andthen we modifyone ofthe parameters,
atedwasamonotonicallydecreasingfunctionofm.However, plota new curveand compareto the controlone. We repeat
as we can see from the Figure 2, there is a value m for the thisprocedureforallparameters.
0
massofthefieldatwhichthetotalnumberofcreatedparticles
N(m )reachesamaximum.
0
) 40
m0 1
·
( 0
N 2
b
b
N40 b b
1
N · b
∆m0 10 b b
b
b b
b
b
b
b b b
0
0 100 200
0 m
0 m0
m FIG. 3: Graphic of the total number of created particles N versus
thefieldmassmfordifferentvaluesofparametersA−B,A+B
FIG.2:Graphicofthetotalnumberofparticlesproducedversusthe andη0. Thecontinuous linerepresentsthecasewhereη0 = 0.01,
massrelatedtothefield. A−B = 1 and A+B = 101. The dashed line represents the
case where η0 = 0.01, A−B = 3.25 and A+B = 101. The
It is convenient to define a width ∆m as it is done in a dottedlinerepresentsthecasewhereη0 = 0.01, A−B = 1and
0
A+B =51. Finally,thecontinuousdottedlinerepresentsthecase
Gaussian distribution. With this purpose, we define m and
m suchthat + whereη0 =0.0115,A−B =1andA+B=101.
−
N(m +m )=N(m −m )=e−1N(m ), (15)
0 + 0 − 0
sothat∆m :=m −m . ChangingtheparametersA−B,A+B andη aswedid
0 + − 0
Notefromequation(13)thatN(m ),m and∆m depend in Figure 3 and then analyzing numericallywhat happensto
0 0 0
onlyonA−B,A+B andη . Despitethefactthatwewere thegraphicofN versusm,suggeststhefollowingbehaviors:
0
notabletogetaclosedanalyticexpressionform ,∆m and N(m ) gets larger and ∆m gets smaller as A−B and η
0 0 0 0 0
N(m )asafunctionofthemetricparametersA−B,A+B decreaseorA+Bincreases.Ontheotherhandm getslarger
0 0
and η , we can see numericallyhow these quantitiesdepend asA−B,A+Borη increase.
0 0
4
III. CONCLUSIONS thedegeneracyfactor(l+1)2 thatappearsinequations(13)
and(14). Thisfactormustbeincludedduetothedegeneracy
Inthisworkwecomputedthetotalnumberofparticlescre- inthequantumnumbersm1andm2. Wethinkourdiscussion
ated andthe totalenergyrelatedto them in a 3+1 spatially maybeofsomehelpintheanalysisofsimilarproblems,asfor
closed Robertson-Walker space-time. By using appropriated example,ifoneconsidersotherfunctionsa(η)withbehaviors
graphics, we also discussed how the parameters that appear slightlydifferentfromthatoneconsideredhere.
in the metric affect the total number of particles produced.
We found an unexpected behavior of N as a function of m,
namely,startingfromzero,itincreasesuntilitreachesamax- IV. ACKNOWLEDGMENTS
imumvalueatm andafterthatitdecreasesmonotonicallyas
0
mincreases. F.PascoalthanksI.WagaandN.Pintoforenlighteningcon-
The essential differencebetween ourcalculationsand that versations, and FAPERJ and CNPq for financialsupport. C.
presentedin[1]isthatthisauthordoesnottakeintoaccount FarinathanksCNPqforapartialfinancialsupport.
[1] M.R.Setare,Int.J.Theor.Phys.432237(2004). [7] A.Erdelyi,W.Magnus,F.OberhettingereF.R.Triconi,Higher
[2] N.D. Birrel and P.C. Davis, Quantum Fields in Curved Space, TranscendentalFunctions,McGraw-Hill,(1953).
CambribgeUniv.Press,Cambridge(1982). [8] W.N. Bailey, Generalised Hypergeometric Series, Cambridge
[3] E.Schro¨dinger,Physica6,899,(1939). UniversityPress,(1935);
[4] L.Parker,Phys.Rev.Lett.21,562(1968); E.W.Barnes,Proc.LondonMath.Soc.6,141(1908).
L.Parker,Phys.Rev.183,1057(1969); [9] Thisproblemalsowasinvestigatedin[6], buttheauthorshave
L.Parker,Phys.Rev.D3,346(1971). notwrittenaclosedanalyticalexpressionforthetotalnumberof
[5] C.BernardeA.Duncan,Ann.Phys.(N.Y.),107,201(1977). particlescreated.
[6] Mostepanenko V M and Trunov N N 1997 The CasimirEffect
anditsApplications(Oxford:Clarendon).