Modified Friedman scenario from the Wheeler-DeWitt equation

0.07 MB·English

by  Michael Maziashvili

Modified Friedman scenario from the Wheeler-DeWitt equation

∗
Michael Maziashvili
Depa rtment of Theoretical Physics, Tbilisi State University,
3 Chavchavadze Ave., Tbilisi 0128, Georgia

Weconsiderthepossible modification oftheFriedmanequation duetooperatororderingparam-
eterentering theWheeler-DeWitt equation.
PACSnumbers: 04.60.Kz,98.80.Qc
The standard approach to quantum cosmology con-
U
sists of quantizing a minisuperspace model. The corre-
sponding Wheeler-DeWitt(WDW) equation[1]contains
an arbitrary operator ordering parameter. In one of our
5
papers[2]itwasnoticedthatthe operatororderingterm
0
for large values of operator ordering parameter can af-
0
fect the matching of false vacuum instanton with the
2
Coleman-De Luccia bounce. Continuing in the spirit of
n
that paper, in this brief note we would like to consider
a how the operator ordering term for large enough values a a
J o
of operator ordering parameter can affect the Friedman
8
equationconsideredasasemiclassicallimit ofthe WDW

2 equation. (Throughout this paper we assume c=1).
v Let us consider a closed universe filled with a vacuum
6 of constant energy density and the radiation,
4
0 ǫ
ρ(a)=ρ + , FIG. 1: A schematic pictureof thepotential U(a).
2 v a4
1
4 where ρv is the vacuum energy density, a is the scale
0 factor and ǫ is a constant characterizing the amount of and replace the momentum in (3) by a differential oper-
/ radiation. The evolution equation for a can be written ator p → −i¯hd/da. Then the WDW equation is written
c
q as down as
-
gr − 3πaa˙2− 3πa+2π2a3ρ(a)=0 . (1) G¯h2 d2 + Gq¯h2 d − 3πa2+2π2a4ρ(a) ψ(a)=0 ,
: 4G 4G 3π da2 3πa da 4G
v (cid:18) (cid:19)
(4)
i Thisequationisidenticaltothatforaparticlemovingin
X where q is an arbitrary parameter corresponding to the
a potential
r operator ordering p2/a=a−(q+1)paqp. Throughout this
a 3π paper q denotes
U(a)= −2π2a2ρ(a) ,
4G
a2
with zero energy, Fig.1. q ≡α 0 ,
¯hG
Wetacitelyassumed256π2G2ρ ǫ/9<1.Theuniverse
v
can start at a=0, expand to a maximum radius a0 and where α is an arbitrary dimensionless parameter. Dis-
thenrecollapseortunnelthroughthepotentialbarrierto regarding the operator ordering term, the eq.(4) has the
the regime of unbounded expansion. Eq.(1) represents formofaone-dimensionalSchr¨odingerequationforapar-
the zero energy condition for the Lagrangian ticledescribedbyacoordinatea,havingzeroenergyand
moving in the potential
3π 3πa
L=− aa˙2+ −2π2a3ρ(a) . (2)
4G 4G 3π
V(a)= a2−2π2a4ρ(a) .
4G
To quantize the model, we find the Hamiltonian
Assuming that α is sufficiently large, in the WKB ap-
G p2 3π
H=− − a+2π2a3ρ(a) , (3) proximation [3]
3π a 4G
ψ(a)=exp{−iW (a)/¯h+W +···} ,
0 1
∗Electronicaddress: [email protected] to the lowest order in h¯ one obtains
2
From the Hamilton-Jacobi equation (5) one obtaines
the following Lagrangian[4]
G dW 2 iGq¯hdW 3πa2
− 0 − 0− +2π2a4ρ(a)=0. (5)
3π da 3πa da 4G 3π iq¯ha˙ Gq2¯h2 3πa
(cid:18) (cid:19) L=− aa˙2− + + −2π2a3ρ(a) .
4G 2 a 12πa3 4G
We have assumed that for large values of α the second
term in eq.(5) may be of the order of the first one. Let
InthisLagrangianonecanomitthesecondtermbecause
