Cosmic recall and the scattering picture of Loop Quantum Cosmology

0.4 MB·English

by  Wojciech Kaminski

Cosmic recall and the scattering picture of Loop Quantum Cosmology
Wojciech Kamin´ski1 and Tomasz Pawl owski2,1
∗ †
1Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoz˙a 69, 00-681 Warszawa, Poland
2Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain
The global dynamics of a homogeneous universe in Loop Quantum Cosmology is viewed as a
scattering process of its geometrodynamical equivalent. This picture is applied to build a flexible
(easy to generalize) and not restricted just to exactly solvable models method of verifying the
preservation of the semiclassicality through the bounce. The devised method is next applied to
two simple examples: (i) the isotropic Friedman Robertson Walker universe, and (ii) the isotropic
sector of the Bianchi I model. For both of them we show, that the dispersions in the logarithm of
the volume ln(v) and scalar field momentum ln(p ) in the distant future and past are related via
φ
strong triangle inequalities. This implies in particular a strict preservation of the semiclassicality
0
1 (in considered degrees of freedom) in both the cases (i) and (ii). Derived inequalities are general:
0 valid for all the physicalstates within theconsidered models.
2
PACSnumbers: 04.60.Pp,04.60.Kz,98.80.Qc
n
a
J
I. INTRODUCTION solutions [37, 38] and spherically symmetric spacetimes
5
[39,40]. Thereexistalsostudiesofdifferentprescriptions
1
within the polymeric quantization [41–43] as well as of
Loop Quantum Gravity [1, 2] and its symmetry re-
the connection between LQC and the noncommutative
]
ducedanalog,knownasLoopQuantumCosmology[3,4]
c geometry [44]. The effects predicted by LQC were also
q haveexperiencedoverrecentyearsadynamicalprogress.
appliedfor the regularizationofthe cosmologicalmodels
- In particular, an application of the latter to the stud-
r not originating from the polymer quantization [45].
g ies of a simplest (isotropic) models of an early Universe
The prediction of the bounce and the existence of the
[ have shown that the quantum nature of the geometry
branchofthe universeevolutionpreceding it haveraised
qualitatively modifies the global picture of its evolution.
1 an interesting question: provided that the expanding
Namely, the big bang singularityis dynamically resolved
v post-bounce branchis semiclassical,whatcanwe deduce
3 as it is replaced by a so called big bounce [5] which con-
aboutthepre-bounceone? Doesithavetobenecessarily
6 nects the current (expanding) Universe with a contract-
semiclassical or can it be completely dispersed and not
6 ing one preceding it. The results initially obtained nu-
2 mericallyfortheflatisotropicmodelwithmasslessscalar possibleto describe byanyclassicalmetric? The prelim-
. inary studies performed in context of the simplification
1 field[6,7]wereshowntobegeneralfeaturesofthatmodel
of LQC and presented in [46] seemed to favor the latter
0 [8, 9] and next extended to more general matter fields
possibility. However more detailed analysis performed
0 [10–12]and topologies[13, 14] as wellas to less symmet-
1 in the framework of the so called solvable LQC [8] have
ric systems: anisotropic [15–21] and some classes of the
: shown, that for the states satisfying quite mild semiclas-
v inhomogeneousones[22]. Also,althoughthetheoryorig-
sicality assumptions for one of the branches the possible
i inates from the canonical framework, a connection with
X growth of the dispersion through the bounce is severely
the spin foam [23] models was made through the studies
r limited[9]. Inconsequence,withinconsideredclassofthe
of the path integral in LQC [24] as well as the analysis
a
states the semiclassicality at the distant future implied
of the cosmological models within the spin foam mod-
the semiclassicality in the distant past [47].
els themselves [25]. Another avenue of extensions is the
The latter results, although firm, strongly rely on the
perturbation theory around the cosmological solutions
analyticsolvabilityofthestudiedmodel,thusaredifficult
[26]. TheelementsofitsmathematicalstructureofLQC,
to extend to the original formulation of it [7] as well as
which was initially formulated in [27], were investigated
tomoregeneralsettings,forwhichonecouldnotapriori
indetail[28–31]. Inadditiontothestudiesperformedon
exclude the loss of semiclassicality through the bounce
the genuine quantum level, there is a fast growing num-
[48]. Finding a definitive answer to the question posed
berofworksemployingaclassicaleffectiveformulationof
above requires construction of the method, which is suf-
the dynamics [32, 33], which often provides qualitatively
ficiently flexible to be adaptable to the situations, where
new predictions [34–36]. The methods of LQC were also
the analytical studies fail. We introduce such method
applied,withvariouslevelsofrigor,outsideofthecosmo-
in this article, next applying it to two simple examples
logical setting, in particular in description of black hole
of flat models with massless scalar field as the sole mat-
tercontent: (i)anisotropicFriedman-Robertson-Walker
universe,and(ii)anisotropicsectorofaBianchiImodel
∗Electronicaddress: [email protected] quantizedasspecifiedin[20]. Inbothcasesourtechnique
†Electronicaddress: [email protected] allows to derive certain (strong) triangle inequalities in-
2
volving the dispersions of (the logarithms of) the total detailed description of the quantization and the proper-
volume and the scalarfield momentum. The inequalities ties of the systems the reader is referred to [7, 27] and
are general, valid for all the physical states admitted by [20].
the model. In the case (i) they imply an exact preserva- Since the considered systems are constrained ones,
tion of the semiclassicality, whereas in (ii) an analogous they are quantized via Dirac program, which consists
resultisonlypartial,asitdoesnotinvolveallthedegrees of the following steps: kinematical quantization ignoring
of freedom describing the anisotropic system. the constraints, promoting the constraints to quantum
The treatment we introduce here is based on the ob- operatorsandfindingthephysicalHilbertspacebuiltout
servation, that the structure of the evolution operator of the states annihilated by the quantum constraint. In
in LQC implies that for each physical state (in cer- accordance to this procedure this section has the follow-
tainsense)thereexiststhegeometrodynamical(Wheeler- ing structure: First we introduce the classical models,
DeWitt) one,whichis its largescale limitin either strict LQC kinematics and quantization of the constraint for
[7, 11] or approximate [10, 13] sense. Furthermore in a the FRW and Bianchi I models separately in Sec. IIA
large class of the models the structure of the physical and IIB respectively. Next we describe the structure of
Hilbertspaceofeachgeometrodynamicaltheorydescrib- the physical Hilbert space and relevant observables for
ing the limit admits a decomposition onto equivalents of LQC theory (in Sec. IIC), as well as its WDW analog
Klein-Gordon plane waves either incoming from or out- (Sec. IID).
