Possible antigravity regions in $F(R)$ theory?

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by  Kazuharu Bamba

Possible antigravity regions in F(R) theory?
Kazuharu Bamba1,∗, Shin’ichi Nojiri1,2,†, Sergei D. Odintsov2,3,4,5‡ and Diego Sa´ez-G´omez6,7,§
1Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 464-8602, Japan
2Department of Physics, Nagoya University, Nagoya 464-8602, Japan
3Instituci`o Catalana de Recerca i Estudis Avanc¸ats (ICREA), Barcelona, Spain
4Institut de Ciencies de l’Espai (CSIC-IEEC), Campus UAB,
Facultat de Ciencies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain
5 Tomsk State Pedagogical University, Kievskaya Avenue, 60, 634061, Tomsk, Russia
6Astrophysics, Cosmology and Gravity Centre (ACGC) and
Department of Mathematics and Applied Mathematics,
University of Cape Town, Rondebosch 7701, Cape Town, South Africa
7Fisika Teorikoaren eta Zientziaren Historia Saila, Zientzia eta Teknologia Fakultatea,
Euskal Herriko Unibertsitatea, 644 Posta Kutxatila, 48080 Bilbao, Spain
4
1 WeconstructanF(R)gravitytheorycorrespondingtotheWeylinvarianttwoscalarfieldtheory.
0 WeinvestigatewhethersuchF(R)gravitycanhavetheantigravityregionswheretheWeylcurvature
2 invariant does not diverge at the Big Bang and Big Crunch singularities. It is revealed that the
n divergence cannot be evaded completely but can be much milder than that in the original Weyl
a invariant two scalar field theory.
J
7 PACSnumbers: 11.30.-j,04.50.Kd,95.36.+x,98.80.-k
2

] I. INTRODUCTION
h
t
- Recentobservations[1–4]including Type Ia Supernovae[5] havesuggestedthe currentcosmic expansionis acceler-
p
e ating. For the universe to be strictly homogeneousand isotropic,there are two major approaches: To introduce dark
h energy within generalrelativity (for reviews,see [6]) and to modify gravityon large distances (for recent reviews, see
[ [7]) Furthermore, it was realized that modified gravity can describe dark energy [8, 9] and also unify dark energy era

with early-time cosmic acceleration [10].
3
v Theoreticalfeatures of such modified gravity theories themselves become important concerns in the literature. For
8 instance, the scale invariance in inflationary cosmology [11, 12] or cyclic cosmologies with the Weyl invariant scalar
2 fields [13]1 have recently been studied. On the other hand, the cosmological transition from gravity to antigravity
3 has been examined in various background space-time including the strictly homogeneous and isotropic Friedmann-
1 Lemaˆıtre-Robertson-Walker (FLRW) universe [15–19]2. Moreover, in Refs. [21–24], it has been explored that in
.
1 extended theories of general relativity with the Weyl invariance (or conformal invariance), antigravity regimes have
0 to be included. Very recently, it has been verified in Ref. [25] that the Weyl invariant becomes infinite at both the
4
Big Bang (Big Crunch) singularity appearing at the transition from antigravity (gravity) and gravity (antigravity).
1
In this Letter, we reconstruct an F(R) gravity theory corresponding to the Weyl invariant two scalar field theory.
:
v OuroriginalmotivationistodemonstratethattheWeylinvarianttwoscalarfieldtheorycanbereformulatedinterms
i of F(R) gravity (see, for instance, Ref. [26]). In addition, we examine whether the F(R) gravity can pass through
X
the antigravity regions. We use units of k = c = ~ = 1, where c is the speed of light, and denote the gravitational
B
r
a constant 8πG by κ2 8π/M 2 with the Planck mass of M =G−1/2 =1.2 1019 GeV. We also adopt the metric
N ≡ Pl Pl N ×
signature diag( ,+,+,+).
