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8 Cosmological implications of an evolutionary quantum gravity
0
0
2
n MarcoValerioBattisti
a ICRA - International Centerfor Relativistic Astrophysics
J Phys. Dept. (G9), Universityof Rome “La Sapienza” P.le A. Moro 5, 00185 Rome, Italy
1
[email protected]
3
] Giovanni Montani
c
ICRA - International Centerfor Relativistic Astrophysics
q
Phys. Dept. (G9), Universityof Rome “La Sapienza” P.le A. Moro 5, 00185 Rome, Italy
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r ENEA C.R. Frascati (Dipartimento F.P.N.), ViaEnrico Fermi45, 00044 Frascati, Rome, Italy
g
ICRANET C.C. Pescara, P.le della Repubblica 10, 65100 Pescara, Italy
[
[email protected]
2
v
0 The cosmological implications of an evolutionary quantum gravity are analyzed in the
9 context ofagenericinhomogeneous model.TheSchro¨dinger problemisformulatedand
6 solved in the presence of a scalar field, an ultrarelativistic matter and a perfect gas
4 regarded asthe dust-clock. Consideringthe actual phenomenology, itisshownhow the
. evolutionaryapproachoverlapstheWheeler-DeWittone.
1
0 Keywords: EvolutionaryQuantum Cosmology,Problemoftime,Inhomogeneous model
8
0 PACSnumbers:04.20.Dw,98.80.Qc
:
v
i Thegenerallycovariantsystem“perexcellence”isthegravitationalfieldinGen-
X
eral Relativity (GR), i.e. it is invariant under arbitrary changes of the space-time
r coordinates(4-dimensionaldiffeomorphisms).Inparticular,theHamiltonianofsuch
a
a theory is a combination of constraints,i.e. is (weakly) zero and one treats on the
samefooting observablesandconstantsofmotion.This nontrivialfeatureofGRis
reflectedatquantumlevelbytheso-calledproblemoftime[1]wheretheSchr¨odinger
equationis replacedby a Wheeler-DeWitt one,inwhichthe time coordinatedisap-
pears from the formalism.
In order to overcame this behavior, the most suitable way is to consider a mat-
ter field to define a relational time, i.e. evolve the gravitational field with respect
to another one. In particular, an useful choice for the matter field resides in an
incoherent dust, i.e. a dust which interacts only gravitationally [2] (see also [3]).
In fact, it is possible to show how there exist a dualism between time evolution
and the presence of a dust fluid in quantum gravity [4,5]. Indeed, starting from a
Schr¨odinger equation for the gravitational field (described by Ψ = Ψ(t, h ) on
ij
{ }
the super-space of the 3-geometries h )
ij
{ }
i∂ Ψ= ˆΨ d3x NHˆ Ψ Hˆχ=ǫχ, Hˆ χ=0, (1)
t i
H ≡ZΣ (cid:16) (cid:17) ⇒
1
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H being the super-Hamiltonian, N = N(t) the lapse function and Σ the compact
Cauchy surface, and taking the zero-order WKB limit for the (stationary) wave
functionoftheUniverse:χ eiS,anewmattercontributioninthesystemdynamics
∼
appears whose energy density is ρ= ǫ(x)/2√h. Therefore,a dust fluid co-moving
−
withtheslicingCauchysurfaces,i.e.withT =ρn n ,isinducedinthetheory.On
µν µ ν
the other hand, if we include ab initio a dust into the Universe dynamics foliating
thespace-timemanifold,weobtainthesameeigenvaluesproblem(1)asabove.This
way, a dust fluid is a good choice to realize a clock in quantum gravity.
