Late-time evolution of cosmological models with fluids obeying a Shan-Chen-like equation of state

0.56 MB·

by  Donato Bini

Late-time evolution of cosmological models with fluids obeying a Shan-Chen-like
equation of state
Donato Bini1,2, Giampiero Esposito2, Andrea Geralico1
1Istituto per le Applicazioni del Calcolo “M. Picone,” CNR, I-00185 Rome, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Napoli,
Complesso Universitario di Monte S. Angelo, Via Cintia, Edificio 6, 80126 Napoli, Italy
(Dated: January 19, 2016)
Classicalaswellasquantumfeaturesofthelate-timeevolutionofcosmological modelswithfluids
obeyinga Shan-Chen-likeequation of stateare studied. The latteris of thetypep=w (ρ)ρ,and
eff
hasbeenusedinpreviousworkstodescribe,e.g.,apossiblescenarioforthegrowthofthedark-energy
contentofthepresentUniverse. AttheclassicallevelthefluiddynamicsinaspatiallyflatFriedmann-
Robertson-Walkerbackgroundimpliestheexistenceoftwopossibleequilibriumsolutionsdepending
6 onthemodelparameters,associated with(asymptotic)finitepressureandenergydensity. Weshow
1 thatnofuturecosmologicalsingularityisdevelopedduringtheevolutionforthisspecificmodel. The
0
corresponding quantum effects in the late-time behavior of the system are also investigated within
2
theframeworkofquantumgeometrodynamics,i.e.,bysolvingthe(minisuperspace)Wheeler-DeWitt

n equation in the Born-Oppenheimer approximation, constructing wave-packets and analyzing their
a behavior.
J
6 PACSnumbers: 04.60.Ds,98.80.Qc
1

] I. INTRODUCTION
c
q
- Most investigations in modern cosmology assume our Universe to be described by a Friedmann-Robertson-Walker
r
g (FRW) backgroundfilled by a cosmologicalfluid endowedwith suitable properties, or equivalently a scalar field with
[ appropriate(self-interacting)kineticandpotentialenergies. Thescalar-fielddynamicsshouldmimicdifferentkindsof

matter-energy during each phase of the cosmologicalevolution up to the present epoch, which is characterized by an
1
accelerating expansion of the Universe. In this respect, we have recently investigated the role of cosmological fluids
v
obeying an equation of state of the type p = w (ρ)ρ, extensively used in the context of lattice kinetic theory to
7 eff
7 modeldynamicphasetransitions,i.e.,theShan-Chen(SC)equationofstate[1]. Itsmainpropertyistosupportphase
1 transitions, so that in a cosmological context different epochs in the evolution of the Universe can be described as a
4 natural consequence of the SC fluid dynamical equations. Furthermore, because of its flexibility, it makes it possible
0 to represent a broad variety of cosmological fluids, from radiation to “exotic” fluids with negative pressure, through
.
1 a smooth variation of its free parameters.
0 SC-basedcosmologicalmodels havebeen successfullyappliedto accountfor the currentdistributionofdarkenergy
6 aswellastodescribe the inflationaryepoch[2,3]. Constraintsonthe freeparametersofthe modelhavebeenderived
1 inRef. [4]accordingtocurrentobservationaldataofthecosmicmicrowavebackgroundandbaryonacousticoscillation
:
v spectra, which enforce the validity of these models. In fact, they were previously tested only on available data from
i measurements of distant type Ia supernovae. The aim of the present work is to study the late-time evolution of such
X
models,henceextending ourpreviousanalysis. We willinvestigatethe propertiesof(equilibrium)attractorsolutions,
r
including their stabilityagainstsmallperturbations,aswellasthe possible occurrenceoffuture singularities. We will
a
also develop a quantum analysis based on the Wheeler-DeWitt (WDW) equation [5] in the framework of quantum
geometrodynamics, in order to investigate the role of the associated quantum effects for large values of the scale
factor. This approach is largely used in the literature to study the quantum avoidance of classical singularities as
well as quantum effects at large cosmological scales [6–9]. Other approaches include path-integral quantization, loop
quantum cosmology,and string theory (see, e.g., Ref. [10] for a recent review).