going to an infinite volume. Since the WDW limits of
LQC states areformedoutof “standingwaves”coupling
both the incoming and outgoing ones, one can perceive A. Isotropic flat FRW universe
the LQC evolution as a scattering process, which trans-
forms the WDW incoming state (contracting universe) 1. Classical theory
onto the outgoing one (expanding). Known properties
of the limit allow to explicitly determine the scattering
On the classical level spacetimes of this class admit
matrix and thus to relate the properties of the incoming
a (parametrized by a time t) foliation by homogeneous
andoutgoingstates. Theprocedureisexplainedindetail surfaces =Σ R, where Σ is topologically R3. Their
inSectionIVthroughanapplicationtothe cases(i)and M ×
metric tensor is
(ii) listed above.
The paper is organized as follows: First, in Sec. II g = N2dt2+a2(t)oq, (2.1)
we briefly introduce the mainaspects of the LQC frame- −
workusedtocharacterizeconsideredmodels,aswellasits where N is a lapse function, a is a scalefactor (or equiv-
WDW analog. Next, in Sec. III we analyze in detail the alently a size of certainselected region , see the discus-
exact WDW limit of LQC states, which is next used in sion below (2.3)) and oq is a fiducial CVartesian metric.
Sec. IV to construct the scattering picture mentioned in To describe the system further we apply the canonical
the previous paragraph. That picture is then employed formalism,expressingthe geometryin terms ofAshtekar
in Sec. V to the analysis of the dispersions of the in- variables: connectionsandtriads,selectingthegaugefix-
coming/outgoing WDW states, which allows in turn to ing in which they can be expressed in terms of the real
derive the triangle inequalities relating them. We also connection and triad coefficients c,p
show there (Sec. VB), that the dispersions of the outgo-
ing/incoming components of the WDW limit equal the Aia =cVo−31 oωia, Eia =pVo−32√oqœai, (2.2)
dispersionsof agenuine LQC state in the asymptotic fu-
ture and past, thus extending the inequalities to these where œa/oωi is an orthonormal triad/cotriad corre-
quantities. We conclude with Section VI with the dis- i a
sponding to the fiducial metric oq and V is the fidu-
o
cussion of the main results as well as the possibilities
cial volume of . The variables c,p are canonically con-
of their extensions to more general settings. In order to V
jugated with c,p = 8πGγ/3 (where γ is a Barbero-
makethepresentationofthestudiesclearertothereader, { }
Immirzi parameter, which value has been set following
some details of the mathematical studies as well as the
[49]) and are global degrees of freedom of the geometry.
details of the numerical analysis used in the article are
In particular p=a2.
moved to Appendix A and B respectively.
Within selected gauge all the constraints except the
Hamiltonianoneareautomaticallysatisfied. Theremain-
ing constraint takes the form C =N(C +C ) where
gr φ
II. THE FRAMEWORK OF LQC AND ITS
WDW ANALOG 1
Cgr =−γ2 d3xǫijke−1EaiEbjFakb, (2.3)
Inthissectionwebrieflysketchthespecification,quan- ZV
tization program and relevant properties of the models with e := det(E) and F being a curvature of a con-
| |
we aregoingto study. We discuss only thoseelements of nection A. To deal with the noncompactness of Σ the
p
the theory, which are relevant for our analysis. For the integrationofa Hamiltoniandensity wasperformedonly
3
over a chosen cubic cell constant in comoving coor- Presently in the literature there exists several pre-
V
dinates, which is an equivalent to the infrared cut-off. scriptionsofconstructingthequantumHamiltoniancon-
Despite this, the physical predictions are invariant with straint differing by fine details, like the lapse, the factor
respect to the choice of the cell [7]. orderingandthesymmetrization. Herewefocusonthree
The remaining matter part of the constraint equals of them, defined in [7], [8] and [18, 51] and denoted, re-
spectively,astheAPS,sLQC andMMO prescription. In
C =8πGp2/p3/2, (2.4) all these cases the resulting operator can be brought to
φ φ
the form
where φ and p are, respectively, the value of the scalar
φ
field and its conjugate momentum with φ,p =1. 11 ∂2+Θ 11, (2.7)
{ φ} ⊗ φ ⊗
where an action of the operator Θ equals
2. Loop quantization
[Θψ](v)=f (v)ψ(v 4) f (v)ψ(v)
+ o
− − − (2.8)
+f (v)ψ(v+4),
The classical system specified in Sec. IIA1 is next
−
quantized via methods of Loop Quantum Gravity. In
with the form of f depending on the particular pre-
particular, the Dirac program is employed to construct o,
scription used and gi±ven respectively by
the physical Hilbert space. It consists of the following
steps:
APS:
Quantization on the kinematical level (ignoring the •
•
constraints). Here, the as the basic objects instead of f (v)=[B(v 4)]−21f˜(v 2)[B(v)]−21, (2.9a)
Ai, Ea weselectthe holonomiesofAi alongthe straight ± ± ±
linaes aind fluxes of Ea along the unitasquare 2-surfaces, fo(v)=[B(v)]−1[f+(v)+f (v)], (2.9b)
i −
which form a closed algebra. The direct implementation
where [52]
of the procedure used in LQG [2] leads to the gravita-
tional Hilbert space = L2(R¯,dµ ), where R¯ is
Hgr Bohr f˜(v)=(3πG/8)v v+1 v 1 (2.10a)
the Bohr compactification of the real line. The basic | | | |−| − |
operators are respectively the holonomies hˆ(λ) along the B(v)=(27/8)v v(cid:12)+11/3 v 1(cid:12)1/3 3 (2.10b)
| | |(cid:12) | −| − (cid:12)|
edge of the length λ and triad pˆ (corresponding to the
(cid:12) (cid:12)
flux across the unit square). The basis of gr is built sLQC: (cid:12) (cid:12)
H
of the eigenstates of pˆand parametrized by v such that •
pˆv =(2πγℓ2 √∆)2/3sgn(v)v 2/3 v ,wheretheparame-
| i Pl | | | i f (v)=(3πG/4) v(v 4)(v 2), (2.11a)
ter∆isthesocalledarea gap specifiedinthenextpoint. ± ± ±
The inner product on gr is given by fo(v)=(3πG/2)pv2, (2.11b)
H
ψ χ = ψ¯(v)χ(v). (2.5) MMO:
h | i •
v R
X∈ f (v)=Cg(v 4)s (v 2)g2(v 2)s (v)g(v),
± ± ± ± ± ±
The matter degreesoffreedomarequantizedvia stan- f (v)=Cg2(v)[g2(v 2)s2(v)+g2(v+2)s2(v)],
dard methods, thus attaining the Schroedinger-like rep- o − − + (2.12)
resentation. In consequence the full kinematical Hilbert
space takes the form where
kin = gr φ, φ :=L2(R,dφ). (2.6) g(v)= 1+1/v 1/3 1 1/v 1/3 −1/2, (2.13a)
H H ⊗H H | | −| − |
The basic operators on φ are the field value φˆ and its s±(v)=(cid:12)(cid:12)sgn(v±2)+sgn(v), (cid:12)(cid:12) (2.13b)
momentum pˆ = i~∂ .H C =πG/12. (2.13c)
φ φ
−
Promoting the constraint to quantum operator. For
•
thatallthegeometriccomponentsin(2.3)and(2.4)have TheoperatorΘisdenoted,respectively,byΘ ,Θ
APS sLQC
to be expressed first in terms of the holonomies and and Θ and is well defined in all the listed prescrip-
MMO
fluxes, which is essentially done via methods specified tions in particular for ε = 0 (see the detailed discussion
in [50]. The field strength Fk is in particular repre- in [31] for APS and [51] for MMO).
