Eur. Phys. J. C (2016) 76:629
DOI 10.1140/epjc/s10052-016-4491-0
Regular Article - Theoretical Physics
Noether symmetries in Gauss–Bonnet-teleparallel cosmology
Salvatore Capozziello
1,2,3,4,a
, Mariafelicia De Laurentis
2,4,5,6,b
, Konstantinos F. Dialektopoulos
1,2,c
1
Dipartimento di Fisica “E. Pancini”, Universita’ di Napoli“Federico II”, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia,
80126 Napoli, Italy
2
INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Napoli, Italy
3
Gran Sasso Science Institute (INFN), Via F. Crispi 7, 67100 L’ Aquila, Italy
4
Tomsk State Pedagogical University, 634061 Tomsk, Russia
5
Institute for Theoretical Physics, Goethe University, Max-von-Laue-Str. 1, 60438 Frankfurt, Germany
6
Laboratory of Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia
Received: 30 September 2016 / Accepted: 7 November 2016 / Published online: 18 November 2016
© The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract A generalized teleparallel cosmological model,
f (T
G
, T ), containing the torsion scalar T and the teleparallel
counterpart of the Gauss–Bonnet topological invariant T
G
,is
studied in the framework of the Noether symmetry approach.
As f (G, R) gravity, where G is the Gauss–Bonnet topological
invariant and R is the Ricci curvature scalar, exhausts all
the curvature information that one can construct from the
Riemann tensor, in the same way, f (T
G
, T ) contains all the
possible information directly related to the torsion tensor. In
this paper, we discuss how the Noether symmetry approach
allows one to fix the form of the function f (T
G
, T ) and to
derive exact cosmological solutions.
1 Introduction
Extended theories of gravity are semi-classical approaches
where the effective gravitational Lagrangian is modified,
with respect to the Hilbert–Einstein one, by considering
higher-order terms of curvature invariants, torsion tensor,
derivatives of curvature invariants and scalar fields (see for
example [1–4]). In particular, taking into account the Ricci,
Riemann, and Weyl invariants, one can construct terms like
R
2
, R
μν
R
μν
, R
μνδσ
R
μνδσ
, W
μνδσ
W
μνδσ
, that give rise to
fourth-order theories in the metric formalism [5,6]. Consid-
ering minimally or nonminimally coupled scalar fields to
the geometry, we deal with scalar–tensor theories of gravity
[7,8]. Considering terms like RR, R
k
R, we are dealing
with higher-than fourth-order theories [9,10]. f (R) gravity
is the simplest class of these models where a generic func-
a
e-mail: capozziello@na.infn.it
b
e-mail: laurentis@th.physik.uni-frankfurt.de
c
e-mail: dialektopoulos@na.infn.it
tion of the Ricci scalar R is considered. The interest for these
extended models is related both to the problem of quantum
gravity [2] and to the possibility to explain the accelerated
expansion of the universe, as well as the structure forma-
tion, without invoking new particles in the matter/energy
content of the universe [4–15]. In other words, the attempt
is to address the dark side of the universe by changing the
geometric sector and remaining unaltered the matter sources
with respect to the Standard Model of particles. However, in
the framework of this “geometric picture”, the debate is very
broad involving the fundamental structures of gravitational
interaction. Just to summarize some points, gravity could be
described only by metric (in this case we deal with a met-
ric approach), or by metric and connections (in this case,
we are considering a metric-affine approach [16]), or by a
purely affine approach [17]. Furthermore, dynamics could
be related to curvature tensor, as in the original Einstein the-
ory, to both curvature and torsion [18], or to torsion only, as
in the so-called teleparallel gravity [19].
Starting from these original theories and motivations, one
can build more complex Lagrangians, by using different
combinations of curvature scalars and their derivatives, or
topological invariants, such us the Gauss–Bonnet term, G,as
well as the torsion scalar T . Many theories have been pro-
posed considering generic functions of such terms, like f (G),
f (T ), f (R, G), and f (R, T ) [20–44]. However, the problem
is how many and what kind of geometric invariants can be
used, and, furthermore, what kind of physical information
one can derive from them. For example, it is well known that
f (R) gravity is the straightforward extension of the Hilbert–
Einstein case which is f (R) = R, and f (T ) is the extension
of teleparallel gravity which is f (T ) = T . However, if one
wants to consider the whole information contained in the cur-
vature invariants, one has to take into account also combina-
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629 Page 2 of 6 Eur. Phys. J. C (2016) 76 :629
tions of Riemann, Ricci, and Weyl tensors.