Thepaperisorganizedasfollows. SectionIIbrieflyrecallsthemainpropertiesofSCcosmologicalmodelsandtheir
equivalent formulation in terms of an ordinary canonical scalar field, describing then both scalar-field dynamics and
late-time evolution. The asymptotic dynamics determines the profile of the interaction potential. Section III studies
the quantum-cosmology counterpart, when the WDW equation can be reduced to a partial differential equation,
and the wave function can be expanded in terms of “fast” and “slow” degrees of freedom according to the Born-
Oppenheimer(BO)approximation. ExplicitformsofsuchdegreesoffreedomarefoundforbothSCattractorsolutions
and wave packets are constructed. Concluding remarks and open problems are summarized in Sec. IV.
2
II. SCALAR FIELD SHAN-CHEN MODEL
Let us consider a spatially flat FRW universe, i.e., with squared line element ds2 = dt2+a2(dx2 +dy2+dz2),
−
wherea=a(t)is the scalefactorandc=1,filled bya perfectfluid obeyinga nonidealSC-likeequationofstate,that
we write in the form
ρ g
p = w ρ + ψ2 ,
(in) (crit),0 ρ 2
(cid:20) (crit),0 (cid:21)
ψ = 1 e−αscρ(crρit),0, (2.1)
−
whereρ =3H2/κ2 isthe presentvalueofthe criticaldensity(H denotingthe Hubble constantandκ2 =8πG).
(crit),0 0 0
The dimensionless quantities w , g and α are free parameters of the model, each of them carrying a well-defined
(in) sc
physical meaning: w describes the nature of matter, ordinary (positive) or exotic (negative), in both high-density
(in)
and low-density limits; g 0 measures the strength of nonideal interactions within the fluid; α 0 sets the ratio
sc
≤ ≥
between the actualcriticaldensity and a thresholddensity abovewhich the excess pressure(with respect to the ideal
gasbehavior)saturatestoaconstantvalue,aregimeoftenassociatedwith“asymptoticfreedom,”asitcorrespondsto
a vanishing contribution of nonideal forces to the momentum budget of the fluid. The idea of asymptotic freedom at
high-density short-distance regimes has played a crucial role in the development of quantum field theory, specifically
in the framework of quantum chromodynamics [11].
Underthe assumptionthatthe SCfluiddominatesoverothermatter-energydegreesoffreedom,thefieldequations
are
κ2
H2 = ρ, ρ˙ = 3H(ρ+p), (2.2)
3 −
where H =a˙/a is the Hubble parameter, and an overdot indicates derivative with respect to the cosmic time t. It is
convenient to introduce the following set of dimensionless variables:
ρ a
ξ , x , τ H t, (2.3)
0
≡ ρ ≡ a ≡
(crit),0 0
so that Eqs. (2.2) become
dx dξ
=x ξ, = 3 ξ[ξ+w (ξ)], (2.4)
dτ dτ − (in)P
p p
where
1
(ξ)=ξ+ gψ2(ξ), ψ(ξ)=1 e−αscξ. (2.5)
P 2 −
The (rescaled, dimensionless) pressure (ξ) reduces to the ideal gas expression (ξ) ξ in the low-density limit,
whereasinthe high-densitylimit it givesP ξ+1g,i.e., the excesspressureis jusPt aco∼nstant,often associatedwith
P ∼ 2
vacuum fluctuations. Noticeably, by changing the parameters of the model, i.e., w , g and α , the SC equation of
(in) sc
state can attain a broad range of values of cosmologicalinterest for the ratio
p gψ2
w =w 1+ . (2.6)
eff ≡ ρ (in) 2 ξ
(cid:18) (cid:19)
In fact, because of the Einstein’s equations fluid dynamics, the SC fluid might undergo a transition from ordinary
matter (w >0) to exotic matter (w <0) and viceversa, as shown in Fig. 1. We refer to Ref. [2] for more details
eff eff
on SC thermodynamics and the microscopic foundations of the SC equation of state.
