Nonstandard approach to gravity for the dark sector of the Universe

We summarize the present state of research on the darkon fuid as a model for the dark sector of the Universe. Nonrelativistic massless particles are introduced as a realization of the Galilei group in an enlarged phase space. The additional degrees of freedom allow for a nonstandard, minimal coupling to gravity respecting Einstein's equivalence principle. Extended to a self-gravitating fluid the Poisson equation for the gravitational potential contains a dynamically generated effective gravitational mass density of either sign. The equations of motion (EOMs) contain no free parameters and are invariant w.r.t. Milne gauge transformations. Fixing the gauge eliminates the unphysical degrees of freedom. The resulting Lagrangian possesses no free particle limit. The particles it describes, darkons, exist only as fluid particles of a self-gravitating fluid. This darkon fluid realizes the zero-mass Galilean algebra extended by dilations with dynamical exponent z = 5/3. We reduce the EOMs to Friedmann-like equations, derive conserved quantities and a unique Hamiltonian dynamics by implementing dilation symmetry. By the Casimir of the Poisson-bracket (PB)-algebra we foliate the phase space and construct a Lagrangian in reduced phase space. We solve the Friedmann-like equations with the transition redshift and the value of the Casimir as integration constants. We obtain a deceleration phase for the early Universe and an acceleration phase for the late Universe in agreement with observations. Steady state equations in the spherically symmetric case may model a galactic halo. Numerical solutions of a nonlinear differential equation for the gravitational potential lead to predictions for the dark matter (DM) part of the rotation curves (RCs) of galaxies in qualitative agreement with observational data. We also present a general covariant generalization of the model.

Fixing the gauge eliminates the unphysical degrees of freedom. The resulting Lagrangian possesses no free particle limit. The particles it describes, darkons, exist only as fluid particles of a self-gravitating fluid. This darkon fluid realizes the zero-mass Galilean algebra extended by dilations with dynamical exponent z = 5 3 . We reduce the EOMs to Friedmann-like equations, derive conserved quantities and a unique Hamiltonian dynamics by implementing dilation symmetry. By the Casimir of the Poisson-bracket (PB)-algebra we foliate the phase space and construct a Lagrangian in reduced phase space. We solve the Friedmann-like equations with the transition redshift and the value of the Casimir as integration constants. We obtain a deceleration phase for the early Universe and an acceleration phase for the late Universe in agreement with observations. Steady state equations in the spherically symmetric case may model a galactic halo. Numerical solutions of a nonlinear differential equation for the gravitational potential lead to predictions for the dark matter (DM) part of the rotation curves (RCs) of galaxies in qualitative agreement with observational data. We also present a general covariant generalization of the model. * Accepted for publication by 'Entropy' in the special issue 'Modified gravity: From black holes entropy to current cosmology' Contents • 1 Introduction 3 • 2 Galilean massless particles and their nonstandard coupling to gravity 5 -2.1 Galilean massless particles 5 -2.2 Coupling to gravity 8 • 3 A self-gravitating (darkon) fluid 9 -3.1 Lagrangian formulation 9 -3.2 Relabeling symmetry and the transport equation for the gravitational field 11 -3.3 Eulerian formulation 11 -3. 4 Coupling with baryonic matter 12 • 4 Hamiltonian dynamics 13 • 5 Space-time symmetries 15