1
As discussed in
[26], assuming a f (R, G theory means to consider the whole
curvature budget and then all the degrees of freedom related
to curvature.
Assuming the teleparallel formalism, a f (T
G
, T ) theory,
where T
G
is the torsional counterpart of the Gauss–Bonnet
topological invariant, means to exhaust all the degrees of
freedom related to torsion and then completely extend f (T )
gravity. It is important to stress that the Gauss–Bonnet invari-
ant derived from curvature differs from the same topological
invariant derived from torsion in less than a total derivative,
as we will show below, and then the dynamical information
is the same in both representations. According to this result,
the topological invariant allows a regularization of dynam-
ics also in the teleparallel torsion picture (see [26,45]fora
discussion in the curvature representation).
The layout of the paper is the following. In Sect. 2,we
sketch the basic ingredients of the f (T
G
, T ) theory showing,
in particular, the equivalence between T
G
and G. Section 3 is
devoted to a derivation of the cosmological counterpart of the
theory and to the derivation of the Noether symmetry. The
specific forms of f (T
G
, T ) function, selected by the Noether
symmetry, are discussed in Sect. 4. Cosmological solutions
are given in Sect. 5. Conclusions are drawn in Sect. 6.
2 f (T
G
, T ) gravity
In order to incorporate spin in a geometric description, as
well as to bring gravity closer to its gauge formulation,
people started, some years ago, to study torsion in gravity
[18,19]. An extensive review of torsional theories (teleparal-
lel, Einstein–Cartan, metric-affine, etc.) is presented in [1].
If in considering the action of the teleparallel theory, i.e. in
a curvature-free vierbein formulation, we replace the torsion
scalar, T , with a generic function of it, we obtain the so-called
f (T ) gravity [46–49],
In this paper, we will study a theory whose Lagrangian
is a generic function of the Gauss–Bonnet-teleparallel term,
T
G
and the torsion scalar, T , i.e.
A =
1
2κ
d
4
x
√
−g
f (T
G
, T ) + L
m
, (1)
which is a straightforward generalization of
A =
1
2κ
d
4
x
√
−g
[
f (T ) + L
m
]
, (2)
where L
m
is the standard matter that, in the following consid-
erations, we will discard. It is important to note that the field
equations of f (T ) gravity are of second order in the met-
1
Clearly, this means that we are not considering higher-order derivative
terms like R, or derivative combinations of curvature invariants.
ric derivatives and thus simpler than those of f (R) gravity,
which are of fourth order [1].
The metric determinant
√
−g can be derived from the
determinant of the vierbeins h as follows. We have
h
μ
i
h
i
ν
= δ
μ
ν
, h
μ
i
h
j
μ
= δ
i
j
. (3)
The relation between metric and vierbeins is given by
g
μν
= η
ab
h
a
μ
h
b
ν
, (4)
where η
ab
is the flat Minkowski metric. Finally, it is |h|≡
det
h
i
μ
=
√
−g. More details on how the two formalisms
are related can be found in [29].
The torsion scalar is given by the contraction
T = S
μν
ρ
T
ρ
μν
(5)
where
S
ρ
μν
=
1
2
K
μν
ρ
+ δ
μ
ρ
T
σν
σ
− δ
ν
ρ
T
σμ
σ
, (6)
K
μν
ρ
=−
1
2
T
μν
ρ
− T
νμ
ρ
− T
ρ
μν
, (7)
T
α
μν
=
α
μν
−
˜
α
μν
, (8)
are, respectively, the superpotential, the contorsion tensor,
the torsion tensor and
˜
α
μν
is the Weitzenböck connection.