A. Late-time evolution
We areinterestedhereininvestigatingthe late-timeevolutionofSC cosmologicalmodels. The fieldequations(2.4)
imply that there exist attractor (equilibrium) solutions corresponding to ξ =0 and ξ = ξ = 0 given by the roots of
the equation ∗ 6
ξ +w (ξ )=0, (2.7)
(in)
∗ P ∗
3
FIG.1: Thebehavioroftheeffectiveequationofstatew p/ρasafunctionofthedimensionlessscalefactorxisshownfor
eff
≡
a SC model with parameters w =1/3, g = 8 and different values of α =[1,2,2.4,3,5]. Initial conditions are chosen so
(in) − sc
thatatthepresenttimeτ =τ onehasx(τ )=1andξ(τ )=1. Noticethatforlargevaluesofthescalefactorw w for
0 0 0 eff → (in)
α =[1,2,2.4], whereas w 1 for α =[3,5], showing theexistence of a critical value of α (αcrit 2.455), as discussed
sc eff →− sc sc sc ≈
in thetext. For x 0, instead, w approaches thecommon valuew =1/3.
→ eff (in)
whichcan be at mosttwo,for fixed values ofthe parameters,asalreadydiscussedin Ref. [2]. The previous equation,
in fact, yields the condition
ξ w g
(in)
Fαsc(ξ∗)≡ (1−e−∗αscξ∗)2 = 2(1+w|(i|n)), (2.8)
forfixedvaluesofα andw >0. We willnotconsiderbelowthe caseofa scalarfieldwhichbehavesasa phantom
sc (in)
field in both low and high density regimes, i.e., w < 1. If 1 w 0, instead, the condition (2.7) cannot
(in) (in)
− − ≤ ≤
be satisfied, so that there exists the attractor solution ξ = 0 only. The function F (ξ) exhibits a minimum at
αsc
ξmin = 1 2W 1 +1 1.256/α , withvalueF (ξmin) 2.455/α , whereW (z)denotesthe branch
∗ −2αsc −1 −2√e ≈ sc αsc ∗ ≈ sc −1
of the Lambehrt W fu(cid:16)nction(cid:17)satisifying W(z) 1. The existence of two roots for Eq. (2.8) is thus guaranteed if
≤ −
2.455/α /w g /2(1+w ). Figure2showsthepairofattractorsolutionsξ =ξ foraselectedparameterchoice,
sc (in) (in)
only one of them|be|ing reachedduring the evolution, depending on the chosen initia∗l conditions. Therefore, for fixed
values of the parameters w and g, there exists a critical value of α which discriminates between the two possible
(in) sc
asymptotic states, i.e.,
2(1+w )
αcrit 2.455 (in) , (2.9)
sc ≈ w g
(in)
| |
such that ξ 0 for α <αcrit and ξ ξ for α >αcrit. For instance, for the choice of parameters w =1/3 and
g = 8 we g→et αcrit sc2.455s(csee Fig. →2). ∗The cosrcresposncding set of equilibrium solutions ξ =ξ for diffe(rine)nt values of
α i−s listed in Tascble≈I. ∗
sc
The late-time evolution of the effective equation of state parameter (2.6) is thus given by
g
w = w 1+ α2 ξ +O(ξ2), ξ 0,
eff (in) 2 sc →
(cid:16) gψ2 (cid:17)
weff = w(in) 1+ 2 ξ∗ +O(ξ−ξ∗), ξ →ξ∗, (2.10)
(cid:18) ∗(cid:19)
where ψ = ψ(ξ ), so that w w for ξ 0, while w 1 for ξ ξ . Therefore, the SC models evolve
eff (in) eff
towards∗a univer∗sefilled by anor→dinaryfluid wi→th barotropicequ→at−ionofstat→e p∗=w ρ in the formercase,whereas
(in)
in the latter case they approach a de Sitter universe with p = ρ = ρ . The stability properties of the attractor
solutions are discussed in Appendix A. − − ∗
4
(a) (b)
FIG.2: Panel(a). ThebehaviorofthefunctionFαsc (seeEq. (2.8))determiningtheattractorsolutionsξ =ξ∗ (seeEq. (2.8))
is shown for different values of α =[1,2,2.4,3,5]. The existence of two solutions is evident by drawing horizontal lines. The
sc
