E Koorambas

G Resconi
email: resconi@speedyposta.it

Chapter COSMIC ACCELERATION FOR HARMONIC GRAVITATIONAL CONNECTIONS

Keywords: Alternative gravity theory, Non conservative gravity, Dark energy

The formalism of Symbolic Gauge Theory (SGT) for the case of General Relativity (GR) has been proposed by R. Mignani, E. Pessa and G. Resconi (1997). This has enabled us not only to recover the usual Einstein theory, but to articulate a new theory of gravitation, characterized by its open nature. This theoretical framework is based on a few principles related to different orders of commutators between covariant derivatives. We show that the non-conservative Gravitational equations can be represented by a wave equation with particular source where the variables are symmetric and anti-symmetric gravitational connections. Since the gravitational connections are harmonic in spacetime, we find that cosmic acceleration can be derived by these harmonic gravitational connections (solutions of the wave equation) at the case of a perfect fluid and active vacuum.

INTRODUCTION

Observations of type Ia supernovae and large-scale structure (LSS), in combination with measurements of the characteristic angular size of fluctuations in the cosmic microwave background (CMB) ( [1][2][3][4][START_REF] Filippenko | Carnegie Observatories Astrophysics Series[END_REF][START_REF] Leibundgut | [END_REF][7][8][9][10]) provide evidence that the expansion of the Universe is accelerating. This acceleration is attributed to the "dark energy", a hypothetical energy with negative pressure ( [1][2][3][4][START_REF] Filippenko | Carnegie Observatories Astrophysics Series[END_REF][START_REF] Leibundgut | [END_REF]). Evidence for the presence of dark energy is also provided by an independent, albeit more tentative, investigation of the integrated Sachs-Wolfe (ISW) ( [11]).

Dark energy may result from Einstein's cosmological constant (which has a phenomenally small value); from evolving scalar fields [12]; and from a weakening of gravity in our 3 + 1 dimensions by leaking into the higher dimensions, as required in string theories [13]. These phenomena may have crucial implications for the perturbation of gravitational waves caused by dark energy. This has stimulated further efforts to confirm the initial results on dark energy, test possible sources of error, and extend our empirical knowledge of this newly discovered component of the Universe.

General relativity (GR), published by Albert Einstein in 1915 [START_REF] O'connor | General relativity[END_REF], generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or space-time. The curvature of space-time, in particular, is directly related to the energy and momentum of whatever matter and radiation are present. This relation is specified by a system of partial differential equations, the Einstein field equations.

In the framework of geometric extensions of GR, a geometric unification between traditional gauge treatments of gravity, represented by a metric field, and dark energy, which arises as a corresponding gauge potential from the single SU (2) group has been proposed by one of the authors [15]. The perturbation of gravitational waves caused by dark energy has also been studied [15].

Various workers have attempted to derive GR from a gauge-like principle, involving invariance of physics under transformations of the locally (i.e. in the tangent space at each point) acting Lorentz or Poincare group [START_REF] Utiyama | Invariant theoretical interpretation of interaction[END_REF][START_REF] Tw | Lorentz invariance and the gravitational field[END_REF][START_REF] Wilczek | Riemann-Einstein structure from volume and gauge symmetry[END_REF] N. Wu (2003) [19][20][21][22] proposed a Quantum Gauge Theory of Gravity (QGTG) based on the gravitational gauge group (G). In Wu's theory, the gravitational interaction is considered as a fundamental interaction in a flat Minkowski space-time and not as space-time geometry.

Models of dark energy resulting from interacting massive gauge gravitons and dark matter consisting of the heavy gauge graviton resulting from shell decay of Higgs bosons have been developed recently by one of us within the framework of QGTG [23,[START_REF] Koorambas | Recent Developments in Bosons Research[END_REF][START_REF] Koorambas | [END_REF].