Imposing the teleparallelism condition, the torsion scalar
can be expressed as the sum of the Ricci scalar plus a total
derivative term, i.e.
hT =−h
¯
R + 2
hT
ν
νμ
,
μ
⇒ T =−
¯
R + 2T
ν
νμ
,
μ
, (9)
where
¯
R here is the Ricci scalar corresponding to the Levi-
Civita connection and h, as above, is the determinant of
the metric. Following [28], the teleparallel equivalent of the
Gauss–Bonnet topological invariant can be obtained:
hG = hT
G
+ total derivative, (10)
where the Gauss–Bonnet invariant, in terms of curvature, is
G = R
2
− 4R
μν
R
μν
+ R
μνρσ
R
μνρσ
, (11)
and the teleparallel T
G
invariant is given by
T
G
= (K
α
1
ea
K
eα
2
b
K
α
3
fc
K
f α
4
d
− 2K
α
1
α
2
a
K
α
3
eb
K
e
fc
K
f α
4
d
+ 2K
α
1
α
2
a
K
α
3
eb
K
eα
4
f
K
f
cd
+ 2K
α
1
α
2
a
K
α
3
eb
K
eα
4
c,d
)δ
a
α
1
b
α
2
c
α
3
d
α
4
. (12)
In a four dimensional spacetime, the term T
G
is a topological
invariant, constructed out of torsion and contorsion tensor.
2
In order to simplify the notation, we will identify T
G
with G
from now on.
2
See Sect. 3 of [28] for the detailed derivation and discussion.
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Eur. Phys. J. C (2016) 76 :629 Page 3 of 6 629
The field equations from the action (1) are then
2 f
T
∂
ν
hh
ρ
κ
S
ρ
μν
− 2hf
T
h
γ
κ
S
ρβμ
T
ρβγ
+ 2hh
ρ
κ
S
ρ
μν
∂
ν
f
T
+ 4hh
κ
ν
RR
μν
f
G
−
1
2
fhh
μ
κ
+ 4hh
κ
ν
g
μν
−∇
μ
∇
ν
Rf
G
+ 16hh
κ
ν
∇
λ
∇
(μ
f
G
R
ν)λ
− 8hh
κ
ν
g
μν
∇
α
∇
β
f
G
R
αβ
− 8hh
κ
ν
f
G
R
μν
− 16hh
κ
ν
f
G
R
να
R
α
μ
+ 4hh
κ
ν
f
G
R
ν
αβγ
R
μαβγ
+ 8hh
κ
ν
∇
(ρ
∇
σ)
f
G
R
μνρσ
= 0 (13)
where f
A
= ∂ f /∂ A being A = T, G.
In the discussion below, we will consider the Friedmann–
Robertson–Walker (FRW) cosmology related to f (T
G
, T ),
i.e. f (G, T ), and we search for Noether symmetries in order
to fix the form of the function f and to derive exact cosmo-
logical solutions.
3 Searching for Noether symmetries
Let us consider a a spatially flat FRW cosmology defined by
the line element
ds
2
=−dt
2
+ a
2
(t)(dx
2
+ dy
2
+ dz
2
), (14)
from which we can express the teleparallel Gauss–Bonnet
term as a function of the scale factor a(t) [50]
T
G
= G = 24
˙a
2
(t) ¨a(t)
a(t)
3
. (15)
As said above, we can discard the total derivative term (see
also [30]) The torsion scalar is
T =−6
˙a
2
(t)
a
2
(t)
. (16)
We can reduce (1) to a canonical point-like action by using
the Lagrange multipliers as
A =
1
2κ
dt
a
3
f (G, T ) − λ
1
G −
¯
G
− λ
2
T −
¯
T
,
(17)
where
¯
G and
¯
T are the Gauss–Bonnet term and the torsion
scalar expressed by (15) and (16). The Lagrange multipliers
are given by λ
1
= a
3
∂
G
f = a
3
f
G
and λ
2
= a
3
∂
T
f = a
3
f
T
and are obtained by varying the action with respect to G and
T , respectively. We can rewrite the action (17)as
A =
dta
3
2κ
f (G, T ) − f
G
G −
24 ˙a
2
¨a
a
3
− f
T
T + 6
˙
a
2
a
2
(18)
and, discarding total derivative terms, the final Lagrangian is
L =a
3
f − G f
G
−Tf
T
−8 ˙a
3
˙
G f
GG
+
˙
Tf
GT
−6 f
T
a ˙a
2
.