ab
sented via holonomies along the closed square loop. The Building a physical space out of states annihilated
•
requirement for our theory to mimic the properties of by the constraint. Since the operator (2.7) is essentially
LQG and the discreteness of the area operator Aˆr there self-adjoint [29], this step can be realized via system-
forcesustofixthe physicalarea∆ofthisloopasthe 1st atic procedure of the group averaging [53, 54]. On the
nonzero eigenvalue of Aˆr, which is the unique physically other hand its form selects another natural way of find-
consistent choice for that technique [35]. ing the solutions [7, 30] which in this case is equivalent
4
toit,namelythereinterpretationoftheconstraintasthe 1. The classical model
Klein-Gordon-like equation
[∂2Ψ](v,φ)= [ΘΨ](v,φ), (2.14) Classically the Bianchi I flat spacetime admits the
φ − samefoliationbyhomogeneoussurfacesastheFRWone.
defining the evolution of a free system along φ, that is The general form of the metric is
the mapping betweenthe the spacesofthe “initialdata”
– restrictions of Ψ to the surfaces of constant φ g = N2dt2+a2(t)dx 2+a2(t)dx 2+a2(t)dx 2 (2.17)
− 1 1 2 2 3 3
R φ Ψ(,φ) . (2.15)
gr where x are chosen(comoving)coordinates defining the
∋ 7→ · ∈H i
fiducialCartesianmetric oq :=dx 2+dx 2+dx 2withor-
The quite simple formofthe operatorΘallowsto eas- 1 2 3
thonormal (co)triad (oωi) œa . To construct the Hamil-
ilydefine thephysicalHilbertspace phy viaitsspectral a i
H tonian formalism one has to introduce again a fiducial
decomposition. This step, as well as the notion of evolu-
cell, which here is described by three fiducial (i.e. with
tion will be described in more detail in Sec. IIC.
respect to the metric oq) lengths L and fiducial volume
Before going to it let us note, that the structure of Θ i
V := L L L . Here it is again possible to fix a gauge
and (2.7) provides the natural division of the domain of o 1 2 3
in which the Ashtekar connections and triads are rep-
v onto the subsets (the lattices)
resented by three pairs of canonically conjugated coeffi-
Lε ={ε+4n; n∈Z}, ε∈[0,4[ (2.16) cients ci,pi
ptrraensesrfevrerdedbytoththeeascptiloitntinofgΘof. This donivtiositohneissunpaetrusrealellcy- Aia =ciL−i 1oωia, Eia =piLiVo−1√oqœai, (2.18)
phy
H
tionsectors. Inconsequenceitisenoughtofixparticular
and all the constraints except the Hamiltonian one are
value of ε and work just with the restriction of the do-
automatically solved. The Poisson brackets between co-
cmlaarinityofwΘe wtoillfucnonctsiiodnesr sjuusptpothrteedseoctnorLεε o=nl0y., hFoowretvheer efficients equal {ci,pj}=8πGγδji.
Following [20] we choose the lapse N = p p p .
the presented treatment and its results generalize easily | 1 2 3|
The Hamiltonian constrainthas the same form as in the
(at the qualitative level) to all the sectors. p
FRW case, and its components are given by (2.3) and
Further simplification comes from the fact, that the
(2.4) (with the basic variables given by (2.18) instead of
considered system does not admit parity violating in-
(2.2)), where in (2.4) p:= p p p 1/3.
teractions. In consequence the triad orientation reflec- | 1 2 3|
tion v v being the large symmetry provides another
7→−
natural division onto superselection sectors, namely the
2. Loop quantization
spaces of symmetric and antisymmetric states. For the
selected sector ε = 0 this particular choice allows to
further restrict the support of the functions to + := To quantize the system we follow the program spec-
+ R+. L0 ified in Sect. IIA2, in particular choosing the polymer
L0 ∩ representationforthegeometrydegreesoffreedom,while
keeping the Schroedinger one for the scalar field.