−
The Letter is organized as follows. In Sec. II, we introduce the Weyl invariant scalar theory and present it as the
correspondingF(R)gravitytheory. InSec.III,weexplorehowthecorrespondingF(R)gravitytheoryobtainedabove
∗ E-mailaddress: [email protected]
† E-mailaddress: [email protected]
‡ E-mailaddress: [email protected]
§ E-mailaddress: [email protected]
1 FortheearlyuniversecosmologyinthecaseoftwoscalarfieldsnottheWeylinvariantlycoupledtothescalarcurvature,see[14].
2 We remark that generally speaking the antigravity regime is possible in F(R) when its first derivative is negative. Of course, it leads
to number of unpleasant consequences like the possibility of only static universe due to the change of gravitational coupling constant
sign in the FLRW equations. Hence, such possibility seems to be rather speculative one which may occur somewhere before the Big
Bang. Onecanspeculate thattheBigBangitselfisthetransitionpointfromantigravitytogravityregimesasduetopassingthrough
zeroofgravitationalcouplingconstantsomesingularitymaybeexpected. Also,notethatevencurrentlysomevariationofgravitational
couplingconstantmaybeexpected asdiscussedinrecentRef.[20].
2
can be connected with antigravity regions. In Sec. IV, some conclusions are presented.
II. THE WEYL TRANSFORMATION IN F(R) GRAVITY
A. The Weyl invariant scalar field theory
An action for the Weyl invariant scalar field theory in the presence of matter is given by [27]
ω
S = d4x√ g ωf(φ)R gµν φ φ V(φ) + d4x (g ,Ψ ) , (2.1)
µ ν M µν M
− − − 2 ∇ ∇ − L
Z (cid:16) (cid:17) Z
with
1 1
f(φ) = ξφ2, ξ = , (2.2)
2 6
ω = 1, (2.3)
±
λ
V(φ) = φ4, λ=1. (2.4)
4
Here, R is the scalar curvature, g is the determinant of the metric g , is the covariant derivative operator
µν µ
∇
associated with g (for its operation on a scalar field, φ = ∂ φ), f(φ) is a non-minimal gravitational coupling
µν µ µ
∇
termof φ, ω =+1( 1)is the coefficientof kinetic termofthe canonical(non-canonicalscalar)fieldφ, ξ is a constant
−
determining whether the theory respects the Weyl invariance, V(φ) is the potential for a scalar field φ, and λ is a
constant. ξ is dimensionless and φ has the [mass] dimension. Moreover, is the matter Lagrangian, where Ψ
M M
L
denotesallthematterfieldssuchasthoseinthestandardmodelofparticlephysics(anditdoesnotinclude thescalar
field φ).
It is significant to remark that since the scalar curvature R is represented as R = T +2 µTρ [28], where
Tρ is the torsion tensor and T is the torsion scalar in telleparalelism [28, 29], F(R−) gravity∇is coµnρsidered to be
equµiρvalenttoF(T +2 µTρ ), andthat the Weylinvariantscalarfieldtheory couplingto(cid:0)the scalarcurv(cid:1)atureis also
∇ µρ
equivalent to that with its coupling to the torsion scalar [30].
B. The Weyl transformation
If the Weyl transformation in terms of the action in Eq. (2.1) is made as g gˆ = Ω2g , where Ω f(φ),
µν µν µν
→ ≡
the action in the so-called Jordan frame can be transformed into that in the Einstein frame [31, 32]. Here, the hat
p
denotes quantities in the Einstein frame for the present case. On the other hand, it is known that a non-minimal
scalar field theory corresponding to an F(R) gravity theory is the Brans-Dicke theory [33] which has the potential
term and does not the kinetic term, i.e., the Brans-Dicke parameter ω =0.
BD
We examine an F(R) gravity theory corresponding to the Weyl invariant scalar field theory. We now consider the
following action given by Eq. (2.1) with ω = 1 and without the matter part
−
φ2 1 λ
S = d4x√ g R+ gµν∂ φ∂ φ φ4 . (2.5)
µ ν
− 12 2 − 4
Z (cid:20) (cid:21)
First looking this action, one may think the field φ is ghost since the kinetic term is not canonical. We can, how-
ever, remove the ghost because the action (2.5) is invariant under the Weyl transformation. By using the Weyl
transformation, we may fix the scalar field φ to be a constant,
6
φ2 = . (2.6)
κ2
Then we obtain the action of the Einstein gravity with cosmologicalconstant:
1 9λ
S = d4x√ g R . (2.7)
− 2κ2 − κ4
Z (cid:20) (cid:21)
The action (2.7) can be also reproduced by using the scale transformation g = 6 gˆ . In this case, the scalar
µν κ2φ2 µν
curvature is transformed as (cid:16) (cid:17)
φ2 6(cid:3)ˆφ 12gˆµν∂ φ∂ φ
R= Rˆ+ µ ν , (2.8)
6κ2 φ − φ2 !