(19)
This is a point-like, canonical Lagrangian whose configu-
ration space is Q ={a, G, T } and tangent space is TQ =
{a, ˙a, G,
˙
G, T,
˙
T }. The Euler–Lagrange equations for a, G
and T are, respectively,
a
2
f − G f
G
− Tf
T
+ 2 f
T
˙a
2
+ 16˙a ¨a
˙
f
G
+ 8˙a
2
¨
f
G
+ 4
˙
f
T
a ˙a + 4 f
T
a ¨a = 0, (20)
a
3
G − 24 ˙a
2
¨a
f
GG
+
a
3
T + 6a ˙a
2
f
T G
= 0, (21)
a
2
T − 6˙a
2
af
TT
−
a
3
G − 24 ˙a
2
¨a
f
GT
= 0. (22)
As expected, for f
GG
= 0 and f
GT
= 0, we obtain, from (21)
and (22), Eqs. (15) and (16) for the Gauss–Bonnet term and
the torsion scalar. The energy condition E
L
= 0, associated
with Lagrangian (19), is
E
L
=
∂L
∂ ˙a
˙a +
∂L
∂
˙
T
˙
T +
∂L
∂
˙
G
˙
G − L = 0,
corresponding to the 00-Einstein equation
24 ˙a
3
˙
f
G
+ 6 f
T
a ˙a
2
+ a
3
f − G f
G
− Tf
T
= 0. (23)
Alternatively, the system (20)–(23) can be derived from the
field equation (13).
Let us now use the Noether symmetry approach [31]to
find possible symmetries for the dynamical system given by
the Lagrangian (19).
In general, a Lagrangian admits a Noether symmetry if its
Lie derivative, along a vector field X , vanishes
3
L
X
L = 0 ⇒ XL = 0. (24)
Alternatively, the existence of a symmetry depends on the
existence of a vector (a “complete lift”), which is defined on
the tangent space of the Lagrangian, i.e.
X = α
i
(q)
∂
∂q
i
+
dα
i
(q)
dt
∂
∂ ˙q
i
, (25)
q
i
being the configuration variables, ˙q
i
the generalized veloc-
ities, and α
i
(q
j
) the components of the Noether vector. In
our case, the Lagrangian admits three degrees of freedom
and then the symmetry generator (25) reads
X = α
∂
∂a
+ β
∂
∂G
+ γ
∂
∂T
+˙α
∂
∂ ˙a
+
˙
β
∂
∂
˙
G
+˙γ
∂
∂
˙
T
. (26)
The system derived from Eq. (24) consists of 10 partial dif-
ferential equations (see [31] for details), for α, β, γ , and
3
There exists a symmetry even if the Lagrangian changes by a total
derivative term, but we will discuss the simplest case.
123
629 Page 4 of 6 Eur. Phys. J. C (2016) 76 :629
f (G, T ). It is overdetermined and, if solved, it allows us to
determine the components of the Noether vector and the form
of f (G, T ).Itis
∂
a
β f
GG
+ ∂
a
γ f
GT
= 0 (27)
β f
GGG
+ γ f
GGT
+ 3∂
a
α f
GG
+ ∂
G
β f
GG
+ ∂
G
γ f
GT
= 0,
(28)
β f
GT G
+ γ f
GTT
+ 3∂
a
α f
GT
+ ∂
T
β f
GG
+ ∂
T
γ f
GT
= 0,
(29)
α f
T
+ β f
T G
a + γ f
TT
a + 2 f
T
a∂
a
α = 0, (30)
af
T
∂
G
α = 0, (31)
af
T
∂
T
α = 0, (32)
f
GG
∂
G
α = 0, (33)
f
GT
∂
T
α = 0, (34)
∂
G
α f
GT
+ ∂
T
α f
GG
= 0, (35)
3α
f − G f
G
− Tf
T
− aβ
G f
GG
+ Tf
T G
− aγ
G f
GT
+ Tf
TT
= 0. (36)
Clearly, it being a system of partial differential equations, a
theorem of existence and unicity for the solutions does not
hold. However, if only one of the functions α, β, γ is different
from zero, a Noether symmetry exists. Below, we will show
that the existence of the symmetry selects the form of the
function f (G, T ) and allows one to get exact solutions for
the dynamical system (20)–(23).