B. Flat Bianchi I universe In the geometry part the basic operators are
holonomies along straight edges generated by œa and
i
The first step in the generalization of the model pre- the fluxes across 2-dimensional rectangles spanned by
sented in previous Section is an extension to the flat œai. The gravitationalkinematicalHilbert spaceconsists
Bianchi I model, describing the universe with the same of the product of three copies of gr of the FRW sys-
matter content andtopology,which howeverwhile being tem, each corresponding to one diHrection of œai Hgr =
still homogeneous is not necessarily isotropic. Its (pre- 3 L2(R¯,dµ ). The basis of this space can be built
i=1 Bohr
liminary)analysiswithinLQC has beeninitiated in[15]. out of eigenstates of the triad (or unit flux) operators
N
Later more detailed analysis of its kinematics and dy- pˆ and parametrized by three real variables λ such that
i i
namics was performed in [16] and [17] (see also [18, 19] pˆi|λ1,λ2,λ3i = sgn(λi)(4πγ√∆ℓ3Pl)23 λ2i |λ1,λ2,λ3i. Al-
for a vacuum case), although the quantization prescrip- ternatively,oneofλ canbereplacedwiththeparameter
i
tionusedthereisnotapplicabletothenoncompactcases v :=2λ λ λ . Here for thatpurpose we selectλ ,finally
1 2 3 3
[36]. The first description valid also in noncompact sit- labeling the basis elements as λ ,λ ,v .
1 2
| i
uation was constructed in [20], which we will follow in The space and the set of basic operators corre-
φ
H
this article. In this section we briefly introduce those spondingtothe matterarethe sameonesasinthe FRW
elements of the framework, which are needed as a basis case. ThefullkinematicalHilbertspaceisalsoaproduct
for our analysis. The treatment is in fact an extension = .
kin gr φ
H H ⊗H
of the one applied to the FRW universe. Therefore, for The quantum Hamiltonian constraint is constructed
shortness, here we will focus just on these aspects of it, out of the classical one by, first reexpressing it in terms
which differ from the description presented in Sec. IIA. of the holonomies and fluxes, and next promoting these
Forthedetaileddescriptionthe readerisreferredto[20]. componentstooperators. Inparticularthefieldstrength
5
Fk isagainrepresentedviaholonomiesalongclosedrect- via an isotropic model constructed via averaging over
ab
angular loops of the physical area equal to ∆. Unlike in anisotropies and equivalent to the sLQC one described
isotropic case however fixing the loop area does not al- in Sect. IIA. However, one has to be aware, that some
low to uniquely fix the fiducial lengths of its edges. This of the physical states might in principle be in the kernel
apparent “ambiguity” gave rise to several distinct pre- of the projection operator Pˆ. Thus, certain care needs
scriptions present currently in the literature (including to be taken,whenrelatingthe propertiesofthe isotropic
the one of [16]). On the other hand the relation of the sector defined above with the full Bianchi I model. In
LQC degrees of freedom with the full LQG ones con- particular it is not confirmed, whether the expectation
structed in [20] allowed to fix the relation uniquely. The values and dispersions of the total volume Bianchi I op-
constraint resulting from this operation (defined on the eratorsagreewiththeanalogousquantitiesofthevolume
dense domainin ) is ofthe Klein-Gordonform operators acting on the averaged states. This issue will
kin φ
H ⊗H
require further studies.
Cˆ =1ˆ1⊗∂φ2+1ˆ1⊗ΘB1. (2.19)
Similarly to FRW model we can restrict our interest C. Physical Hilbert space, observables
to just the symmetric sector, that is those states Ψ,
which satisfy Ψ(λ ,λ ,v,φ) = Ψ(λ , λ , v ,φ). This
1 2 | 1| | 2| | | KnownformoftheHamiltonianconstraint(2.7)andin
allows to restrict the studies just to the positive octant
particularthe evolutionoperatorΘ (2.8) allowsto easily
λ ,λ ,v >0,onwhichanactionofΘ isgivenby(quite
1 2 B1 extract the Hilbert structure of the space of states an-
complicated)Eqs.(3.35)-(3.37)of[20]. Itsimportantfea-
nihilated by the constraint. The exact construction of
ture is, that, analogously to the isotropic one, it divides
isdoneviagroupaveraging(see[6]forthedetails).
onto the superselection sectors built of the states Hphy
Hsupphpyorted just on the sets (λ ,λ ,v);λ ,λ R,v To start with, we note, that the spectrum of Θ is for
L+ε} with L+ε := {v = ε+4n{; n1 ∈2N}, p1rese2rv∈ed by a∈n anlolntdheegecnaeserastceonansiddeerqeduahlserSepa(bΘs)ol=uteRly+cont0inuo[2u9s].[56I]n,
action of ΘB1. Therefore to extract the physics one can consequence one can build a base of ∪[5{7]}out of the
considerjustoneofthosesectors. Here,forsimplicitywe Hgr
eigenfunctionse correspondingtononnegativeeigenval-
choose ε=0. k
ues
As we show in Appendix A1 the operator Θ admits
B1
self-adjoint extensions. Knowing its action one can in [Θe ](v)=ω2(k)e (v), (2.22)
k k
principlefindthephysicalHilbertspace(s)corresponding
to the model by analyzing the spectral properties of the (where ω(k) = √12πGk, k > 0) and normalized such
extensions. Ontheotherhandthereexistsawelldefined
that ek ek′ =δ(k′ k). Inthesuperselectionsectorsε=
procedure of the averaging over anisotropies (defined in h | i −
0theremainingfreedomofglobalrotationisfurthermore
[20] to build an embedding of the isotropic model in the
fixed by the requirement, that
homogeneous anisotropic one). In this article we will
fpohcyussicjaulstproonptehrteiessp.aceofaveragedstatesH¯phy andtheir ek(v =4)∈R+. (2.23)
Applying the simplest form of the group averaging pre-
sented in [6] and abovespectral decompositionwe arrive
3. The isotropic sector to the following representation of the elements of
phy
H
Following [20] and the ideas of [55] we consider a pro- Ψ(v,φ)= dkΨ˜(k)e (v)eiω(k)φ, (2.24)
jection Pˆ mapping from the dense domain in Hgr of the ZR+ k
BianchiImodeltothe onetheisotropicmodelasfollows
where Ψ˜ L2(R+,dk) is the spectral profile of Ψ. The
∈
ψ(λ ,λ ,v) ψ(v)=[Pˆψ](v):= ψ(λ ,λ ,v). physical inner product is given by
1 2 1 2
7→
λX1,λ2 (2.20) ΨΦ = dkΨ¯˜(k)Φ˜(k). (2.25)
Through the direct inspection one can check that there h | i R+
Z
exists an operator Θ¯ such that
B1
Sinceherewearedealingwiththeconstrainedsystem,
[PˆΘ ψ](v)=Θ¯ [Pˆψ](v). (2.21) there is no natural notion of time and evolution. It can
B1 B1
be providedvia the unitary mapping (2.15). An alterna-
An action of that operator equals exactly the one of tive wayto define anevolutionis the constructionof the
Θ defined via (2.8, 2.11). family ofpartial observables [58], parametrizedby oneof
sLQC
The consequence of the above observation is, that at the dynamical variables and of the elements related via
least to some extent those of the aspects of the Bianchi unitary transformation. Here we construct family ln vˆ
φ
| |
I model, which are related exclusively to the behavior [6]interpretedasln v atgiven“time”φ. Thesystematic
| |
of the isotropic degrees of freedom, can be investigated wayofconstructingsuchobservablesispresentedin[30].