Introduction
Understanding the dark sector of the Universe is one of the greatest challenges of today's theoretical physics. The main question is whether the underlying astrophysical observations (missing gravitation in galactic systems resp. the accelerated expansion of the Universe) can be understood by means of known physical concepts, e.g. General Relativity (GR), or whether some kind of "new" physics is needed. Indeed, cosmic acceleration could be an apparent effect due to the averaging over large scale inhomogeneities in the Universe, but a reliable quantification of such an effect is not yet available (see the recent review [1]). On the other hand there exists an overwhelming evidence for the existence of gravitational effects on all cosmological scales (termed "dark matter" (DM)) which cannot be explained by the gravitation of standard matter in the framework of GR (see the review [2]).
The current picture of cosmological structure formation assumes for DM some pressure less dust containing massive, perhaps weakly interacting cold particles (CDM). The present standard cosmological model (ΛCDM-model) uses besides CDM a positive cosmological constant Λ as the cause for the accelerated expansion of the Universe (see recent reviews on dark energy (DE), e.g.
[3], [4] and [5]). But, "favoured by a number of observations" [3], the ΛCDM-model suffers at least from the following insufficiencies: • Interpreted as the energy density of the vacuum, the experimental value of Λ turns out to be a factor of about 10 54 too small (see the recent estimate in [6]).
• Rotation curves (RCs) of DM-dominated galaxies behave in their inner part in sharp contrast to the CDM-based simulations (known as the core-cusp problem [8]).
In a very recent paper P. Kroupa [9] lists a large set of extragalactic observations which falsify the ΛCDM-model.
In this paper we will neither comment on the huge number of phenomenological models for a dynamical DE (see [3]- [5]), which all rest on at least one unknown function (e.g. a scalar field potential), nor on those modified gravity theories which explain DM-effects only (e.g. modified Newtonian dynamics (MOND) in its simplest, Milgromian form [10]). But we know of at least two kinds of modified gravity models which explain both, DE and DM i.e. the whole dark sector of the Universe: • For MOND-models see section 9 of the very recent review [11]. But all these models are very phenomenological in that they are based on one unknown "MOND function".
• Modified theory of gravity (MOG) by Moffat [12] in which, besides the Einstein-Hilbert and matter actions, a massive vector field is introduced whose mass, coupling constant to matter and gravitational constant are promoted to scalar fields. The self-interaction potentials of these four fields are not fixed a priori and play the role of DE [13]. This model can explain galactic phenomena as e.g. RCs [14] as well as the accelerated expansion of the Universe and other cosmological observations [13].
In summary, all present attempts to explain quantitatively the new cosmological observations (e.g. late time cosmic acceleration) as well as the galactic RCs and other galactic phenomena contain either some new parameters or even free functions. If we assume that the cosmic acceleration is a real effect and not an apparent one (see above) then, obviously, we need some new (i.e. unconventional) physics which, however, should be based on known fundamental physical principles (e.g. symmetry).
In this paper we review some ongoing research on a model which is a first building block for a new theory describing the dark sector of the Universe. This model, introduced in [15] and further developed in [16] and [17], contains no new parameters in its Lagrangian and is based on well-known physical principles (Galilei symmetry and Einstein's equivalence principle).
The only free parameters appearing in our model are some integration constants (see sections 6 and 7). The basic idea it uses involves nonrelativistic massless particles. The use of such particles may appear very strange at a first sight as it seems to contradict Special Relativity according to which massless particles must move with the velocity of light. However, as we have shown before, we can introduce nonrelativistic massless particles if we enlarge the dimension of phase space; i.e. such particles are not described by position and momentum only. The additional degrees of freedom allow for a nonstandard, minimal coupling of these "exotic" particles [15] to gravity in accordance with Einstein's equivalence principle. This coupling, for a self-gravitating fluid, leads to a dynamically generated effective gravitational mass density of either sign which is the source of the corresponding gravitational field. This fact leads to the possibility of using such a model as a first building block of a new theory describing the dark sector. In our approach we formulate the equivalence principle (local equivalence between gravitation and acceleration) as usual as an invariance of the dynamics w.r.t. arbitrary time-dependent translations (Milne gauge transformations). By fixing the gauge we can reduce the dimension of phase space and remove the unphysical variables. The Lagrangian of the resultant model does not possess a free particle limit and hence the particles it describes, called darkons, exist only as fluid particles of a self-gravitating fluid. This darkon fluid presents a dynamical realization of the zero-mass Galilean algebra extended by anisotropic dilations with the dynamical exponent z = 5 3 . This dilation symmetry emerges in a natural way from the minimal gravitational coupling. The cosmological EOMs are derived, as usual, from the darkon fluid EOMs by restricting the form of their solutions through the cosmological principle. However, the corresponding Hamiltonian cannot be derived from the darkon fluid Hamiltonian. So we have to derive a Hamiltonian and the corresponding Poisson-brackets (PBs) from the EOMs, a procedure which, as is well known, is not unique. So we construct, using the constants of motion, a Hamiltonian which has the correct scaling dimension and obtain the unique PBs by implementing the dilation symmetry. Moreover, our procedure allows us to construct a Lagrangian having foliated the phase space by means of the Casimir of the PB-aLgebra. We solve the cosmological EOMs and obtain, by fixing two integration constants by their values determined from experimental data, a prediction for the Hubble parameter H(z) (z is the redshift) in general agreement with the existing data.
Furthermore, we demonstrate that our model not only gives a proper description of the late time cosmic acceleration but it also predicts galactic halos and, qualitatively correct, the corresponding flat rotation curves (RCs). To do this we derive for the spherically symmetric steady state case a nonlinear ordinary differential equation for the gravitational potential which is then solved numerically.
Almost all cosmological models are based on General Relativity (GR) or on some of its modifications which, however, preserve general covariance. The appearance of anisotropic dilations in our model seems, at first sight, to forbid a generalization of our model to a general covariant theory. But the dilation symmetry with z = 5 3 is an emergent symmetry and not an input of our model. So we are able to propose a generalization of our model showing general covariance without imposing any form of scale symmetry.
One of the aims of the present paper is to show in detail the main ideas of our model. A comparison with observational data (Hubble parameter resp. galactic rotation curves) serves only as an illustration that it might be meaningful to develop the present model to a proper theory of gravitation for the dark sector of the Universe. For this reason we do not produce any least square fits for the undetermined integration constants appearing in the cosmological context.
The paper is organized as follows: In section 2 we introduce Galilean massless particles and their coupling to gravity. In section 3 we describe the generalization of this particle picture to a self-gravitating fluid, fix the gauge and so obtain the darkon fluid. 2 Galilean massless particles and their nonstandard coupling to gravity.
As stated earlier the introduction of nonrelativistic massless particles may appear very strange at first sight. It seems to contradict Special Relativity according to which massless particles must move with the velocity of light. But in Special Relativity spinless particles are described by the vectors of position and momentum only. In our recent papers [15]- [17] we have shown that we can introduce nonrelativistic massless particles if we enlarge the dimension of the phase space from six to twelve. In subsection 2.1 we will show that this enlargement of phase space is also necessary: The 12-dimensional phase space is the minimal one allowing the dynamical realization of nonrelativistic massless particles.
How to couple gravity if there is no mass? Our nonrelativistic framework forbids the coupling to energy, as then this would be a relativistic effect. So we take Einstein s equivalence principle as a starting point in subsection 2.2. Then, within the Lagrangian framework, the gravitational field will be coupled to one of the new degrees of freedom.