dashed line corresponds to the choice of parameters w(in) = 1/3, g = −8, leading to Fαsc = 1, so that two roots do exist for
αsc ' 2.455. Panel (b). The set of equilibrium solutions ξ = ξ∗ is shown as a function of αsc greater than the critical value
αcscrit ≈2.455 (dashed horizontal line). The solutions ξ∗ →0,1 are approached in thelimit αsc →∞.
B. Scalar field description
LettheSCfluidbedynamicallyimplementedbyahomogeneousscalarfieldφ(t)minimallycoupledtogravity,with
energy density ρ and pressure p related to the kinetic and potential energy in a standard way, i.e.,
φ˙2 φ˙2
ρ= +V, p= V, (2.11)
2 2 −
so that
1
V = (ρ p), φ˙2 =ρ+p. (2.12)
2 −
The field equations (2.2) imply that the dynamics of the scalar field is governed by the equation
φ¨+3Hφ˙ +V =0, (2.13)
′
where a prime denotes differentiation with respect to the scalar field φ.
In terms of the dimensionless variables (2.3) the potential takes the form
V 1
= ξ w (ξ) , (2.14)
ρ 2 − (in)P
(crit),0
(cid:0) (cid:1)
whereas the evolution equation for the scalar field reads as
dφ
κ = 3(ξ+w ). (2.15)
dτ ± (in)P
q
An example of numerical integration of the model equations (2.4) and (2.15) is shown in Figs. 3 and 4 for the same
choice of parameters as well as initial conditions as in Fig. 1.
Figure 3 shows the behavior of φ and φ˙2 as a function of the dimensionless scale factor x for selected values of the
parameters. The scalar field has two branches, each of them either extending to infinity (φ ) or approaching
a finite constant value (φ φ ) for large x, depending on the chosen value of α (for±fi→xed±∞w and g). The
∗ sc (in)
associated kinetic energy i±s a→lway±s positive in the entire domain and vanishes asymptotically.
Figure4shows,instead,theprofileV =V(φ)oftheSCpotential. Foreveryfixedvalueofα ,theleft(right)branch
sc
V (V ) corresponds to φ (φ ). Therefore, the potential either vanishes asymptotically (V 0 for φ ) or
+ +
tends−to a constant (V V −for φ φ ). ± → ± →±∞
∗
± → ∗ ± → ±
5
2T.A45B5L4EsuIc:hTthheatpaξi∗−r ofξe∗+quilib0.r5iu1m17.soNluottieonthsaξt=ξ ξ∗±ξ∗i+s ashtolwatneftoimr desiff.erent values of αsc greater than the critical value αcscrit ≈
≃ ≈ →
αsc ξ∗− ξ∗+
2.45540748 0.51169966 0.51169969
2.4555 0.50659526 0.51680963
2.456 0.49879508 0.52463974
2.46 0.47591246 0.54776068
2.5 0.40262914 0.62332204
3 0.19258836 0.84988278
3.5 0.12405732 0.92230203
4 0.08793621 0.95696028
4.5 0.06596326 0.97532713
5 0.05145050 0.98556820
5.5 0.04131315 0.99145133
6 0.03393271 0.99489482
6.5 0.02838350 0.99693497
7 0.02410150 0.99815336
7.5 0.02072581 0.99888485
8 0.01801619 0.99932556
9 0.01398286 0.99975265
10 0.01117017 0.99990912
15 0.00477222 0.99999939
C. Asymptotic profile of the SC potential
In order to find an explicit form of the potential when approaching the asymptotic regime, let us solve the SC
dynamical equations in this limit. They are given by
dξ 3w gψ2(ξ)
= λ2ξ 1+ (in) ,
dα − 2 λ2 ξ
(cid:20) (cid:21)
dφ 3w gψ2(ξ)
(in)
κ ± = λ 1+ , (2.16)
dα ± s 2 λ2 ξ
where α = lnx such that dα/dτ = √ξ and λ2 = 3(1+w ) > 0, since we are considering the case of a scalar field
(in)
which behaves as an ordinary field in both low and high density regimes only, so that w > 1. The special case
(in)
−
w = 1 will be treated separately.