This paper is based on the formalism of Symbolic Gauge Theory (SGT) as applied on the case of General Relativity (GR), as proposed by R. Mignani, E. Pessa and G. Resconi [START_REF] Mignani | Commutative diagrams and tensor calculus in Riemann spaces[END_REF]27,[START_REF] Resconi | Geometry for a brain. Optimal control in a network of adaptive memristor[END_REF][START_REF] Jessel | A general system logical theory[END_REF][START_REF] Mignani | Electromagnetic-like generation of unified-Gauge theories[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Licata | Information as Environment Changings. Classical and Quantum Morphic Computation[END_REF]. We show that the non-conservative gravitational equations can be represented by a wave equation with particular source, where the variables are symmetric and anti-symmetric gravitational connections. Since the gravitational connections (solutions of the wave equation) are harmonic in space-time, we find that cosmic acceleration can be derived by these gravitational connections in the case of a perfect fluid and active vacuum.

FUNDAMENTAL CONCEPTS OF THE NON-CONSERVE THEORY OF GRAVITY

Following, Manuel E. Rodrigues et al. [28], let us first rewrite some equations derived from GR, to be used as inspiration to the modification made by Resconi et al. [START_REF] Mignani | Commutative diagrams and tensor calculus in Riemann spaces[END_REF]27,[START_REF] Resconi | Geometry for a brain. Optimal control in a network of adaptive memristor[END_REF][START_REF] Jessel | A general system logical theory[END_REF][START_REF] Mignani | Electromagnetic-like generation of unified-Gauge theories[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Licata | Information as Environment Changings. Classical and Quantum Morphic Computation[END_REF].

Cosmic Acceleration for Harmonic Gravitational Connections 3 Space-time is considered to have the Riemannian manifold, in which the line element is given by (1) where are the components of the metric. The signature of the metric is defined as (+,-,-,-,). For making a parallel transport of the vector field , between two infinitesimally close points, we define the covariant derivative as: (2) where are the components of the Levi-Civita connection. If we apply the commutator of the covariant derivatives and , on a vector we get:

(3)

In (3) the Riemann tensor components are:

Using two times the commutator of the covariant derivatives , and , and making the cyclic subscripts, we get the Jacobi identity:

(5)

In terms of the Riemann tensor in (3), equation ( 5) is written as:

The latter equation is called the first Bianchi identity. Contracting with in (6), we obtain: (7) where are the components of the Ricci tensor. Contracting with , leads to the second Bianchi identity:

2 () ds g x dx dx    () gx  () Vx   V V V                    V  ,, a V R V             a R                                   , , , , , , 0 
                                            0 R R R                  R R R             R    (8)
where is the curvature scalar. Considering the Einstein equations of GR, it is:

Inserting ( 9) in (7) we get: (10) where (11) and T is the trace of the energy-momentum tensor. Equations (10) are called the old quasi-Maxwellian gravitational equations [START_REF] Mitskievich | Relativistic Physics in Arbitrary Reference Frames[END_REF]. All this is established in GR. In the next section, we introduce the fundamental concepts of the theory constructed by Resconi et al. [START_REF] Mignani | Commutative diagrams and tensor calculus in Riemann spaces[END_REF]27,[START_REF] Resconi | Geometry for a brain. Optimal control in a network of adaptive memristor[END_REF][START_REF] Jessel | A general system logical theory[END_REF][START_REF] Mignani | Electromagnetic-like generation of unified-Gauge theories[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Licata | Information as Environment Changings. Classical and Quantum Morphic Computation[END_REF].