Galilean massless particles
Our first task involves finding a minimal dynamical realization of the unextended (m = 0) Galilean algebra G0 in three space dimensions.
G0 is given by the Lie-brackets The translation generator P acts on x resp. p as Therefore we get the identification To show that our minimally enlarged phase space contains, besides x and p, two additional vectors corresponding to the reduced boost q and the velocity y, we proceed in two steps: The boost K acts on x as leading by (2) and (4) to Taking the PB of (4) with pj it is easily seen by the expressions in (1) that the PB of pi and qj vanishes [pi, qj] = 0.
• 2. Next we introduce the velocity vector which is translationally invariant (take the PB of (10) with pj and use the Jacobi identity) to get From (10), (1), (2) and (5) we get the action of the boost generator on yi Inserting now (4) into (12) leads by (11) to We conclude from (13) and by (6), (7) and (9) that y cannot be a function of the other variables x, p or q.) To close the PB-algebra in phase space we choose, in accordance with the principle of a minimal dimension, Note that (14) respects all Jacobi identities.
It remains to construct out of the phase space variables x, p, q, and y the Hamiltonian H and the generator of rotations L.
Our foregoing results lead to the EOMṡ xi = yi,ṗi = 0 andqi = −pi butẏi has not not yet been fixed.
The EOMs (15) together with the canonical PBs [xi, pj] = δij, [yi, qj] = δij (16) imply that the Hamiltonian is given by The principle of a minimal dynamical realization of G0 forces us to put f = 0. Thus we see that the Hamiltonian H is given by which leads, due to (16), to the EOM for yi, namelẏ The conserved generator of rotations Li is then given by The two parts of Li act separately on the ( x, p), resp. ( y, q) parts of the phase space.
The Hamiltonian dynamics given by H (17) and the nonvanishing PBs (16) is equivalent to the dynamics derived from the Lagrangian Finally we conclude • A minimal dynamical realization of the unextended Galilei algebra in three dimensional configuration space requires a 12-dimensional phase space spanned by ( x, p, q, y) • The relation between energy H and the momenta (velocities) is nonstandard, • Momenta pi and velocities yi are dynamically independent, • The qi have no counterpart in standard particle mechanics, • Galilean massless particles move with arbitrary finite velocity.

Dilations
G0 may be enlarged by dilations with arbitrary dynamical exponent z. The corresponding generator has to satisfy the following PB-relations (cp [18] for z = N 2 ) It can easily be checked that D is given by Below we will show that the value of z becomes fixed by the coupling of our massless particles to gravity.