3
and hence the action (2.5) is represented as
1 6(cid:3)ˆφ 12gˆµν∂ φ∂ φ 6 18λ
S = d4x gˆ Rˆ+ µ ν + gˆµν∂ φ∂ φ
2κ2 − " φ − φ2 φ2 µ ν − κ2 #
Z
p
1 9λ
= d4x gˆ Rˆ . (2.9)
− 2κ2 − κ4
Z (cid:20) (cid:21)
p
Thus, the correspondingF(R) gravitytheory is F(R)=(R 2Λ)/ 2κ2 with Λ 9λ/κ2 as in Eq.(2.9), that is, the
− ≡
Einstein-Hilbert action including cosmologicalconstant.
(cid:0) (cid:1)
We mention that it is meaningful to explore the reason why the corresponding F(R) gravity theory in Eq. (2.9),
into which the Weyl invariant scalar field theory is transformed, has no Weyl invariance. This is because that when
we write gˆ as gˆ = κ2φ2 g , the theory is trivially invariant under the Weyl transformation: φ Ω−1φ and
µν µν 6 µν →
g Ω2g . (cid:16) (cid:17)
µν µν
→
III. CONNECTION WITH ANTIGRAVITY
A. The Weyl invariantly coupled two scalar field theory
We investigate the Weyl invariantly coupled two scalar field theory. This was first proposed in Ref. [34] and
cosmology in it was explored in Ref. [35]. Recently, the connection with the region of antigravity has also been
examined in Ref. [25]. The action is described as [21–23, 25, 34–36]3
φ2 u2 1
S = d4x√ g − R+ gµν(∂ φ∂ φ ∂ u∂ u) φ4J(u/φ)
µ ν µ ν
Z − "(cid:0) 12 (cid:1) 2 − − #
+ d4x (g ,Ψ ) , (3.1)
M µν M
L
Z
where u is anotherscalarfield andJ is a function ofa quantity u/φ. The importantpoint is that this actionrespects
the Weyl symmetry, even though the coefficient of the kinetic term for the scalar field φ is a wrong sign, namely, φ is
not the canonical scalar field in this action. Indeed, this action is invariant under the Weyl transformations φ Ωφ,
u Ωu, and g Ω−2g . This implies that there does not exist any ghost. →
µν µν
→ →
B. Representation as single scalar field theory with its Weyl invariant coupling
We rewritethe actioninEq.(3.1)withtwoscalarfieldstothe onedescribedbysinglescalarfieldthroughthe Weyl
transformation.
For the action (3.1) without the matter part, given by
φ2 u2 1
S = d4x√ g − R+ gµν(∂ φ∂ φ ∂ u∂ u) φ4J(u/φ) , (3.2)
µ ν µ ν
Z − "(cid:0) 12 (cid:1) 2 − − #
we may consider the Weyl transformationg =φ−2gˆ . The scalar curvature is transformed as
µν µν
6(cid:3)ˆφ 12gˆµν∂ φ∂ φ
R=φ2 Rˆ+ µ ν . (3.3)
φ − φ2 !
Accordingly, the action (3.2) is reduced to
1 u2 u2 (cid:3)ˆφ gˆµν∂ φ∂ φ 1
S = d4x gˆ 1 Rˆ+ 1 µ ν + gˆµν(∂ φ∂ φ ∂ u∂ u) J(u/φ)
Z − "12(cid:18) − φ2(cid:19) (cid:18) − φ2(cid:19) 2φ − φ2 ! 2φ2 µ ν − µ ν − #
p
3 Notethattheactionofsuchasortassumingphantom-likekinetictermforumaybeobtainedfrommoregeneralnon-conformaltheory
duetotheasymptotical conformalinvariance[37,38]. Thisphenomenon oftenoccursinasymptotically-freetheories.