4 Selecting the form of f (G, T ) by symmetries
In order to solve the above system, we have to make some
assumptions. There are two ways to look for solutions: the
first is to assume specific families of f (G, T ) and derive
symmetries accordingly, i.e. find the components of the sym-
metry vector. The second approach consists in imposing a
specific form for the symmetry vector and then finding the
form of f (G, T ). However, in the second case, the chosen
functions α, β, γ must be a solution of the system (27)–(36).
This approach is straightforward and more mathematically
consistent. It consists in reducing one of the above equations,
e.g. Eq. (36), to a differential equation for f (G, T ), once the
components of the Noether vector are assigned. As reported
in [31] for the case of a scalar–tensor theory with only a
scalar field φ, by assigning α and β, it is possible
4
to derive
a constraint differential equation for the coupling F(φ) and
then for the potential V (φ). The general solution has to be
discarded since it is an implicit function of F(φ) without a
physical meaning and then useless. On the other hand, the
4
In this case, the configuration space is Q ={a,φ} and the tangent
space is TQ ={a, ˙a,φ,
˙
φ}. and then we need only two components α
and β for the Noether vector.
particular solution F(φ) = ξφ
2
is physically meaningful
and then can be used to find relevant cosmological solutions
(see [31] for details). In the present case, the constraint for
f (G, T ) is a differential equation in two variables. Achiev-
ing general solutions does not give physically relevant and
workable models.
To obtain physically reliable models, the first route is more
convenient. In this preliminary paper, we will adopt this strat-
egy to find solutions choosing classes of f (G, T ) function.
4.1 The case f (G, T ) = g
0
G
k
+ t
0
T
m
We substitute this form of f (G, T ) in the system (27)–(36)
and find that, for k = 1 and arbitrary m, the only possible
Noether vector is the trivial one, X = (0, 0, 0), which means
that there is no symmetry. However, for k = 1 and arbitrary
m, i.e. f (G, T ) = g
0
G + t
0
T
m
, the vector assume the non-
trivial form
X ≡
α
0
a
1−
3
2m
,β(a, G, T ), −
3α
0
Ta
−
3
2m
m
, (37)
with α
0
being an arbitrary integration constant and any non
singular β. This means that this theory admits a symmetry
with the conserved quantity being
0
=−12α
0
mt
0
˙a
a
3
2m
−2
T
m−1
, (38)
which coincides with the case f (T ) = t
0
T
m
and then the
contribution of the Gauss–Bonnet invariant is trivial.
5
This
is expected since, in a 4-dimensional manifold, the linear
Gauss–Bonnet term is vanishing in the action and thus this
model is not different from f (T ) gravity.
4.2 The case f (G, T ) = f
0
G
k
T
m
In this case, the system (27)–(36) becomes slightly more
complicated. As previously, we have two possible choices
of the powers k, m.Ifm = 1 − k, f (G, T ) reduces to pure
f (T ), i.e. we have to set k = 0 and therefore we have the
same symmetries as before. Nevertheless, if m = 1 − k,
the model becomes f (G, T ) = f
0
G
k
T
1−k
and it admits a
Noether symmetry denoted by the vector
X = (0,β(a, G, T ),
T
G
β(a, G, T )), (39)
where β is a non-singular function. It is interesting to point
out the analogy with the curvature case, where the Noether
symmetry approach selects the form f (G, R) = f
0
G
1−k
R
k
as discussed in [50]. In some sense, symmetries preserve
5
See Eqs. (455)–(457) in the review paper [1] and the discussion in
[12].
123
Eur. Phys. J. C (2016) 76 :629 Page 5 of 6 629
the structure of gravitational theories independently of the
teleparallel or metric formulation.
6
5 Cosmological solutions
Starting from the model f (G, T ) = f
0
G
k
T
1−k
, let us find
cosmological solutions for any values of k. The Lagrangian
(19) assumes the form
L = f
0
(k − 1) ˙a
2
G
k−2
T
−k
4k ˙a
G
˙
T − T
˙
G
+ 3aG
2
.