(in)
−
According to the analysis done previously, if α < αcrit the energy density vanishes asymptotically (ξ 0), so
that ψ(ξ)=α ξ+O(ξ2), whereas the scalar fieldsics posistcively/negatively diverging with α. In fact, in this →limit Eq.
sc
(2.16) gives the equilibrium solution
ξ =0, κ(φ φ0)= λ(α α ), (2.17)
0
±− ± ± −
where one can set φ0 = 0 = α without any loss of generality. If α > αcrit, instead, both the energy density and
0 sc sc
the scalarfield tend±to a constantvalue (ξ ξ , φ φ ), so that ψ(ξ)=ψ +(1 ψ )α (ξ ξ )+O[(ξ ξ )2] and
∗ sc
the equilibrium solution is → ∗ → ± ∗ − ∗ − ∗ − ∗
ξ =ξ , φ =φ . (2.18)
∗
∗ ± ±
1. Case 1: α <αcrit
sc sc
Linear perturbations ξ =ǫξ and κφ = λα+ǫκφ of Eq. (2.16) around the equilibrium solution (2.17) give
(1) (1)
± ±
dξ dφ
(1) = λ2ξ , κ (1) = λσξ , (2.19)
dα − (1) dα ∓ (1)
6
(a) (b)
FIG. 3: The behavior of φ (in units of κ, panel (a)) and φ˙2 (in units of ρ , panel (b)) as a function of the dimensionless
(crit),0
scale factor x is shown for a SC model with the same choice of parameters as in Fig. 1. Initial conditions are chosen as in
Fig. 1, with in addition φ−(τ0) = φ+(τ0) = 5. The two branches φ+ (lower) and φ− (upper) asymptotically either diverge
−
crossing each other (φ± ) or approach a finite constant value for αsc ' 2.455. Notice that the labels refer to their
asymptotic character in t→he±di∞verging case. In particular, limx→∞φ± =φ∗± 4.47 for αsc =3, and φ∗± 4±.87 for αsc =5.
≈∓ ≈∓
The quantityφ˙2, instead, vanishesasymptotically for every fixed valueof α .
sc
(a) (b)
FIG. 4: The behavior of the SC potentials V± (in unitsof ρ(crit),0, panels (a) and (b), respectively) corresponding to the two
branches φ± (in units of κ) of the scalar field is shown for the same choice as well as initial conditions as in Fig. 3. Arrows
indicate thedirection of approach to theasymptotic regime.
where ǫ denotes a smallness indicator and
3w gα2
σ = (in) sc, (2.20)
−4 λ2
implying that
σ
ξ(1) =ξ¯e−λ2α, κφ(1) =±λξ(1). (2.21)
Inverting the relation φ=φ(α) to the lowest order then gives
ξ =ξ¯e κλφ±, (2.22)
∓
7
where ξ¯is an integration constant and we have set ǫ=1.