4
1 u2 1 u u
= d4x gˆ 1 Rˆ gˆµν∂ ∂ J(u/φ) . (3.4)
− 12 − φ2 − 2 µ φ ν φ −
Z (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
p
Therefore if we define a new scalar field ϕ u/φ, the action has the following form:
≡
1 1
S = d4x gˆ 1 ϕ2 Rˆ gˆµν∂ ϕ∂ ϕ J(ϕ) . (3.5)
µ ν
− 12 − − 2 −
Z (cid:20) (cid:21)
p (cid:0) (cid:1)
The obtained action has no Weyl invariance because gˆ and ϕ are invariant under the Weyl transformation. The
µν
Weyl invariance appears because we write gˆ = φ2g and ϕ = u/φ. Therefore the Weyl invariance is artificial or
µν µν
fake, or hidden local symmetry. Conversely even in an arbitrary F(R) gravity,if we write the metric as g =φ2g˜ ,
µν µν
there always appears the Weyl invariance.
C. Corresponding F(R) gravity
We may relate the action (3.5) with F(R) gravity. By the further Weyl transforming the metric as gˆ =eη(ϕ)g¯
µν µν
with a function η(ϕ), we rewrite the action (3.5) in the following form:
eη(ϕ) eη(ϕ)
S = d4x√ g¯ 1 ϕ2 R¯ g¯µν 1+2ϕ2 η′(ϕ)2 4η′(ϕ) 4 ∂ ϕ∂ ϕ e2η(ϕ)J(ϕ) , (3.6)
µ ν
− 12 − − 8 − − −
Z (cid:20) (cid:21)
(cid:0) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1)
where the prime means the derivative with respect to ϕ, and the bar shows the quantities after the above Weyl
transformation. Then if choose η(ϕ) by
2ϕ 2 3ϕ2+1
1+2ϕ2 η′(ϕ)2 4η′(ϕ) 4=0, that is η′(ϕ)= ± , (3.7)
− − 1+2ϕ2
p
(cid:0) (cid:1)
the kinetic term of ϕ vanishes and we obtain
eη(ϕ)
S = d4x√ g¯ 1 ϕ2 R¯ e2η(ϕ)J(ϕ) . (3.8)
− 12 − −
Z (cid:20) (cid:21)
(cid:0) (cid:1)
Then by the variation of the action with respect to ϕ, we obtain an algebraic equation, which can be solved with
respect to ϕ as a function of R¯, ϕ = ϕ(R¯). Then by substituting the expression into the action (3.8), we obtain an
F(R) gravity:
S = d4x√ g¯F(R¯), F(R¯)= eη(ϕ(R¯)) 1 ϕ R¯ 2 R¯ e2η(ϕ(R¯))J ϕ R¯ . (3.9)
− 12 − −
Z (cid:16) (cid:0) (cid:1) (cid:17) (cid:0) (cid:0) (cid:1)(cid:1)
D. Finite-time future singularities
Now let us examine the reconstructionof the above model when a singular flat FLRW cosmology is considered. In
this background, the metric is given by ds2 = dt2+a(t)2 3 dxi 2, where a(t) the scale factor. Depending on
− i=1
thenatureofthe singularity,aclassificationoffinite-timefuturesingularitiesintheFLRWcosmologieswaspresented
P (cid:0) (cid:1)
in Ref. [39] as follows.
Type I (“Big Rip”): For t t , a and ρ , P .
s
• → →∞ →∞ | |→∞
Type II (“Sudden”): For t t , a a and ρ ρ , P .
s s s
• → → → | |→∞
Type III: For t t , a a and ρ , P .
s s
• → → →∞ | |→∞
Type IV: For t t , a a and ρ ρ , P P but higher derivatives of the Hubble parameter diverge.
s s s s
• → → → →
Here, ρ and P are the energy density and pressure of the universe, respectively. We might now study a simple case,
where the Hubble parameter H a˙/a is described by
≡
α
H = . (3.10)
t t
s
−
5
This solution describes a Big Rip singularity that occurs in a time t . Then, by the F(R) FLRW equations, the
s
corresponding action (3.9) with the matter action can be reconstructed as
1 RF F
H2 = κ2ρ + R− 3HR˙F ,
M RR
3F 2 −
R (cid:20) (cid:21)
1 1
3H2 2H˙ = κ2p +R˙2F +2HR˙F +R¨F + (F RF ) , (3.11)
M RRR RR RR R
− − F 2 −
R (cid:20) (cid:21)
where the subscripts correspondto derivativeswithrespectto R, andρ andp arethe energydensity andpressure
M M
of all the matters, respectively.