A modification of GR by Camenzind [START_REF] Camenzind | [END_REF] was constructed as an analogy to the Yang-Mills theory, with the symmetry group SO (3.1). In the same vein, inspired by the foundations of gauge theory, Resconi et al. [START_REF] Mignani | Commutative diagrams and tensor calculus in Riemann spaces[END_REF]27,[START_REF] Resconi | Geometry for a brain. Optimal control in a network of adaptive memristor[END_REF][START_REF] Jessel | A general system logical theory[END_REF][START_REF] Mignani | Electromagnetic-like generation of unified-Gauge theories[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Licata | Information as Environment Changings. Classical and Quantum Morphic Computation[END_REF] constructed a modified theory of gravity, in which the algebra of the covariant derivatives operators adds a term to the equations of motion. To present some basic concepts of this theory, we start by applying the commutator of with the commutator of with to a vector field . Doing this we obtain: (12) and then,

This relationship is demonstrated by Resconi et al. [START_REF] Mignani | Commutative diagrams and tensor calculus in Riemann spaces[END_REF]27,[START_REF] Resconi | Geometry for a brain. Optimal control in a network of adaptive memristor[END_REF][START_REF] Jessel | A general system logical theory[END_REF][START_REF] Mignani | Electromagnetic-like generation of unified-Gauge theories[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Licata | Information as Environment Changings. Classical and Quantum Morphic Computation[END_REF] by means of commutative diagrams, as used in category theory. Using the matter source in (11), we can derive the Einstein equations of GR, for the particular case where the covariant derivative of the vector field vanishes identically. But firstly, we must obtain the general equation of motion for this theory.

1 0 2 RR            R g R    1 2 R g R xT     RJ      1 () 2 J T T g T g T                           K          , , , , K K K R K R K                                                          ,, K R K R K                       J  K 
Cosmic Acceleration for Harmonic Gravitational Connections 5 Considering as matter source the (11), (the same as in GR) we can match the expression (13), with the new source condition, [START_REF] O'connor | General relativity[END_REF] which leads to:

Contracting with and with , we obtain the following general equation of motion:

(16a)

In the particular case where , considering the validity of Einstein's equations ( 9), we re-obtain the second Bianchi identity ( 8), for a non-zero vector field . These motion equations are stemming directly from a Lagrangian or variational principle, as in the case of Rastall theory [31]. All matter fields, (scalar, vector and spinorial fields), are contained in the energy-momentum tensor, which gives us the interaction between matter and geometry, just as in RG.

Equation ( 16) can be interpreted as the equation of motion of a theory which can be viewed as a generalization of GR. This theory is derived from the foundations of gauge theory, with an added term with respect to GR, coming from the covariant derivatives commutator. We call this theory Symbolic Gauge Theory of Gravity (SGTG). The vector field can be interpreted as a field that describes the interaction with a substratum [START_REF] Mignani | Commutative diagrams and tensor calculus in Riemann spaces[END_REF]27,[START_REF] Resconi | Geometry for a brain. Optimal control in a network of adaptive memristor[END_REF][START_REF] Jessel | A general system logical theory[END_REF][START_REF] Mignani | Electromagnetic-like generation of unified-Gauge theories[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Licata | Information as Environment Changings. Classical and Quantum Morphic Computation[END_REF]. As in gauge theories, the vector field can be a gauge field of a certain local (as SU(2) gauge group for example) or global (as SU(3)) gauge group [START_REF] Camenzind | [END_REF]. In the current case, the second term of the sum in ( 16) is the main coupling term between the substratum and the gravity, in such a way that if this term vanishes, GR is recovered. The crucial difference with GR here is the consideration of the source of the Riemann tensor in (10) for the GR: In SGTG by contrast this is a source of another operator, like an eigenvalue equation of the double commutator of covariant derivatives in [START_REF] O'connor | General relativity[END_REF].

Physical Meaning of the Double Commutator

Considering the validity of the Einstein Field Equations for the matter-sources, the double commutator acting on a non-vanishing vector field K can be expressed as a term proportional to a matter-source current J. The latter can in turn be related to the quadridivergence of the Riemann tensor.