6
For the models considered in the article the expectation and the physical states (positive frequency solutions to
values and dispersions of ln vˆ for the physical state Ψ (2.30)) have the form
φ
| |
equaltheanalogousquantitiesofthekinematicalobserv-
able ln|vˆ| acting on the initial data ψφ :=Ψ(·,φ)∈Hkin Ψ(v,φ)= dkΨ˜(k)ek(v)eiω(k)φ, (2.32)
R
Z
[ln(v)φoΨ](v)=ei√Θ2(φ−φo)ln(v)Ψ(v,φo). (2.26) where Ψ˜ L2(R,dk) and ω(k) = √12πGk . The inner
∈ | |
Forcompletenessweintroduceonemoreobservable,cor- product has the same form as (2.25) but now k runs the
respondingtotheconstantofmotionln(ω)–anoperator entire R.
ln(ωˆ) acting as follows To characterize the states and define a physical evo-
lution we use the observables ln(ωˆ) and ln vˆ given, re-
φ
| |
[ln(ωˆ/√G)Ψ˜](k)=ln(ω(k)/√G)Ψ˜(k). (2.27) spectively,byfullanalogsof (2.27)and(2.26). Thelatter
ones can be expressed as quite simple differential opera-
This operator will be useful later in the paper as (it will tors acting directly on Hphy
be shown that) its dispersion bounds the growth of the
spread in ln|vˆ|φ. ln|vˆ|φΨ˜ =−ieiω(k)φ∂ke−iω(k)φΨ˜ (2.33)
=[ i∂k (∂kω(k))φ1ˆ1]Ψ˜.
− −
D. Wheeler-DeWitt analog This fact will be very useful in the following sections,
where we will use it to derive the relation between the
The systems studied in this article can be also quan- dispersion of the components of the WDW limits of the
tized via methods of the geometrodynamics. Indeed, the LQC states.
geometriccomponent(2.3)oftheHamiltonianconstraint
canbeexpressedentirelyintermsofthecoefficients(c,p)
defined in (2.2) III. WDW LIMIT OF AN LQC STATE
C = 6 c2√p, (2.28) The comparison of the forms of the operators Θ (2.8)
gr −γ2 and Θ specified via (2.30) shows that under certain con-
ditions (slowly changing functions) one of the operators
and the entire system can be treated just as an abstract can be approximated by the other. Therefore one may
one of the phase space coordinatized by (c,p,φ,pφ) and expect, thatthe solutionsto (2.7)convergein certainre-
quantized via standard methods of quantum mechanics. gions to some solutions of (2.30). Indeed, it was shown
AstheresultthekinematicalHilbertspacetakestheform via numerical methods in [7] that the eigenfunctions e
= L2(R,dφ),where :=L2(R,dv)withthe k
Hkin Hgr⊗ Hgr convergetocertaincombinationsofek ande k. Further-
inner product between ψ,χ∈Hgr moretheanalyticproperties(reality)ofΘim−plythatthis
limithasthe formofa“standingwave”thatisitis com-
ψ χ = dvψ¯(v)χ(v). (2.29) posed equally of incoming (k > 0) and outgoing (k <0)
h | i R plane waves (2.31). More precisely [59]
Z
The quantum Hamiltonian constraint can be expressed e (v)=r(k)(eiα(k)e (v)+e iα(k)e (v))
k k − k
as a differential analog of (2.7) −
+O(e (v)(k/v)2) (3.1)
| k |
∂2Ψ(v,φ)= ΘΨ(v,φ) =:ψ (v)+O(e (v)(k/v)2),
φ − (2.30) k | k |
:=12πG v ∂ v ∂ v Ψ(v,φ).
| | v| | v | | where r(k) R+ can be determined analytically via the
∈
p p relations between norms of the LQC and WDW states
To arriveto aboveequationwe selectedthe factororder-
(see Appendix A2) and α(k) S1 is a phase shift.
ing in (2.28) consistent with the one of (2.3) (see [7] for
∈
In this section we analyze the WDW limits ψ of the
the detailed explanation). k
LQC eigenfunctions e in detail. First, in Sec. IIIA we
The physical Hilbert space can be again constructed k
provide an analytic proof of the convergence for all the
via group averaging and it is an almost complete analog
forms of Θ considered in this paper, as well as recall
of the one of LQC models, with just slight differences
theargumentsallowingtodeterminethestructureofthe
being a consequences of the two-fold degeneracy of the
limit. Second, in Sec. IIIB we perform a detailed an-
eigenspaces of the operator Θ. Here the orthonormal
alytical and numerical analysis of the phase shifts α(k)
basis of (the symmetric states on) consists of the
Hgr defined in (3.1). The properties of these shifts are the
functions
critical components allowing to arrive to the triangle in-
1 equalities relating the dispersions of the physical state
e (v)= eikln v , k R, (2.31)
k √2πv | | ∈ and being the main result of this paper.