For the solution (3.10), it is straightforwardto check that the F(R) function yields
1 3n+2n2
F(R)=Rn , where − =α , (3.12)
n 2
−
with n a constant Then, by (3.9) the corresponding scalar-tensortheory is obtained as
eη(ϕ) ∂F
1 ϕ2 = =nRn−1
12 − ∂R
(cid:0) (cid:1) ∂F
e2η(ϕ)J(ϕ) = R F(R)=(n 1)Rn . (3.13)
∂R − −
Thus, the cosmological evolution for the scalar field ϕ is obtained as well as its self-interacting term J(ϕ), such that
the corresponding action is obtained. Note that in such a case, the antigravity regime is never crossed, since R > 0
for (3.10) which leads to ϕ < 1. Nevertheless, for other kind of singular solutions within the FLRW metrics, the
| |
antigravity regime might be expected.
E. Connection with antigravity
It is clear from the action of a scalar field theory in Eq. (3.5) that if ϕ2 >1, there emerges antigravity. When this
conditionissatisfied,itfollowsfromtheformofthecorrespondingF(R)gravityinEq.(3.9)thatthecoefficientofR¯can
bepositiveas eη(ϕ(R¯))/12 1 ϕ R¯ 2 <0,andthusantigravitycanappear. Inotherwords,theeffectiveNewton
−
couplinginthehactionoftheic(cid:16)orrespo(cid:0)nd(cid:1)in(cid:17)gF(R)gravitytheoryinEq.(3.9)isdescribedasG¯N 6e−η(ϕ)GN/ 1 ϕ2 .
≡ −
Accordingly, when ϕ= 1 and ϕ=+1, there happens transitions between gravity and antigravity.
− (cid:0) (cid:1)
We investigate what happens in the travel to the antigravity region for the F(R) gravity theory in Eq. (3.9)
corresponding to the Weyl invariantly coupled two scalar field theory in Eq. (3.2). By following the procedures in
Ref.[11],weexplorethebehaviorsofsolutionsintheanisotropicbackgroundmetricsothathomogeneousandisotropic
solutionsshouldnotbesingulararoundtheboundarybetweengravityandantigravityregions(forthedetailedanalysis
on homogeneous and isotropic solutions in non-minimally coupled scalar field theories, see, e.g., [40]).
Provided that the background metric of the space-time is expressed as [41] ds2 = a2(τ) dτ2+ 3 eβidx2
− i=1 i
with β1 2/3α1(τ)+√2α2(τ), β2 2/3α1(τ) √2α2(τ), and β3 2 2/3α1(τ), wh(cid:16)ere τ is tPhe conforma(cid:17)l
≡ ≡ − ≡ −
time, and an anisotropy function α (i = 1, ,3) only depends on τ. In the following, the so-called γ gauge of
i
p p ··· p
g detg =1 [21]. Moreover,we take into account the existence of radiation. In this gauge, φ and u are given
µν
− ≡−
by [22]
√ τ q 2 q τ τ −q
φ = C −q + |C| q , (3.14)
|C|q (cid:12)√6(cid:12) A|A| ! √C √6(cid:12)√6(cid:12) |A| !
(cid:12) (cid:12) (cid:12) (cid:12)
√ (cid:12) τ (cid:12)q 2 q τ (cid:12) τ (cid:12)−q
u = C (cid:12) (cid:12) −q |C| (cid:12) (cid:12) q , (3.15)
|C|q (cid:12)√6(cid:12) A|A| !− √C √6(cid:12)√6(cid:12) |A| !
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
with (cid:12) (cid:12) (cid:12) (cid:12)
ρ ρ
r r
p+ τ = (τ τ ) , (3.16)
BC
A ≡ √6 √6 −
√6p
τ , (3.17)
BC
≡ − ρ
r
6
1 p
σ
q 1+ , (3.18)
≡ 2 p2 +p2+p2!
σ 1 2
p p2 +pp2+p2, (3.19)
≡ σ 1 2
q
where is a constant, ρ is a constantoriginating from the existence of radiation,and (p , p , p ) are constants (the
r σ 1 2
C
casep =p =0is notconsidered,becauseinthatcaseα andα becomesconstants). Inthe limitofτ 0,namely,
1 2 1 2
→
the Big Bang singularity, τ/√6 0, while in the limit of τ τ , namely, the Big Crunch singularity, 0.
BC
→ → A →
Furthermore, from Eq. (3.18) we find 0 q 1.