J  ,, K kJ K                   kJ K R K R K                    1 0 2 R k T g T K R K                     0 K   K  K  K 
As is well known, through a contraction of the Bianchi identities this quadri-divergence of the Riemann tensor can be expressed, as a difference between quadri-divergences of the Ricci tensor. Using these expressions is easy to see that the quadri-divergence of the Riemann tensor vanishes for the Friedmann-Lemaitre-Robertson-Walker universe models (FLRW): indeed, the space-time metric is highly symmetrical, and the only non-vanishing components of the Ricci tensor are those with equal indices.

This implies that the matter source term J vanishes for the FLRW models. Note that the FLRW are conformally flat models (the Weyl curvature tensor vanishes). In these cases there is indeed no non-uniformity of the gravitational field, in the sense that there are no tidal effects. In conclusion, in this context the double commutator has to vanish for the FLRW models.

Note, as well, that the de Sitter space-time solution also leads to a vanishing double commutator because of the high degree of symmetry of the model.

On the other hand, the situation turns out to be very different for anisotropic and homogenous Bianchi type models as well as for inhomogeneous cosmological solutions. Essentially this occurs because the non-vanishing components of the Ricci tensor contain both diagonal and non-diagonal terms: in this way, the quadri-divergence of the Riemann tensor is non-zero and the double commutator is also non-vanishing. For Bianchi-type models the Weyl curvature tensor is non-vanishing. As a consequence, these models are conformally non-flat. General inhomogeneous models can be shown to lead to a non-zero double commutator as well.

In addition, the Schwarzchild space-time solution also leads to a non-zero double commutator: these solutions are not conformally flat and thus are characterized by nonvanishing tidal effects.

The physical aspect which seems to be relevant for the interpretation of the double commutator is related to the non-uniformity of the gravitational field: if there would be deep or fundamental requirements for a non-vanishing double commutator then this would in effect, motivate the necessary presence of non-vanishing tidal effects in the physical models.

In earlier work, a heuristic analysis suggested that one has indeed to consider as viable only those models which are characterized by an intrinsically non-uniform gravitational field: otherwise the basic principles of thermodynamics would be contravened. These arguments also imply relations to a quantum theory of gravity and to the holographic principle, as well as to the possible entropic nature of gravity Y. Gaspar and G. Acquaviva (2010) [] . Connections with R. Penrose's ideas on Quantum Gravity and the fundamental role of the Weyl curvature tensor are also worth investigating.

CLASSIFICATIONS OF MANIFOLDS AND THE NON METRIC VACUUM PHASES

Following Moshe Carmeli (1999) [32], we consider the following three generic phases leading to metric spacetimes [START_REF] Schr¨odinger | Space-Time Structure[END_REF]:

(1) General differentiable manifold (unconnected), (2) Affinely connected manifold, (3) Metrically connected manifold. In addition, there is an intermediate phase between (2) and (3): space-time with a rudimentary metric structure (STRMS): in metrically connected spacetimes:

(16b)
Hence in non metric, gravitational affinely connected space-times, it is possible, to define the pressure p as the potential of , and let play the role of √-g.

Equation ( 2) determines the volume element up to an overall multiplicative factor. STRMS is metric only in the sense of having a volume element; ds 2 is not defined in it [32].

Perhaps surprisingly, a lot can be done on general differentiable manifolds. Covariant, contravariant and mixed tensors, tensor densities and spinors can be defined. (However, the usual relationship between covariant and contravariant components does not hold: there is no metric tensor to connect them); There are invariant integrals which are obtained by integrating over tensor densities: If £ is a density, then is invariant, and hence we can have a Lagrangian formalism, with Euler-Lagrange equations.

The Non-Metric Cosmological Vacuum Phases (Active Vacuum)

The closer we get to the real vacuum, the more relevant are the quantum features. Such quantum effects may result to the self-organized vacuum state where metric time and space lose their meaning. This leads to the following question: Is there a non-metric cosmological vacuum phase, generically as well as temporally, prior to metric spacetime? The fact is that in our approach, in the case of the vacuum gravitational field, and unlike electromagnetic and other fields, we don't have a spacetime of a definite structure on which the field is defined. The thrust of the present approach is to show that the answer is, "yes": There is a non-metric cosmological vacuum phase which is generically prior to ordinary metric vacuum. We call this phase Active Vacuum (AC).