Fromsuchapointofview,thismattercontributionhastoemergeinanysystem
which undergoes a classical limit and therefore it must concern the history of the
Universe. Since it is expected the quantum behavior of the Universe takes place in
thePlanckera,onlyPlanckmassparticlescanbereliablyinferredandappropriately
described by a perfect gas contribution. In particular, we construct a solution in
which sucha termis included into the dynamics and we link the quantumnumber,
associatedtoitsenergydensity,totheeigenvalueǫ.Indoingthat,weusethePlanck
massparticlesasaclockforthequantumdynamicsand,atthesamelevel,weinduce
bythemanon-relativisticmattercomponentintotheearlyUniverse.Ofcourse,the
phenomenological implications of such an approach have to be analyzed in order
to comprehend if this matter field (related to the the non-zero eigenvalue of the
super-Hamiltonian) is observable or not in the actual Universe. As we will see, no
phenomenology can came out today from our dust fluid (see [5]).
Inquantumgravity,theearlyUniversehastobedescribedbyagenericinhomo-
geneousmodel [6]since no symmetries canbe assumedin the model because of the
(expected) quantum fluctuation of the metric structure concerning the space-time
manifold. Such a model is classically described by the line element [7]
ds2 =N2dt2 h (dxi+Nidt)(dxj +Njdt), (2)
ij
−
the spatialmetric h being h =eqaδ OaOd∂ yb∂ yc, while q and ya are the dy-
ij ij ad b c i j a
namicalvariablesandOa(x)areSO(3)matrices.Inparticular,theya variablescan
b
be used as new spatial variables in order to identically solve the super-momentum
constraint H (x) 0. This way, all the dynamical information about this model
i
≈
are contained in the scalar constraint H(x) 0. As we said, we add a perfect gas
≈
contribution into the gravitationaldynamics in order to representthe Planck-mass
particles. Furthermore, since we want to describe the early Universe evolution, we
have to add also a free scalar field φ in order to account for the inflaton field and
an ultrarelativistic term describing the thermal bath of all fundamental particles
(whose mass is negligible with respect to the temperature). Therefore the scalar
constraint reads in the Misner variables (a,β±) [8]
p2 1 3 p2 a3
H(x)=κ(cid:20)− aa + a3 p2++p2− (cid:21)+8πaφ3 −4κl2 V(β±)+a3(ρur+ρpg)≈0, (3)
(cid:0) (cid:1) in
κ = 8πlP2 being the Einstein constant, V(β±) the potential term which accounts
for the spatial curvature of the model and ρ = µ2(x)/a4 and ρ =σ2(x)/a5 are
ur pg
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theenergydensityoftheultrarelativisticandperfectgascontributionsrespectively.
Thevariablesadescribestheisotropicexpansionwhileβ± the shapechangesofthe
Universe and the cosmological singularity appears for a 0. An important point
→
hasnowtobestressed.Sincetherelation∂ ya =Ni∂ ya holds[7],inasynchronous
t i
reference (N = 1, Ni = 0) the variables ya become passive functions and the
dynamics reduces, point by point, to the one of a Bianchi IX model [9]. This way
wecanworkintheminisuperspace representationandthe fieldtheoryisreducedto
a 4-dimensional mechanical system.