7
A. The convergence of the bases as well as it confirms the rate of convergence specified
in (3.1). This limit can be expressed in terms of the
In order to explicitly show the convergence (3.1) we coefficients introduced in (3.1) in the following form
comparethe eigenfunctionse –solutionsto (2.22),with
k eiα(k)
the solutions (2.31) to the WDW analog of (2.22). To χ~ =r(k) , (3.10)
k e iα(k)
startwith,wenote,thattheEq.(2.22)isa2ndorderdif- −
(cid:18) (cid:19)
ference equation, however, due to the specific properties
which is a consequence of the observation, that all the
of Θ characteristic for each of the prescriptions consid-
coefficientsf (v)in(2.22)arereal,so(by (2.23))is e .
ered here, the whole solution is determined just by the The scalingo,±factor r(k) can be easily determined fromk
single value e (v = 4) (see [7, 8, 51] for the details on
k the relation between norms in LQC and WDW theory
respective prescriptions).
discussed in Appendix A2 and equals
Toanalyzethesolutionitismoreconvenienttorewrite
that equation in the 1st order form [60]. For that we
r(k)=2. (3.11)
introduce the vector notation, defining
The behaviorof the phase shift function α(k) is however
e (v)
~e (v):= k . (3.2) much less trivial and requires detailed studies.
k e (v 4)
k
(cid:18) − (cid:19)
Using it we can rewrite (2.22) in the following form
B. The phase shifts
~e (v+4)=A(v)~e (v), (3.3)
k k
To extract the properties of the phase shifts α(k) de-
where the matrix A can be expressed (with use of the fined in (3.10) we combine the analytical and numerical
notation introduced in Eq. (2.8)) as methods. We focus on the behavior of the derivative
α := ∂ α, as it is exactly the quantity which will be
′ k
A(v)= fo(vf)+−(ωv)2(k) −ff−+((vv)) . (3.4) relevant in the further studies. First we derive analyti-
1 0 ! cally the asymptotic behavior of α(k) for k for the
→∞
sLQC prescription (Sec. IIIB1). The analytical results
To relate e with e we note that the value of e at are then strengthened and generalized to other prescrip-
k k k
each pair of consecu±tive points v,v+4 can be encoded tionbymeansofthenumericalmethodsdescribedinAp-
as values of the (specific for the chosen pair of points) pendix B. The results of that analysis are presented in
combination of e , that is Sec. IIIB2.
k
±
~e (v)=B(v)χ~ (v), (3.5)
k k
1. Asymptotics in sLQC
where the transformation matrix B is defined as follows
B(v):= ek(v+4) ek(v+4) . (3.6) AmongtheprescriptionsconsideredherethesLQCone
e (v) e (v) is somehow distinguished, as the bases e expressed as
(cid:18) k k (cid:19) k
functions of an appropriately defined canonical momen-
Havingathandthe objectsdefinedabovewecanrewrite
tum b of v have a simple analyticalform [8]. This allows
theequation(3.3)astheiterativeequationforthevectors
for a quite high level of control over their properties, a
of coefficients χ~
k factwhichwewillexploitbelow. To startwith, letusfix
the definition of b, choosing it to equal
χ~k(v+4)=B−1(v)A(v)B(v 4)χ~k(v)
− (3.7)
=:M(v)χ~k(v). b:=bocp−21, v,b =2, (3.12)
| | { }
The exact coefficients of the matrix M(v) can be calcu- wherec,paregivenby(2.2)andtheproportionalitycon-
lated explicitly for each of the prescriptions specified in stant b is fixed via the righthand side equality. This
o
Sect. IIA2. An important feature (found by direct in- quantity can be next used as a configuration variable
spection) of it is, that in all three cases they have the on the quantum level. A particularly convenient choice
following asymptotic behavior of the representation of the quantum states using this
variable is provided by the following transformation op-
M(v)=11+O(v−3), (3.8) erators chosen respectively for WDW ( ) and LQC ( )
F F
where O(v n) denotes the matrix, all the coefficients of framework
−
whichbehavelikeO(v n). Thatconvergenceimplies(via
− 1 ivb
application of the methods presented in [60] Sec. 4) the [ ψ](b)= dv v −2e 2 ψ(v), (3.13a)
F R | |
existence of the limit Z
1 ivb
[ ψ](b)= v −2e 2 ψ(v), (3.13b)
F | |
lim χ~ (4n)=χ~ , (3.9)
n k k vX∈Lε
→∞
8
where the part of ψ supported on v < 0 is determined e correspond to the respective components e ikln b/2
k ∓ | |
by the symmetry requirement. The form of these trans- o±f e .
′k
formations implies, that the domain of b is the entire R Bringing together this two observations we see, that
in the case of WDW and the circle of the radius 1/2 in to find the desired phase shifts one just needs to find
LQC. the transform of the functions e (b). As they are the
k
TheevolutionoperatorsΘandΘ transformedvia eigenfunctions of Θ, they are proportional to e (v)
sLQC k
(3.13) take the form
1[e ikln b/2](v)=N˜(k)e (v) (3.19)
Θ2 = 12πG[b∂ ]2, (3.14a) F− ∓ | | ±k
b
− and the factor of proportionality N˜(k) equals the trans-
Θ2 = 12πG[sin(b)∂ ]2, (3.14b)
sLQC − b form of √2πe∓ikln|b/2| at the points v = 1. Selecting
±
and their (symmetric) eigenfunctions corresponding to forthecomponentproportionaltoek(b)thepointv =−1
the eigenvalue ω2 = 12πGk2 (and in case of sLQC cor- we get
responding to the sector ε = 0) are combinations of the
orthonormal basis elements N˜(k)=√2π dbe−i(kln|b/2|−b/2)
R
Z (3.20)
ek(b)=N(k)e−ikln|b/2|, (3.15a) =√8πke−ikln(k) dye−ik(ln|y|−y),
e (b)=N(k)cos( kln(tan b/2)), (3.15b) R
k Z
− | |
where to arrive to the latter equality we introduced the
where N(k),N(k) are the normalization factors deter-
change of variables b = 2ky. The last integral can be
mined by the physical inner product [8] and (3.15b) is
computed in the regime k by a stationary phase
written in the chart b [ π/2,π/2]. The sign in the → ∞
∈ − − method (see Appendix A3 for the proof of the applica-
exponents comes from the comparison of the spectrum
bility of the method). The result is
and bases of the operator √Θ in v and b representation
(see for example [19]). y
Toretrievethe largev behaviorofthe functions (3.15) N˜(k) √8πke−ikln(k)√ 2πi o e−ik(ln|yo|−yo)
≈ − √k
one needs to perform the transform inverse to (3.13). (cid:20) (cid:21)yo=1
Since the large v correspond to high frequencies in b, 4π√ i√ke−i(kln(k)−k).