The abovefirstordersolutiondoes notdepend onthe SC parametersg andα . Therefore,itis worthgoingto the
sc
next order looking for solutions of the form ξ = ǫξ +ǫ2ξ and κφ = λα+ǫκφ +ǫ2κφ . Equations (2.16)
(1) (2) (1) (2)
then imply ± ±
dξ dφ σ
(2) = λ2 ξ 2σξ2 , κ (2) = λσ ξ α ξ2 . (2.23)
dα − (2)− (1) dα ∓ (2)− sc− 2 (1)
(cid:16) (cid:17) h (cid:16) (cid:17) i
Using the first order solution (2.21) we find
2α +3σ
ξ = 2σξ¯2e 2λ2α, κφ = sc ξ . (2.24)
(2) − − (2) ± 2λ (2)
Expressing α in terms of φ by inverting perturbatively the relation φ=φ(α) finally leads to
ξ =ξ¯e κλφ± 1 σξ¯e κλφ± . (2.25)
∓ ∓
−
Note that the solution(2.25) canalso be obtained by e(cid:0)xpanding up to(cid:1)the second order in ξ both equations (2.16),
which can be combined to yield
1 dξ
= λξ(1 σξ), (2.26)
κdφ ∓ −
±
with solution
1
ξ = 1ξ¯e±λκφ± +σ − . (2.27)
(cid:18) (cid:19)
Therefore, the profile of the potential
V 1
= ξ(1 w )+O(ξ2) (2.28)
ρ 2 − (in)
(crit),0
is of the form
V 1
V(φ)= 0 , B =σξ¯, V = ξ¯(1 w )ρ , (2.29)
eλκφ+B 0 2 − (in) (crit),0
where we have dropped the sign indicator, since the signs correlate with those of φ /φ which are posi-
+
tively/negatively increasing during the evolution. Very cl±ose to the attractor solution one can ne−glect the constant
term B, so that the SC potential is well approximated by a decreasing exponential, i.e.,
V(φ)=V e λκφ, (2.30)
0 −
whichcorrespondstothelowestordersolution(2.22). Therefore,inthisregimetheSCfluidexhibitsthesamefeatures
of an ordinary fluid with a barotropic equation of state p=w ρ with constant parameter w .
(in) (in)
2. Case 2: α >αcrit
sc sc
Linear perturbations ξ =ξ +ǫξ and φ =φ +ǫφ of Eq. (2.16) around the equilibrium solution give
(1) ∗ (1)
∗ ± ±
dξ dφ
(1) (1)
= Kξ , κ =0, (2.31)
dα − (1) dα
implying that
ξ ξ +ξ¯e Kα, φ φ , (2.32)
− ∗
≃ ∗ ± ≃ ±
where ξ¯is an integration constant and
K =λ2(1+2α ξ )+3w gα ψ , (2.33)
sc (in) sc
∗ ∗
8
FIG.5: ThebehaviorofthequantityK givenbyEq. (2.33)as afunction of w is shown for g= 8and differentvaluesof
(in) −
αsc =[2.5,3,4,5]. The values of ξ∗ corresponding to each pair (αsc,w(in)) are computed byEq. (2.8).
which has to be positive to ensure stability. Its behavior as a function of w is shown in Fig. 5 for g = 8 and
(in)
−
different values of α (for instance, for α = 3 and w = 1/3 we get ξ 0.85 and K 2.27). As a result, the
sc sc (in)
potential does not depend on φ, but is a function of α ∗ ≈ ≈
V K
ξ +ξ¯ 1+ e−Kα, (2.34)
ρ(crit),0 ≃ ∗ (cid:18) 6 (cid:19)
and the corresponding profile is
ξ¯ K
V(α)=V 1+qe Kα , q = 1+ , V =ξ ρ . (2.35)
∗ − ξ 6 ∗ ∗ (crit),0
∗ (cid:18) (cid:19)
(cid:0) (cid:1)
For very large values of the scale factor, one can neglect the exponential term, so that the SC potential is well
approximated by a constant, i.e., V(α)=V . Therefore, in this regime the SC fluid acts as a cosmological constant.