Coupling to gravity
To couple our massless particles to the gravitational field strength gi( x, t) we have to be consistent with Einstein s equivalence principle: Locally, the gravitational field strength is equivalent to an accelerating frame. The only known nonrelativistic EOM for the particle trajectory satisfying this form of the equivalence principle is given by the Newton law: because (23) is invariant w.r.t. arbitrary time-dependent translations (Milne gauge transformations) [19] xi provided that gi transforms to The EOM (23) is realized if we add to L0 an interaction part (minimal coupling) Then the EOMqi = 0 gets replaced bÿ The total Lagrangian L = L0+Lint now becomes invariant w.r.t. the gauge transformations This gauge invariance, resp. Einstein s equivalence principle still holds if we add in (26) further terms containing spatial derivatives of gi i.e. by replacing gi → gi − κ1 gi + higher order terms, (29) where is the laplacian and K1 is a constant.

Lagrangian formulation
In the last subsection (2.2) we have introduced a model for the motion of one Galilean massless particle in an external gravitational field g( x, t).
Now we want to generalize this model in two respects: • 1. Instead of one particle we consider a continuum (fluid) labeled by comoving coordinates ξ ∈ R 3 (Lagrangian formulation of fluid dynamics, cp. [20]).
• 2. We treat the gravitational field g as a dynamical variable in order to get a selfgravitating fluid.
To realize the second point we add to our previous Lagrangian L0 + Lint a kinetic part L f ield for the gravitational field. We choose as usual where G is Newton's gravitational constant.
Then our Lagrangian, formulated in terms of (xi,ẋi), (qi,qi) and g, becomes where n0 is the (constant) particle density in ξ-space.
Varying qi , resp., xi leads to the continuum generalization of the previous EOMs (23) and Now we have two possibilities to go further: In the first case we take gi = −∂iφ and vary φ getting with n( x, t) = n0 J −1 | ξ= ξ( x,t) (particle density) (36) and where ξ( x, t) denotes the function inverse to x( ξ, t).
The right hand side of the Poisson equation (35) describes a dynamically generated effective gravitational mass density which may be of either sign with + sign leading to an attractive gravitation, -sign leading to a repulsive gravitation.
This promotes the self-gravitating fluid to being a possible candidate for the dark sector of the Universe.
Note that our EOMs are invariant w.r.t the gauge transformations (see (24), (25)) when φ transforms as and qi and n remain invariant.
In the second case we vary gi instead of φ and this leads to a linear relation between qi and gi: Note that the expression for gi can be rewritten as The expression (41) allows to eliminate, from the EOMs, the unphysical phase space variables qi in favour of the gravitational field strength gi.
This removes the unphysical degrees of freedom. However, we have to pay a prize for this as now • the gravitational field strength is not automatically given by the gradient of a potential, • the expression (41) destroys the gauge symmetry i.e. it fixes the gauge.
Remark The gauge symmetry can be restored by considering a curl-free velocity field u = ∇u and treating u as the variable to be varied [16].
Inserting qi from (41) into the previous EOMs we get two coupled EOMs for x( ξ, t) and (initial condition x( ξ, 0) = ξ) and Furthermore by inserting (41) into the Lagrangian (32) we obtain where we have used the identity It can be easily checked that the Euler-Lagrange EOMs following from (44) are identical with (42) and (43).
The EOMs do not contain the gravitational constant G. The Lagrangian contains it only as a common factor. Moreover, our modified Lagrangian (44) does not split into two parts involving a free and an interacting term. Clearly this strange property is a consequence of the phase space reduction due to (41). We know only one other physical system possessing a similar property: the interacting Chaplygin gas if one eliminates the particle density from the Lagrangian (see [21], Sect. 2.1, item (i)).

Relabeling symmetry and the transport equation for the gravitational field
Any fluid dynamics in the Lagrangian formulation has to be invariant under infinitesimal relabeling transformations (volume preserving diffeomorphisms) (see [20]) For the corresponding Noether charge Q we obtain from the Lagrangian (44) where we have alerady performed one partial integration. But from the EOMs (42), (43) it follows that the integrand in (46) is already conserved i.e. that is a conserved quantity and so is a function of ξ only. By solving (47) for d dt g k n we obtain the once-integrated form of the EOM (43) which is a transport equation for the gravitational field (cf. [22] for the standard selfgravitating dust fluid).

Eulerian formulation
In the Lagrangian formulation the fluid particles are labeled by comoving coordinates ξ. So a Lagrangian observer moves with the particle. On the other hand an Eulerian observer is located at a fixed position x (see [20]). Therefore we obtain the fluid fields in the Eulerian formulation from the ones given in the Lagrangian formulation by the following change of independent variables Correspondingly, we have to transform the time derivatives with The EOMs then become: Euler eq.
Transport eq. for the gravitational field: where the auxiliary variable obeys the EOM Dt θi + θ k ∂i u k = 0 (54) and the particle density n obeys the continuity eq.: All these equations can be derived from the Lagrangian Here θ, α and β give the Clebsch-parameterization of the vector field θi( x, t) They obey the Euler-Lagrange EOMs Operating with Dt on (57) leads then to the EOM (54) for θi. All the remaining EOMs are Euler-Lagrange EOMs derived from (56).