≈ −
the particular form of the functions implies, that only (3.21)
the domainnearb=0willgivetherelevantcontribution
to the asymptotics in v . Analogously, one can calculate N˜(k) for the component
Inordertobe ableto→com∞parethe functions ek(b) and proportional to e k(b) by selecting the point v = 1.
e (b) one first needs to to deal with the fact, that the Thesetworesultsa−llowustoextractthephaseshiftα(k),
k
inversetransformof (3.15b),involvesanintegrationover which equals
the domain [ π/2,π/2] whereas for (3.15a) one need to
perform an in−tegration over R. α(k)=−k(ln(k)−1)− 34π+o(k0). (3.22)
Todoso,letusfirstconsideron[ π/2,π/2]afunction Via the same method one can compute the derivative
−
ξ(b)ek(b), where ξ(b) is some smooth function with sup- ∂kN˜ =:N˜′.
portin( π/2,π/2)andequalto1insomeneighborhood
of 0. Reg−arding this function as defined on the entire R N˜ =√2π db[iln b/2]e i(kln b/2 b/2)
′ − | |−
we note that the difference R | |
Z
g (b):=e (b) ξ(b)e (b) (3.16) =√8πke−ikln(k) dy[ iln y ]e−ik(ln|y|−y)
k k − k R − | | (3.23)
Z
isasmoothfunctionwithappropriatebehavioratinfinity √8πikln(k)e ikln(k) dye ik(ln y y)
− − | |−
[61]. Hence its Fourier transform is of the order O(v−N) − ZR
for any N N. iln(k)N˜(k),
On the o∈ther hand, for ξ(b)e (b) considered as a func- ≈−
k
tion on a circle, the difference where the last estimate follows from the fact, that de-
composing N˜(k) =: A(k)eiα(k) (where A(k) R+) one
fk(b):=e−ikln(tan|b/2|) ξ(b)e−ikln|b/2|. (3.17) can express its derivative as ∈
−
is of the class C0, thus by Lebesgue-Riemann lemma its N˜′(k)=iα′(k)A(k)eiα(k) +A(k)′eiα(k)
transform 1f is of the order o(e (v)). In conse-
quence theFfu−nctkion | k | ≈−i√8π√−iln(k)√ke−i(kln(k)−k) (3.24)
+O(k−1/2ln(k)).
e′k(b)=ξ(b)cos(kln|b/2|)+(fk(b)+f−k(b))/2, (3.18) In consequence the phase shift derivative equals
supported on R has the same WDW limit as e . Fur-
k
thermore the components of this limit proportional to α′(k)= ln(k)+O(k−1ln(k)). (3.25)
−
9
2. Numerical generalization (represented by Ψ ) to the expanding one (denoted as
| iin
Ψ
| iout
In the case of the remaining two prescriptions repeat-
Ψ Ψ =ρˆΨ , (4.2)
ing the analytical calculations preformed for the sLQC | iin 7→| iout | iin
oneisnotpossible,astheeigenfunctionsofΘdonothave
Inconsequence,lookingat the evolution“fromthe infin-
manageable analytic form in either of v or b representa-
ity” (in the configuration space or in cosmic time) one
tions. We note however, that between the prescriptions
caninterprettheevolutionastheprocessofscatteringof
the operatorsΘdiffer justbyacompactoperator. Thus,
the contracting geometrodynamical universe. The form
it is expected that the asymptotic behavior of both α(k)
of the limit (4.1) immediately allows to write down the
andα(k)correspondingtothemisagaingivenby (3.22)
′ scattering matrix
and(3.25)uptothe resttermsdecayingwithk. We ver-
iofdysthwihsicehxpaercetadteisocnrifboerdαi′n(kd)e,tuasiilnigntAhpepneunmdiexricBa.lTmheothse- ρ(k,k′)=(ek|ρˆ|ek′)=e−sgn(k′)α(|k′|)δ(k+k′), (4.3)
methodsallowtodeterminethevaluesofα inquitewide
′ which form encodes in particular the fact, that the con-
range of k as well as to verify its asymptotic behavior
tractinguniversetotally“reflects”intotheexpandingone
(within the limitations of appliednumerics). The results
for different prescriptions are presented in Fig. 1. Al- Ψ˜(k) e2sgn(k′)α(k′)Ψ˜( k). (4.4)
| |
thoughthe exactformofα dependsontheprescription, 7→ −
′
especially for small k, one can observe the following fea- Thispictureallowstoaddressinaquitenaturalandintu-
tures common for all of the prescriptions considered in itivewaythequestionsregardingtherelationoftheprop-
this article: erties of the pre bounce branch (universe in the asymp-
totic past)andthe postbounce one (asymptoticfuture).
(i) For large k the derivative α converges to the limit
′ In particular, we will apply it in the next section to de-
specified in (3.25) with the rate
termine how muchthe bouncing universecandisperse in
the distant future of the bounce in comparison to the
α(k)= ln(k)+O(k 2), (3.26)
′ − − initial spread in its distant past.
Whenconsideringtheabovepictureonehastoremem-
(ii) The scaled 2nd order derivative of α is bounded ber that, although the LQC basis functions converge to
the combinations of the WDW ones, this is not neces-
[k∂2α](k) 1 (3.27) sarily the truth for the general physical states, as the
| k |≤
convergence of the bases is not uniform with respect to
for every value of k. k. Nonetheless, once the attention is restricted just to
the states localized with respect to the observable kˆ de-
These properties will be crucial for building the relation
finedanalogouslyto(2.27)(thatisofthefinitedispersion
between the dispersion growth through the bounce.
in k) the uniformity is restored and the WDW limit is
definedintheprecisesense. Thisfactisusedforexample
inSec.VAwherethe expectationvaluesanddispersions
IV. THE SCATTERING PICTURE of the LQC states are related with the ones of its WDW
limit.