∗
3. Case 3: w = 1
(in) −
For w = 1 the dynamical equations (2.16) reduce to
(in)
−
dξ 3 dφ 3 ψ(ξ)
= g ψ2(ξ), κ ± = g . (2.36)
dα −2| | dα ± 2| | √ξ
r
The equation for ξ can be solved exactly (see also Ref. [3])
1 1
ξ = ln 1+ , y =W y eσ˜α+y0 , (2.37)
0
α y
sc (cid:18) (cid:19)
(cid:0) (cid:1)
where σ˜ = 3 g α and W(z) denotes the principal branch of the Lambert W function such that W(z)eW(z) = z.
2| | sc
Its asymptotic behavior for z is W(z) lnz, implying that for large values of the scale factor the solution
→ ∞ ∼
behaves as y σ˜α and α ξ 1/(σ˜α). Substituting then into the equation for φ = φ gives κφ 2√α, so that
sc
ξ 4/(σ˜α κ2∼φ2). Equivalent∼ly, the solution for ξ as a function of φ can be obtained b±y re-expre±ssi∼ng±Eq. (2.36) as
sc
∼
1 dξ 3
= g ξψ(ξ). (2.38)
κdφ ∓ 2| |
r
±
9
Close to the attractor solution ξ 0 the above equation becomes
→
1 dξ
σ˜α ξ3/2, (2.39)
sc
κdφ ≃∓
± p
with solution
2
1 1 −
ξ σ˜α κφ , (2.40)
≃" ξ¯± 2 sc ±#
p
where ξ¯is an integration constant such that ξ =pξ¯for φ=0. Therefore, the associated potential
V
=ξ+O(ξ2) (2.41)
ρ
(crit),0
is an inverse square function of φ at late times (where φ ), i.e.,
± →±∞
V 4
0
V(φ) , V = ρ . (2.42)
≃ (κφ)2 0 σ˜α (crit),0
sc
III. QUANTUM FEATURES
Letusstudynowthequantumfeaturesofthelate-timeevolutionofSCcosmologicalmodelsdescribedinSec. II.We
willfollowtheapproachofquantumgeometrodynamicsbasedontheWDWequation,whichreads Ψ(α,φ)=0,
(wdw)
L
where
~2 κ2
= ∂ ∂ +a6e6αV(φ), (3.1)
L(wdw) 2 6 αα− φφ 0
(cid:20) (cid:21)
with κ2 8πG and α ln(a/a ), as specified before. Hereafter, we shall set a = 1 and κ2 = 6, for simplicity.
0 0
≡ ≡
As a second order differential equation (of the hyperbolic type in minisuperspace applications), the WDW equation
requires two data at the Cauchy surface, i.e., the value of the function and its “time” derivative. A way to select
physically meaningful solutions for the wave function Ψ(α,φ) matching with quantum initial value data has been
recently proposed in Ref. [12]. At the semiclassical level, these solutions are represented by oscillating waves in
classically allowed domains of superspace, and exponential fall off in classically forbidden regions.
The WDW equation can be explicitly solved only for particular potentials or under some approximation. We will
adopt the BO approximation, briefly reviewed in Appendix B (see, e.g., Ref. [13]), so that
Ψ(α,φ)= φ (α,φ)C (α), (3.2)
k k
k
X
whereφ andC aretheslowandfastdegreesoffreedom,respectively,satisfyingEq. (B6). Sinceweareinterestedin
k k
the asymptotic behaviortowardsfuture attractorsolutions,wewill limit ourinvestigationsto approximatepotentials
in the neighborhood of such attractors. Therefore, we will consider below the two asymptotic SC potentials (2.29),
(2.35)and(2.42),correspondingtoCase1,Case2andCase3,respectively,discussedinSectionIIB.Veryclosetothe
attractor solutions the above potentials are well approximated by an exponential potential and a constant potential,
respectively, which allow to find exact solutions to the WDW equation, as recalled in Appendix C.