Coupling with baryonic matter
We describe baryonic matter (B) by a standard self-gravitating fluid coupled to the darkon fluid by the jointly created gravitational field φ. Then, within the Lagrangian formulation, we have to add to (32) the following baryonic Lagrangian where m is the (constant) baryonic mass density in ξ -space.
Varying x B i leads to the Newton equation whereas varying φ in the total Lagrangian where LD is the darkon fluid Lagrangian (first term in (34)), leads to the Poisson equation whose r.h.s. consists of the sum of the darkon fluid source and the usual B-source. The B-mass density ρB is defined by where J is the Jacobian defined by (37). We note that ρB satisfies the local conservation law (continuity equation) where u B is defined analogously to (51).
Adding baryonic matter leads to the following changes in the foregoing subsections: The gravitational force appearing in LD is the total one gi = −∂iφ and so is it in the case of the EOMs (33) and (34). The solution of the Poisson equation (62) is a superposition of the gravitational potentials created by the B and the D-sources and So (67) is now the substitute for (41) and, therefore, the conserved quantity (47) becomes (note that LB is separately invariant w.r.t. relabeling transformations (45)) Then, in the Eulerian formulation, we obtain finally instead of (53) the transport equation where θi obeys the EOM (54).

Hamiltonian dynamics
The derivation of the Hamiltonian dynamics, ie the formulation of classical dynamics in terms of a Hamiltonian and a PB-algebra, is an indispensable prerequisite for any quantization procedure. We start with the Lagrangian (56) from which we obtain immediately the Hamiltonian where we have used the identity [21] and then neglected the boundary contribution arising from the last term in (71). For the we choose as independent field variables A ∈ (n, θ, nα, β, ui, gi). To read off from (70) and (72) the PBs for the A-fields we next rewrite their EOM as follows : The EOM (73) have the Hamiltonian form (72) if the fluid fields A obey the following nonvanishing PBs and For these PBs all Jacobi identities are trivially fulfilled. It is interesting to note that the PB-algebra (74-76) decomposes into three irreducible parts. Each component has a canonical structure which allows quantization by applying Dirac s rule (details will be given in a separate paper) whereÂ(B) denotes the quantum operator corresponding to the classical function A(B) . The PB-algebra (74-76) possesses a Casimir (being a quantity whose PBs with all elements of the PB-algebra vanish) defined by To prove that C is a Casimir we proceed in two steps: • We choose a particular relabeling transformation (45) with αi( ξ) = − ikl ∂θ l ( ξ) ∂ξ k . The corresponding Noether charge (46) is then given by But Eulerian variables do not respond to a relabeling of the Lagrange variable ξ [21]. So Q is a Casimir of the PB-algebra (74-76).
• It remains to express Q in terms of Eulerian variables. Do do this we get from the definition of θi( x, t) However, using (80) and the identity Hence we finally have Of course the Casimir property of C can be shown also directly by bracketing (78) with all fluid fields and using the PB-algebra (74-76).
We note that the vortex helicity where ω ≡ ∇ × u which is a Casimir in standard fluid dynamics, is not a Casimir in the case of our model. He is a conserved quantity if g is the gradient of a potential, but it has a non-trivial PB with g, namely: [gi( x, t), He] = 8π G ωi( x, t).
Up to now we have used, within the Hamiltonian framework, those fluid fields which lead to the simplest possible form (74-76) for the PB-algebra. But in many applications it is more convenient to introduce instead of the auxiliary field θi the momentum field pi where we have used the definition of pi in the Lagrangian formulation and then changed to the Eulerian formulation. In terms of the momentum density npi we obtain for our PB-algebra, besides the PB-relation (76), which can be obtained from the transformation formulae and the canonical PBs in the Lagrange formulation (continuum generalization of (16)) and if, in addition, we use the relation (41) for gj.

Space-time symmetries
In this section we will show • The darkon fluid possesses the unextended (zero mass) Galilei algebra G0 as a symmetry algebra. The corresponding energy-momentum tensor is not symmetric and contains a negative pressure term.
• G0 may be enlarged by dilations with dynamical exponent z = 5 3 . This particular value of z is determined by the minimal coupling to gravity as given by eq. (26).
• Expansions are not a symmetry of the darkon fluid.