usjudgethevalidityofthisassumption. Fromeq.(5)one
itcontainsthetotaltimederivativeandtherebydoesnot
obtains
affect the equation of motion [4]. Therefore one gets
dW q¯h f(a)
0 =−i ±i , (6) 3π Gq2¯h2 3πa
da 2a p2 L=− aa˙2+ + −2π2a3ρ(a) ,
4G 12πa3 4G
where
for which the zero-energy condition takes the form
q2¯h2 9π2a2 24π3a4ρ(a)
f(a)= + − .
a2 G2 G
In what follows, in the region a < a0, we assume the − 3πaa˙2− α2a40 − 3πa+2π2a3ρ(a)=0 . (10)
following boundary condition 4G 12Gπa3 4G
dW0 q¯h f(a) The modified Friedman equation(10) containes a new
=−i −i . (7)
da 2a 2 term, which decays very fast in the course of expansion
p
of the universeand therefore cannot affect the late time
Under this assumption, in the region a < a , for large
0 cosmology. But, however, the appearance of this new
enoughvaluesofαtheratioofthefirstandsecondterms
term is quite important for it can avoid the collapse of
in eq.(4) takes the form
the universe.
Tosummarize,weconsideredtheinfluenceoftheoper-
a dW 1 1 9π2a4 24Gπ3a6ρ(a)
0 atororderingtermonthesemiclassicallimitoftheWDW
= + 1+ − ∼1 .
S(cid:12)(cid:12)(cid:12)(cid:12)oq,¯hunddaer(cid:12)(cid:12)(cid:12)(cid:12)the(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)b2oun2dsary conαd2iati40on (7), inα2tah40e regio(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)n a(8<) erwaqemuaaesttsieournm. fMeorqorl=aerpgαreaeec20ni/so¯heulGyg,hfrwovhmaelurteehseαovfiesorypaenbreaagtriobnrintorinradgryeirndineimgq.pe(n4a)--
a , the secondterm ineq.(4) for largevalues of α canbe sionless parameter and take the boundary condition (7)
0
oftheorderofthefirstoneandmustbekept. Noticethat in the region a < a0. For large values of α, such that to
forthesolution(6)withthepositivesignthesecondterm ensurevalidityofeqs.(8, 9),the semiclassicallimitofthe
ineq.(5)becomessuppressedincomparisonwiththefirst WDWequationproducesamodificationoftheFriedman
one and should be omitted. equation eq.(10). The new term appearing in equation
LetusnowcheckthestandardWKBvaliditycondition (10)decaysveryfastandthereforedoesnotaffectthecos-
[3]. Forlargevaluesofαintheregiona<a oneobtaines mology in the course of expansion of the universe, but,
0
however,can avoid the collapse of the universe.
dW a2α d2W a2α
0 ∼−i 0 , 0 ∼i 0 ,
da Ga da2 Ga2
and correspondingly Acknowledgments
d2W dW 2
¯h 0 0 ∼q−1 ≪1 . (9) The author is greatly indebted to the Abdus Salam
(cid:12) da2 da (cid:12) International Center for Theoretical Physics, where this
(cid:12) (cid:30)(cid:18) (cid:19) (cid:12)
(cid:12) (cid:12) paper was begun, for hospitality. It is a pleasure to ac-
(cid:12) (cid:12)
Thus, for the(cid:12)solution (7) the ap(cid:12)proximation(5) is justi- knowledge helpful conversations with Professors A. Dol-
fied in the region a<a for large values of α. gov, M. Gogberashvili and V. Rubakov.
0
[1] J. Wheeler, in Battele Renconters, eds. C. DeWitt and Moscow, 1989); J.MathewsandR.Walker,Mathematical
J.Wheeler(Benjamin,NewYork,1968);B.DeWitt,Phys. Methods of Physics, (Atomizdat, Moscow, 1972).
Rev.160 (1967) 1113. [4] L. Landau and E. Lifshitz, Mechanics, (Nauka, Moscow,
[2] M. Maziashvili, Phys.Rev. D69 (2004) 107304. 1965).
[3] L. Landau and E. Lifshitz, Quantum mechanics, (Nauka,