A. Case 1
Let us consider first the potential (2.29). In the BO approximation,the equation for φ reads
k
V e6α
~2φ′k′+2 Ek(α)− eλκ0φ+B φk =0, (3.3)
(cid:20) (cid:21)
which can be cast in the form of a hypergeometric equation
d2 d
z(1 z) +[c (a+b+1)z] ab f =0, (3.4)
− dz2 − dz − k
(cid:20) (cid:21)
10
through the transformation
φk =z−ek(α)/2(z 1)fk, z = Be−λκφ, (3.5)
− −
with
1 1
a=1 [b (α)+e (α)], b=1+ [b (α) e (α)], c=1 e (α)=a+b 1, (3.6)
k k k k k
− 2 2 − − −
and
2 2E (α) 8V e6α
e (α)= − k , b (α)2 =e (α)2+ 0 . (3.7)
k λκ~ k k λ2κ2~2B
p
Equation (3.4) has general solution
fk =c1(α)F(a,b,c;z)+c2(α)z1−cF(1+a c,1+b c,2 c;z). (3.8)
− − −
The latter can be considered as a special case of the confluent Heun function, already shown to play a role in this
context in Ref. [9], i.e.,
1 1
fk = c˜1(α)(z 1)−aHeunC 0,bk(α), ek(α),0, (bk(α)2+ek(α)2);
− − 4 1 z
(cid:18) − (cid:19)
1 1
+c˜ (α)(z 1) bHeunC 0, b (α), e (α),0, (b (α)2+e (α)2); . (3.9)
2 − k k k k
− − − 4 1 z
(cid:18) − (cid:19)
We are interested in normalizable solutions, so as to obtain a discrete spectrum. Therefore, we will consider only
thosesolutionssuchthatthehypergeometricfunctionsreducetoorthogonalpolynomials,includingJacobipolynomials
P(γ,δ) and their special cases (Legendre or Chebyshev or Gegenbauer polynomials), by using the relation
k
k!
F( k,γ+δ+1+k,γ+1;z)= P(γ,δ)(1 2z), (3.10)
− (γ+1) k −
k
where (m) (m+n 1)!/(m 1)! is a Pochhammer symbol. They satisfy the orthogonality condition
n
≡ − −
1 2γ+δ+1 Γ(k+γ+1)Γ(k+δ+1)
(1 x)γ(1+x)δP(γ,δ)(x)P(γ,δ)(x)dx= δ , (3.11)
− j k 2k+γ+δ+1 Γ(k+γ+δ+1)k! jk
Z−1
with γ,δ,γ+δ > 1, and are normalized in such a way that
−
k+γ
P(γ,δ)(1)= . (3.12)
k k
(cid:18) (cid:19)
Therefore, in the limit of large φ (implying z 0) the functions f chosen as the Jacobi polynomials of Eq. (3.10)
k
→
take a constant value. Equation (3.10) then implies
2V e6α
0
γ = e (α), δ =1, 2(k+1)=e (α) b (α), e (α)=k+1 , (3.13)
− k k − k k − (k+1)λ2κ2~2B
and the equation for the functions C (α) becomes
k
d2 1
λ2κ2e (α)2 C =0. (3.14)
dα2 − 4 k k
(cid:20) (cid:21)
This equationcan be solvedin terms ofthe Whittaker functions of the firstand secondkind M (x) and W (x) as
µ,ν µ,ν
C =e 3α[c M (x )+c W (x )], (3.15)
k − 1 µ,ν k 2 µ,ν k
with
1 V e6α
0
µ= (k+1)λκ=ν, x = . (3.16)
12 k 3(k+1)λκ~2B