Space-time translations and the energy-momentum tensor
The linear momentum (generator of space translations) is given by where the momentum density npi obeys the local conservation law Thus the momentum tensor is not symmetric and the pressure P is negative.
The Hamiltonian (70) (generator of time translations), expressed in terms of the momentum density, takes the form with The energy density Ttt, of course, obeys the local conservation law If we identify, as usual, the space-time component of the energy-momentum tensor with the momentum density we may conclude, that the energy-momentum tensor T α,β with α, β ∈ (t, k) obeys the standard conservation law But, because T αβ is not symmetric, we note that We remark that such a nonsymmetric energy-momentum tensor was introduced for the first time in 1947 by Weyssenhoff and Raabe [23].

Zero mass Galilean symmetry
Galilean boosts are generated by which is a conserved quantity because the EOM for the gravitational field gi is given, due to (73) and (85), byġ The PB of the boost generator Ki and the momentum Pj vanishes as is easily see from the fact that Pi is the generator of space translations The vanishing of the right hand side of (105) tells us that the total mass of the darkon fluid is zero.

Angular momentum
The continuum generalization of expression (19) for the one-particle angular momentum is given by By using the identity (41) and by passing to the Eulerian variables we obtain

Anisotropic scaling
Anisotropic scaling, characterized by a dynamical exponent z = 1, implies that the time and In section 2.1 we have shown that the dynamics of free massless particles allows for an implementation of dilations with an arbitrary z. To see what happens in the case of a darkon fluid we consider a continuum generalization of (22) of the generator of dilations D where in the last term we have used the relation (41) between the field q and the gravitational field g. Taking now the time derivative of D (110) and by using the EOMs (73) we obtain Thus D is conserved if z takes the value z = 5 3 . It may easily be checked that with z = 5 3 all PBs between D and the elements of G0 are in agreement with the PB-relations (21).
The dilations are implemented by the following transformation of the fluid fields A where zA denotes the scale dimension of A. . For an infinitesimal dilation we get which leads, due to the explicit form of D (110) and the PBs (76), (86-88), to the following values for the zA: and therefore, due to (85), zp i = 1.
Let us finally prove that our EOMs are not invariant with respect to expansions. We prove this by demonstrating the contradiction. So consider the expansions: Suppose that we would be able to implement such expansions. Then there would exist a generator K of expansions which obeys, in combination with H and D an O(2, 1) algebra.
On the other hand, as is well known [18], the PB-algebra between the elements of G0 and the O(2, 1) algebra would not close for z = 2. The PB between K and the generator of boosts Ki would then imply the existence of a conserved generator of accelerations Fi But we know from the discussion given in section 3 that the relation (41)  Casimir which can be used to obtain a foliation of the three-dimensional phase space into an infinite number of two-dimensional phase spaces. Within the two-dimensional phase space we get a completely integrable system. We are able to construct a corresponding Lagrangian.
Due to the appearance of a constraint it is not possible to quantize the arising Hamiltonian by applying the Dirac quantization rule. An alternative way seems to be the affine quantization (see [26]).
In subsection 6.3 we solve the cosmological EOMs with the cosmological redshift z as the independent variable. By choosing some integration constants to be positive our model shows qualitatively the transition from a deceleration to an acceleration phase of the Universe. Fixing two integration constants by observational values we obtain a prediction for the cosmological history in general agreement with the existing data of the Hubble parameter.

Cosmological EOMs
The realization of the cosmological principle within the Lagrangian formulation leads to the following structure for the field variables xi( ξ, t) = a(t)ξi, where a(t) is the cosmological scale factor, and gi( x, t) = n(t) n0 g(a(t))xi, n0 = n(0).
The split of the r.h.s. of (119) into two factors and defining g as a function of a(t) are a matter of convenience. If we next put all this into the original EOMs we obtain the following cosmological EOMs • From the definition of n (see (36)) we get . (120) • From the EOM (42) and using (120) we geẗ which is a Friedmann-like equation.
• From the EOM for gi (43) and the definition of θi (see (47)) we geṫ where K1 is related to θi by We note that possible derivative terms in the gravitational coupling (29) give no contribution to the cosmological EOMs. Taking into account, additionally, baryonic matter (see 3.4) will not change the cosmological EOMs. In such a case g(a) would be changed by an additive constant (see [15]).