The evolutionary quantization, summarized in the eigenvalues problem (1), of
this system appears to be (considering the right normal ordering [2])
1 1 3 1 R3 µ2 σ2
(cid:26)κ(cid:20)∂aa∂a− a3 ∂+2 +∂−2 (cid:21)− 8πa3∂φ2+−4κl2 V(β±)+ a + a2(cid:27)χ=ǫχ,
(cid:0) (cid:1) in
(4)
where l denotes the co-moving physical scale of inhomogeneities. The solution of
in
the above problem (for details see [5]) (4) can be obtained neglecting the potential
ctearnmbVe(oβm±i)t.teIndafaacnt,dinitthisepreogssioibnlea3to≪faOct(cid:16)olrP2izleintphe<wKa2ve>f/u|Vn¯c(tβio±n)|(cid:17)χ(sau,cβh±a,φt)erams
χ(a,β±,φ)=θ(a)F(β±,φ).Therefore,thefunctionsF(β±,φ)arenothingbutplane
waves with wave number K = kβ2 +kφ2 and V¯(β±) is the averageof the potential
q
overthe width∆β 1/∆k wherethe realwavepacketsaresignificantlynon-zero,
β
∼
i.e. V¯(β±) = ∆β2V(β±)d2β. On the other hand, the function θ(a) satisfies the
equation R
d 1dθ K2 µ2 σ2
κ + κ + + ǫ θ =0 (5)
da(cid:18)ada(cid:19) (cid:18) a3 a a2 − (cid:19)
and it explicitly reads θ(a) = ω(a)exp 1 a+ǫl2/16π 2 where, near the sin-
gularity, ω(a) = ∞ c an+γ (γ = 1h−√2l2P1(cid:0) K2)Pand th(cid:1)eicoefficients c of the
n=0 n − − n
seriesobeyacompPlicatedrecurrencerelation[5].Sincewerequirethewavefunction
χ(a,β±,φ)todecayatlargescalefactora(wherethepotentialtermbecomesimpor-
tant),theseriesω(a)mustthereforeterminate.Thisway,weobtaintheeigenvalues
ofourproblem(4).Moreprecisely,thespectrumoftheUniversesuper-Hamiltonian
H and of the quantum number associatedto the ultrarelativistic term are givenby
the following relations
σ2
ǫ = (6)
n,γ l2(n+γ 1/2)
P −
l ǫ 2 µ2
P n,γ
2(n+γ)= + . (7)
(cid:18) 16π (cid:19) 8π
aIt is possibleto show how the quasi-classical limitof this model is reached before the potential
termbecomesimportantassuringtheself-consistencyofourmodel[5].
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It is to be emphasized that the ground state n = 0 eigenvalue ǫ0,γ =
σ2/l2(1/2 γ), for γ < 1/2 is negative. Thus the real ground state, according
t−o equPation (−7) for µ2 = 0 (γ ≃ 1.8·10−3), ǫ0 ≃ −2σ2/lP2 is associated to a posi-
tive dust energy density. As we can see from (6), the non-zero super-Hamiltonian
eigenvalue ǫ, i.e. the evolutionary feature of the model, is linked to the quantum
number σ2 describing the perfect gas of Planck mass particles. Therefore, as we
said, such dust is regardedas the clock for the Universe dynamics and at the same
levelit appearsas a new matter contributionwhich, in principle, may have reliable
implications at cosmologicallevel.
In order to phenomenologically analyze such cosmological implications of our
clock,we put by hands a cut-offlength in the model. The existence of a fundamen-
talminimalscalehaslongbeenexpectedinquantumgravityanditappearsinallthe
most promising attempts to solve this problem [10,11]. What we do is to require a
minimallengthlperparticleintheperfectgas,i.e.l l .Thisway,theenergyden-
P
≥
sityoftheperfectgasisconstrainedtobe lessthenthePlanckone(ρ (1/l4))
pg ≤O P
andthereforeσ2 (l ). Therefore,two mainconclusionscanbe inferred.(i) The
P
≤O
upper limitforσ2 ensuresthatthe spectrum of the Universe is limited by belowand
in particular it appears as ǫ (1/l ). (ii) From a phenomenological point of
P
| | ≤ O
view, the evolutionary approach to quantum gravity overlaps the Wheeler-DeWitt
scheme and therefore it can be inferred as appropriate to describe early stages of
the (quantum) Universe without significant traces on the later evolution. In fact,
the effects of the new matter to the actual Universe can be easily analyzed by its
contribution to the critical parameter, which is Ω ρ /ρ (10−60).
dust dust today
≃ ≃ O
Therefore,the cosmologicalimplications ofanevolutionaryquantumgravityto the
presentUniversearecompletelynegligible.Foracompletediscussiononthissubject
see [5,12].
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