(40)
and the Euler–Lagrange equation for a(t) and the energy
equation become
2kG
2
a
4Ta
T
+ a
2TT
− 2kT
2
+ aT G
+ 4kGT a
a
2(k − 1)G
T
− TG
− 2Ta
G
+ G
3
T
2aa
+ a
2
− 2kaa
T
− 4(k − 2)kT
2
a
2
G
2
= 0, (41)
4ka
GT
− TG
+ aG
2
= 0, (42)
while the other two, i.e. for G and T , give the Lagrange
multipliers (15) and (16). If we substitute the constraints (15),
(16) into Eq. (41), (42) we get
2a
2
¨a
4
+ k
2
˙a
4
¨a
2
− 2(k − 1)ka
...
a
˙a
3
¨a + 4ka
2
...
a
˙a ¨a
2
+ a ˙a
2
(k − 2)ka
...
a
2
+ (1 −5k) ¨a
3
+ ka
....
a
¨a
= 0, (43)
a ¨a
2
+ ka
...
a
˙a − k ˙a
2
¨a = 0. (44)
These general (for arbitrary k = 1) equations admit power-
law solutions for the scale factor of the form
a(t) = a
0
t
s
, with s = 2k + 1. (45)
It is easy to verify that the Gauss–Bonnet term and the torsion
scalar behave asymptotically as G ∼ 1/t
4
and T ∼ 1/t
2
, for
any k.
From these considerations, it is easy to realize that any
Friedmann-like, power-law solution can be achieved accord-
ingtothevalueofk. For example, a dust solution is recovered
for
a(t) = a
0
t
2/3
, with k =−
1
6
; (46)
a radiation solution is for
a(t) = a
0
t
1/2
, with k =−
1
4
; (47)
and a stiff matter one is for
a(t) = a
0
t
1/3
, with k =−
1
3
. (48)
6
Clearly also the case f (G, T ) = f
0
G
1−k
T
k
gives a symmetry.
Power-law inflationary solutions are achieved, in general, for
s ≥ 1 and then k ≥ 0.
Some comments are necessary at this point. Clearly, the
above solutions cannot track the whole cosmological evolu-
tion but only limited phases. Achieving solutions capable of
tracking completely the cosmic history could not be possi-
ble due to the fact that smooth, derivable solutions cannot
account for phase transitions occurring during cosmic evo-
lution. In [51], a detailed discussion is pursued in view of
providing a unified description of the cosmic evolution rang-
ing from early-times inflation to late-times acceleration. In
particular, the authors consider models like
f (G, T ) =−T + β
1
(T
2
+ β
2
G) + β
3
(T
2
+ β
4
G)
2
(49)
where higher-order terms become significant at an early stage
of evolution and lower-order terms are relevant today, accord-
ing to the values of the parameters β
i
. In such a way, the two
accelerated phases of the cosmic evolutions can be achieved.
Considering the present discussion, this means that, in
order to achieve cosmological models consistent with more
than one phase of cosmic evolution, one need to consider
more general classes of f (G, T ) functions coming from
Noether’s symmetries.
6 Conclusions
In this paper, we discussed a theory of gravity where the inter-
action Lagrangian consists of a generic function f (T
G
, T ) of
the teleparallel Gauss–Bonnet topological invariant, T
G
, and
the torsion scalar T . The physical reason for this approach is
related to the fact that we want to study a theory where the
full budget of torsional degrees of freedom are considered.
Furthermore, it is easy to show that, from a dynamical point
of view, the Gauss–Bonnet invariant, derived from curvature,
G, and the Gauss–Bonnet invariant, derived from torsion, T
G
,
are equivalent and then we can consider a f (G, T ) theory.
After these considerations, we searched for Noether sym-
metries in the cosmology derived from this model. We
showed that specific forms of f (G, T ) admit symmetries and
allow for the reduction of the dynamical system.
In particular, the class f (G, T ) = f
0
G
k
T
1−k
results par-
ticularly interesting and, depending on the value of k,itis
possible to achieve all the behaviors of standard cosmology
as particular solutions.
Clearly, other cases can be considered and a systematic
approach to find other solutions can be pursued. This will be
the argument of a forthcoming paper where a general cos-
mological analysis will be developed.
Acknowledgements The authors acknowledge the COST Action CA-
15117 (CANTATA) and INFN Sez. di Napoli (Iniziative Specifiche
QGSKY and TEONGRAV). M. D. L. is supported by ERC Synergy
123