Hamiltonian dynamics
The cosmological EOMs (121) and (122), when written in 1 st -order form, take the forṁ So we have to consider a dynamical system in a three-dimensional space. This finite dimensional space, called usually mini-superspace, will be the phase space for our Hamiltonian dynamics. The dynamical system (124) possesses two constants of motion where the PBs are required to fulfill the Jacobi-identities and H has to be a function of the two constants of motion Q2 and Q3 only. To reduce the arbitrariness in the construction of Let D be the conserved generator of dilations. Then we get from (127) and (128) that (with no summation over α in the middle term !) leading to which is equivalent to However, this expression tells us that we have zH = 1. Thus neither Q2 nor Q3 but Q2Q3 is a possible candidate for the Hamiltonian H. So we choose where the factor K −3 1 has been inserted in order to get a simple form for H which will be obtained when we rewrite it in terms of some new variables which will be introduced below.
With this H the PBs for the xα have, according to (127), to be chosen such, that holds. But (134) is equivalent to where The matrix A αβ is not invertible (it is anti-symmetric and of odd order). Therefore (135) has no unique solution for the variables y β . Even if we require in addition, the validity of the Jacobi identities, the solutions for the y β will not be unique. To render the PBs unique we which shows that the PB between x and y has to vanish [x, y] = 0.
As the third variable (of our phase space) besides x and y we choose w = g 2 aK 1 which has zw = 1. Then, due to zH = 1 the Hamiltonian H (133) has the structure In order to determine the remaining PBs [x, w] and [y, w] we next rewrite the EOMs (124) in terms of the new variables x, y and ẇ But (142) It is easy to check that the PBs (139) and (144) satisfy all Jacobi-identities.
Expressing now the original variables a, b and g in terms of x, y and w a = K The equations (146) can be summarized by one equation (no summation over α or β.) The PB-algebra (147) is identical with a construction given by Hojman [24] who starts also with some EOMs, a Hamiltonian and a symmetry. But in contrast to [24] our derivation of (147) shows for the particular case of dilation symmetry the uniqueness of the PB-algebra.

One exact solution
The EOMs (142) have a simple power law solution of the form which for t0 = 0 are dilation invariant. We will later show that (148) governs the behaviour of the cosmological solutions at early times. Furthermore the solution (148) determines a relative minimum of the Hamiltonian (140) at H = 0.

Reduced phase space
In the following we will solve the EOMs (142) for x as a function of y. So we reduce the dimension of phase space from three to two. To do that we can either use standard analytical methods or algebraic methods. The latter shows more clearly the physical nature of this reduction. For this reason we will present both methods.

Analytical method
From the first two EOMs in (142) we obtain immediately an ODE of Bernoulli-type for where x (y) ≡ dx(y) dy . As usual we use the transformation to obtain a linear ODE for u(y) The point y = 1 2 is a singular point of the DE (151).Therefore we obtain one solution u−(y) for y < 1 2 and another one u+(y) for y > 1 2 .
with real-valued constants k+ and k−. Note that the first term in (152) represents a particular solution of the inhomogeneous DE (151) whereas the second one represents the most general solution of the homogeneous part. The two solutions u± are not connected to each other by an analytical continuation in y. This implies that the reduced phase space consists of two dynamically disconnected parts with either y < 1 2 or with y > 1 2 . We will show later that the validity of the condition is necessary in order to explain the accelerating expansion of the Universe. So we will use y > 1 2 exclusively in the following.

Algebraic method
The PB-algebra (139), (144) possesses a Casimir where we have used the equivalence between (153) and the condition To prove the Casimir-property of C we observe first that zC = 0. So C can depend only on x and y. By using the definition of the Qi, i = 2, 3 and the transformation formula (145) we But as ∂H ∂w = 0 we see that [w, C] must vanish. This reasoning shows that any function of the independent variables which is a constant of motion and has a vanishing scaling exponent is a Casimir of our PB-algebra. But there exists no other function, functionally independent from (154), which possesses this property. Now we use the fact that the equation defines, for different values of k, a foliation of the three-dimensional phase space (cp [25]). So we may solve (154) for x at fixed k and obtain x(k, y) = 1 in accordance with (152) if we identify k = k+.
This way we see that our system is described by a Hamiltonian dynamics on a twodimensional phase space with variables w and y and the Hamiltonian where for the last equality we have used (159), and the PB [w, y] = 2w leading together to the following Hamiltonian EOṀ The dynamical system (162) Inserting (163) into the first eq. in (162) we finally obtain y(t) by a quadrature.