DYNAMIC EQUATIONS OF NON-CONSERVATIVE GRAVITY

The representation of the dynamical equation (16a) by a wave equation with particular source where the variables are symmetric and anti-symmetric gravitational connections (including torsion in one geometric picture) is achieved through the following steps:

First, let (17)   , , ln a a gp       a a   p e 4 dx   , V D V V x              (18)
So we have and (22,23) and [START_REF] Koorambas | Recent Developments in Bosons Research[END_REF] In conclusion we have: , , (

) -( ) (18) ,,

F D D V D D V D D V v VV D V D V v v xx v                               , , ( ) , ,, ,, , ( ) (19) , , ,, ( ) (20) ,, ( ) (21)
,, B F D D V B B v v x B BB v v x v V BV v x v V BV x                                                               , ( ) (22) , , , ( ( ) (23
) ,, ,,

F D D V R V T D V v v Where v qq R q v q vv xx v                                        , , T v v v          
Now we have that:

The field G is, therefore invariant for the change of . Now we impose the like Lorenz gauge as follows:

(30) Now we have , , , ( ) , ( ) , , , , ( ) (25) ,, , , , , ( ) (26)
, ( ) ( ) , , , , , , , v F D D V x x v qq V q v q v D V G vv where v G V x x v qq V D V q v q v v v                                                                                            (27) (28) , , ( ) ( ) , , ( ) , 2 2 , , ( ) (29)
, v v x where v x x G V x x v v V G x x x x x x v v                                                                   , 0 (30) x       (31) , , , , , () , , , , ( ) ( ) (32)
,, ,

G G G x x x v V x x x v VV x x x x x x v x x x x x x                                                                                        
The Lagrangian gravitational density is where (36) are the Lagrangian density of the free gravitational field and the reaction field of the active vacuum, respectively. The interaction terms connect the gravitation field with the field of the vacuum. Next, the dynamic equation for non conservative gravity can be obtained by the SGT [START_REF] Mignani | Commutative diagrams and tensor calculus in Riemann spaces[END_REF]27,[START_REF] Resconi | Geometry for a brain. Optimal control in a network of adaptive memristor[END_REF][START_REF] Jessel | A general system logical theory[END_REF][START_REF] Mignani | Electromagnetic-like generation of unified-Gauge theories[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Resconi | A novel representation of quantum cybernetics using Lie algebras[END_REF][START_REF] Licata | Information as Environment Changings. Classical and Quantum Morphic Computation[END_REF] in this way: (

G G G x x x v V x x x v VV x x x x x x v x x x x x x x x x x v                                                                                                        , 0 (33) xx        , , 31) , ,, , , () , , , , ( ) ( ) (32) ,, , ,,
, ,

L F F G G G G G                                     , , , 2 , , , G G and G
                , , G      , , , , , (37) , ( ) , (38) , , , ( ) , (39) ,,
,

D D D V D D D V D D D V D F D D D V D G D D D V J D G D D D V J                                                                            (44)
Now we have: See [START_REF] Filippenko | Carnegie Observatories Astrophysics Series[END_REF]. When currents are equal to zero, it is:

[ ) , ] (40) , , , , (41) , , , , , , , [ ) , ] (42) , , , and D
G J D D D D V where G j j j D G G G G j j j v x so G j j G G j j v x j G J D D D D V j and G                                                                                   , ( ) , , , ( ) , ) (43) ,
,

j j j G G G j j j v x J D D D D V                                    , , G j x         2 2 2 2 2 2 2 2 , , , , , ( ) ) ( ) (45) x have , , , , , ( ) ( ) (46)
G v v V V x x x x x x x x v v for x we G v v V V x x x x x x x v but for the like Lorenz gauge                                                                                   2 , , ( ) 0 (47) 2 2 2 , , , , 2 ( )
(48) 2 2 2 v we have x x x x and G v v v V c V J x x x t                                               
The currents are:

where R is a reaction of a virtual matter or AC and J are the ordinary currents for the ordinary matter represented by the energetic tensor. The non-linear reaction of the self-coherent system produces a current that justifies the complexity of the gravitational field and non-linear properties of the non-metric gravitational waves. So, in analogy with non-linear optic we can model gravitational phenomena as optic of tensor potentials