It was shown in Sec. III that the basis functions span- Thescatteringpicturecanobviouslybe constructedin
ningtheLQCphysicalHilbertspaceadmitcertainWDW the context of any LQC model in which the basis func-
limits. Given that one can define a WDW limit of any tions converge explicitly to the WDW ones, like for ex-
physical state by replacing the basis functions in (2.24) amplethemodelswiththepositivecosmologicalconstant
with the limits ψ defined via (3.1). This operation de- [11] or the Bianchi I ones [19]. The applicability is how-
k
fines a relation between the LQC physical Hilbert space evernotrestrictedjusttosuchsystems. Inparticularthe
andtheWDWone,whichintermsofthespectralprofiles models featuring the classical recollapse, like [10, 13], in
can be written as follows LQC admita quasi-periodicevolution. Forthose models
it is also possible to build a correspondence between the
Ψ˜(k ) Ψ˜(k)=2eisgn(k)α(|k|)sgn(k)Ψ˜(k ), (4.1) LQC and WDW states, since the basis elements of phy
| | 7→ | | H
convergetotheirWDW analogsalsothere. Thenew dif-
where k spans the entire real line. ficulty in these cases is the fact that, as the spectra of
That limit consists of two components, the incoming theLQCevolutionoperatorsarediscretewhiletheWDW
wave packet (corresponding to k > 0) and the outgoing onesarecontinuous,the directanalogofthe transforma-
one (k < 0). On the physical level they represent the tion (4.1) leads to the WDW states of the zero norm.
universe whichis, respectively, contractingto big crunch Thisproblemcanbecircumventedbyintroducinganap-
singularity and expanding from the big bang one. The propriate interpolation of the discrete spectral profile of
entireLQCdynamicscanbethusseenasthespecifickind an LQC state. The WDW state constructed this way
of “transition” between the contracting WDW universe represents a single epoch (between the bounces) of the
10
evolution of the LQC one. However, since such interpo- an action of the observables ln v defined by (2.33).
φ
| |
lations are not defined uniquely, there is no direct 1 1 Their expectation values and dispersions equal respec-
−
correspondence between the loop states and their “lim- tively
its”. One can however choose the interpolations which
reproduce the expectation values and the dispersions of v(φ):= ln v φ = aφ Ψ± + i∂k , (5.2a)
h | | i± ± k k h− i±
therelevantobservablesatleastapproximately. Thisway σ := ∆ln v = ∆( i∂ ) , (5.2b)
φ k
± h | | i± h − i±
it is possible build the WDW states well mimicking one
epochoftheloopuniverseevolution. Ithappenshowever where a := √12πG and for any observable Oˆ we define
atthe costof relaxingthe relationsbetweenthe physical Oˆ := Ψ± Oˆ Ψ± .
parameters corresponding to them to approximate ones, h Ti±he mhain |qu|estioin we would like to address here is
without explicit convergence of their values. The reason whetherthereexiststherelationbetweenσ andσ+ and
for that is two-fold: what is its form. The answer to the forme−r is certainly
true as the transformation (4.4) unitarily maps Ψ+
(i) nontrivialcorrectionsduetointerpolationofthedis- Ψ− in the following way →
crete spectral profile, and
Ψ˜−(k)=[UΨ˜+](k)=e−2α(|k|)Ψ˜+( k),
(ii) the fact that the basis elements of the LQC physi- − (5.3)
calHilbertspaceconvergetotheir(rescaled)WDW U :H+phy →H−phy,
analogsonly asymptotically,thus obviously beyond
thus the expectation values and dispersions in (5.2) are
the classical recollapse point.
related as follows
Despite this, such relations can be still quite useful. In
particularthis methodis wellsuitedtoaddressthe ques- i∂k = U−1[ i∂k]U +, (5.4a)
h− i− h − i
tion, how the parameters (for example dispersions) can ∆[ i∂ ] = ∆U 1[ i∂ ]U , (5.4b)
k − k +
change between the epochs. In particular it can be used h − i− h − i
to investigate the issue of the spontaneous coherence of where
the LQC state, that is to address the question whether,
given an initial date describing the state which is not U−1[−i∂k]U =−i∂k−2α′11. (5.5)
semiclassical,the statewilladmitinthe future evolution
Combining together (5.2b), (5.3), (5.4b), (5.5) and ap-
the semiclassical epoch.
plying very general bound on the dispersion of the sum
of operators (A24) we obtain the following inequality
V. THE DISPERSION ANALYSIS σ σ++2 ∆α′11 +. (5.6)
− ≤ h i
Towriteitdownintheusefulformwehavetoexpressthe
This section is dedicated to the main goal of this pa-
per: finding the precise relation between the dispersions quantity2h∆α′11i+ intermsofdispersionsofobservables
commonlyusedtocharacterizethephysicalpropertiesof
ofthephysicalLQCstaterepresentingtheuniverseinthe
the state. For that we exploit the properties of the func-
distantfuture (postbounce)andpast(prebounce). The
tion α found in Sect. IIIB. Namely, by the definition of
studies are divided onto two steps. First, in Sec. VA we ′
the dispersion we can bound the term under considera-
apply the scattering picture to relate the dispersions of
tion via
incoming and outgoing asymptotic WDW states. Found
rlaetlaiotnionbeitswneeexntthtreadnisslpateerdsioinnsSoefca. VgeBnutioneoLbtQaCinstthaeterien- h∆α′11i2+ =h(α′2−hα′i+)211i+ ≤h(α′2−α′⋆)211i+,
(5.7)
the asymptotic past and future. which is true for any value of α⋆. Here we choose it to
′
be
A. Dispersions of the WDW limit components α⋆ =α(exp(λ⋆)), λ⋆ := ln(k) . (5.8)
′ ′ +
h i
Upon that choice, applying the property (3.27) of α we
GiventheWDWlimitΨ(definedby(4.1))oftheLQC
can bound the left-hand side of (5.7) as follows
state described by the spectral profile Ψ˜ let us define its
cdoemcopmopnoensittsiosnucohntthoatthteheinscpoemctinraglΨpr+ofialensdcoorurtegsopionngdΨin−g h∆α′11i2+ ≤h(ln(kˆ)−λ⋆11)2i+ =h∆ln(kˆ)i2+. (5.9)
to them equal Finally, knowing the relation ω(k) we can express the
right-hand side of (5.9) via the dispersion of the WDW
Ψ˜±(k):=θ( k)Ψ˜(k), (5.1) analog of the observable (2.27), which corresponds just
±
to a logarithmic scalar field momentum ln(p /b), where
φ
where θ is a Heaviside step function. Denote the sub- b:=~√G. The result is
spaces of Hphy formed by these components as H±phy re-
spectively. Oneachofthesecomponentsonecanconsider σ σ +2σ , (5.10)
+ ⋆
− ≤