≤ ≤
For the F(R) gravity theory whose action is given by Eq. (3.9), the Weyl curvature invariant is considered to be
eη −2
= 1 ϕ2 C Cµνρσ, (3.20)
µνρσ
I 6 −
(cid:20) (cid:21)
(cid:0) (cid:1)
where Cµ is the Weyl curvaturetensor. As a consequence,weacquire =243e−2ηΥ τ/√6 δ1 δ2 with δ <0and
νρσ I A 1
δ <0,where Υ is a function ofseveralvariablesas Υ=Υ(τ/√6,p,p ,p ,ρ ) [11]. Thus, atthe Big Bang singularity
2 1 2 r (cid:0) (cid:1)
we obtain because of δ < 0, whereas at the Big Crunch singularity we have owing to
I|τ→0 → ∞ 1 I|τ→τBC → ∞
δ <0.
2
It is worthy to emphasize that for the originalWeyl invariantly coupled two scalar field theory [25] whose action is
given by Eq. (3.1), the power of τ/√6 and that of (τ τ ), to which is proportional, are equal to “ 6”, while
BC
− A −
for the present F(R) gravity theory, 6 < δ < 0 and 6 < δ < 0, that is, how singular is can be much milder
1 2
(cid:0) −(cid:1) − I
than that in the original two scalar field theory with those Weyl invariant couplings.
In addition, it is interesting to mention that in Ref. [24], the following counter-discussions to the statements of
Ref. [25] have been presented. For a geodesically complete universe, it is necessary to match the values of all the
physical quantities including the divergent curvatures with continuous geodesics in the two regions, not to prevent
the divergence of the curvature at the transition point4. This has been demonstrated through the identification of
conserved quantities across the transition [24] and it is not specific but generic consequence. Adopting this point of
view, the transitionthroughantigravityregionin the Weyl invariantscalarfield theory (as wellas in the aboveF(R)
theory seems to be possible.
IV. CONCLUSIONS
In the present Letter, we have performed the reconstruction of an F(R) gravity theory corresponding to the
Weyl invariant two scalar field theory. We have also demonstrated how the F(R) gravity theory cannot connect
with antigravity region in order for the Weyl invariant to be finite at the Big Bang and Big Crunch singularities.
Nevertheless, the Weyl invariant divergence at these singularities can be much milder than that in the original
Weyl invariantly invariant two scalar field theory. It would be very interesting to investigate this problem for F(R)
bigravity theories [42] where the above phenomena could qualitatively be different due to possible exchanges of
gravity-antigravityregions between g and f F(R) gravities.
Finally,wementionthewayfortheenergyconditionstobemetinourmodelbyfollowingthediscussionsinRef.[44],
where a novelformulationto dealwith additionaldegreesof freedomappearingin extendedgravitytheorieshas been
made. In this work, we have examined the Weyl invariant (two) scalar field theories. These can be categorized
to non-minimal scalar field theories such as the Brans-Dicke theory [33], into which F(R) gravity theories can be
transformed via the conformal transformation. According to the consequences found in Ref. [44], the four (i.e., null,
dominant, strong, weak) energy conditions can be described as in general relativity, although the physical meanings
become different from those in general relativity. This is because the properties of gravity interactions as well as the
geodesic and causal structures in modified gravity would be changed from those in general relativity. Thus, these
differences are considered to be significant when it is examined whether extended theories of gravity can pass the
solar-systemtests and cosmological constraints.
4 Notethatinthesamespirit,thepossibilitytocontinuetheuniverseevolutionthroughthemildfinite-timefuturesingularitieslikeType
IVsingularityseemstoexistasgeodesics maybecontinuedthroughthesingularity.
7
Acknowledgments
S.D.O. acknowledges the Japan Society for the Promotion of Science (JSPS) Short Term Visitor ProgramS-13131
and the very hearty hospitality at Nagoya University, where the work was developed. The work is supported in part
by the JSPS Grant-in-Aid for Young Scientists (B) # 25800136 (K.B.); that for Scientific Research (S) # 22224003
and(C)#23540296(S.N.);andMINECO(Spain),FIS2010-15640andAGAUR(GeneralitatdeCatalunya),contract
2009SGR-345, and MES project 2.1839.2011 (Russia) (S.D.O.). D. S.-G. also acknowledges the support from the
University of the Basque Country, Project Consolider CPAN Bo. CSD2007-00042 (Spain) and the URC financial
support from the University of Cape Town (South Africa).
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