The above equations in the free field of the medium contain the Proca terms,     , the Chern-Simons terms (     )   and the Maxwell like terms (     ) (     ). In other words, we have the mass terms, the topologic terms and the electromagnetic field-like terms. [34.35].

COSMIC ACCELERATION FOR HARMONIC GRAVITATIONAL CONNECTIONS

Let us begin with a homogeneous and isotropic universe as described by the (standard) FRW metric in co-moving coordinates (see, e.g., [36] [START_REF] Misner | Gravitation[END_REF]): (53) in which k = 0, 1, or -1 is the curvature constant of a three-dimensional spatial hypersurface, a(t) is the "scale factor" and t is the "cosmic" time. The Einstein field equation is:

2 2 2 , , , , 2 ( ) 0 (49) 2 2 2 G v v v V c V x x x t
λ the tensor Γ is harmonic in the space time and is a wave.

v,α                            ) , , , ( ) , (50) , , ) , , , ) , (51)
, , (52)
,, The energy-momentum tensor of the cosmic fluid in its rest-frame is:

j j j j G G G j j j vv J D D D D V j j j GGG j j j v D D D D V J RJ                                                                  j v  2 2
(55)

where ρ and p are its total mass density and pressure, respectively, and u μ = (-c, 0, 0, 0) is its four-velocity. Besides the cosmic fluid, we also consider a contribution of ordinary vacuum, related with the Λ-term and given by:

(56)

From ( 53), ( 54) and ( 55

a kc Gp c a a         2 2 4 3 1 33 a G p c a c           , 8 , 8 G pp G          (62)
The acceleration equation can be written as:

(63) Equation (16b) reveals that pressure p in equation ( 60) is affected by AV affects. Thus our modification in the FRW universe is subject to pressure p. The modification of the accelerating equation (63) in the FRW universe due to the AV is achieved in the following steps.

First, substituting equation (16b), to the homogeneous wave equation (49), we get the following pressure wave equation: .

Integrating equation ( 64):

where c is integrating constant. Here we consider that the integrating constant c is arbitrary small. With this consideration equation (65) has the following solutions: ,

Equation ( 68) is the space part of solution (67), and P is the amplitude of the pressure. ω and t are the angular frequency of the universe, and cosmic time, respectively.

Next, we substitute solutions (67) of the pressure wave equation to the accelareting equation( 63 ( , ) 0 Cosmic accelerated expansion of the universe can be characterized by the equation of state of dark energy. Here, in the simplest case, the equation of state of the cosmological constant is . More generally, the expansion of the universe is accelerating for any equation of state . The accelerated expansion of the Universe has indeed been observed [START_REF] Hogan | Welcome to the Dark Side[END_REF]. According to observations, the value of equation of state of cosmological constant is near -1.

p x t t x                2 2 , , 2 2 2 2 2 2 , , 2 2 2 2 ( , ) ( , ) ( , ) ( , ) ( , ) ( ,
                                             (

CONCLUSION

Based on the formalism of SGT to the case of GR we have shown that the nonconservative gravitational equations can be represented by a wave equation with particular source, where the variables are symmetric and anti-symmetric gravitational connections.

We find that cosmic acceleration can be derived by the harmonic gravitational connections (solutions of the wave equation) at the case of a perfect fluid and active vacuum.

 

  ,W' are the effective amplitudes. For large scales of distances equation (70) becomes: (71)