Generator of dilations
We can also give an explicit expression for the generator of dilations in two-dimensional phase space. Fromḟ = H (131) we getẏ ∂f ∂y = H and so

Lagrangian
The EOMs (162) are Euler-Lagrange EOM derived from the Lagrangian where H(w, y) is given by (160). To prove this it is sufficient to note that is the momentum canonically conjugate to y. Indeed, by using (161) we obtain [y, Π(y, w)] = 1.
We can now use (166) to eliminate w from the Hamiltonian (160) to obtain With (167) and (168) we have finally Hamiltonian dynamics in a two-dimensional phase space expressed in terms of canonical variables y and Π.
How to quantize this system?
We would like to quantize the system (167) and (168) The reason for this is that, for self-adjoint operatorsŷ andΠ (169) has, up to unitary equivalence, only one irreducible representation given bŷ [26]). So the spectrum ofŷ covers the whole real axis. To overcome this difficulty we follow J. Klauder (see [26]) and consider instead of the canonical quantization (169) the affine quantization given by and choose the irreducible representation for q > 0. Details of such a quantization procedure still have to be worked out.

Solution of the EOMs
Let us first put the two constants of motion Q2,3 equal to numbers K2,3. Next we solve the defining equation of K2 (125) forȧ = b and obtaiṅ So, with K2 > 0, equivalent to condition (153), we must have in order to describe an expanding Universe.
Using the definition of K3 (126) we obtain (we consider now g as a function of the redshift z, (1 + z) −1 ≡ a), a cubic equation for g(z): where the transition redshift zt is defined by Let us look now at (173) for K2 > 0 and K3 > 0 together with the Friedmann-like equation For z > zt we have g < 0 and so, due to (175)ä < 0. So, for z > zt we are in the deceleration phase of the early Universe. On the other hand, for z < zt we have g > 0 and then, due to (175),ä > 0. So, for z < zt we are in the acceleration phase of the late Universe.
To get these results we have to require that K2,3 > 0.
To see this note that • From (174) we conclude that K3 > 0.
• Let zt be a zero of g. Then differentiating (173) at z = zt shows that K 2 K 3 g (zt) < 0. But the transition from the decelerating to the accelerating phase requires g (zt) < 0. . Thus we must have K 2 K 3 > 0.
Next, to simplify (173) we introducẽ Then (173) becomes (we recall that k = K3K whereg now depends on the two parameters zt and k only. Using (171) the Hubble parameter H(z) ≡ȧ a (z) may be expressed in terms ofg However, it is more convenient to consider instead of H(z) the normalized Hubble param- for which we find So, if we know H0, the cosmological history is determined in our model completely by the two parameters zt and k.
To obtain k in terms of observable quantities we consider the deceleration parameter which, when expressed in terms ofg, zt and k takes the form: Thus the value of k can be determined from the knowledge of the present deceleration parameter q0 = q(0) and of zt.

Remark:
The expressions forg(z), h(z) and q(z) in our paper [15] can be obtained from the present ones by the substitutionsg(z) → − √ 6g(z) and k → √ 6k. Furthermore, the constants K1 and n0 are related to the constants D and β from [15] by the relations K1 = −3Dβ and n0 = 3D 4π . Asymptotic results for z zt (183) Thus we obtain a power law behaviour for h(z) which shows that asymptotically our cosmological solutions are scale invariant. On the other hand the deceleration parameter reaches asymptotically a constant value The asymptotic behaviour (184)

Distances
We must decide how to relate observational data of distances (e.g. of supernovae) to the Hubble parameter H(z). In standard FLRW cosmology we obtain, in terms of comoving coordinates for a flat Universe and for a radial light ray (see section 1.2 in [27]) the coordinate distance In any relativistic theory of gravitation which possesses a metric the limit of an isotropic, homogeneous and flat Universe is described by the FLRW metric given in comoving coordinates by Then eq. (187) arises from the requirement that light propagates along a null geodesic Note that our darkon fluid model is neither a relativistic nor a metric theory. However, it may be understood as a limiting case of such a theory as shown in section 9. On the other hand, in a nonrelativistic theory we do not have an expanding space but only moving particles in a static space. But both points of view lead to the same observable effects as has been already noted by Milne in 1934 [19]. We need only to assume 'that, relative to a particular observer O the speed of light is a constant c independent of the light source and that the classical formula for the Doppler effect is valid' (McCrea, page 351 in [28]; see also section 5.2 in [29]). So both points of view allow us to use the expression (187) for the coordinate distance in our model.
The dimensionless coordinate distance ξ0, defined by may easily be computed in terms of the two parameters zt and k by using (180) with (177) leading to (cp. Appendix A in [15]) where i = 1, 2, 3 and cyclic permutations and the xi are the roots of the cubic equation: given by In terms of ξ0 we obtain, as usual, the dimensionless luminosity distance DL(z) the related distance modulus µ(z) µ(z) ≡ 5 log 10 DL(z) − 5 log H0| 100kms −1 M pc −1 + 42. 38 (195) and the angular diameter distance DA(z)

Predictions versus observations
In subsection 6.3 we have shown already that our model explains qualitatively the transition from a deceleration phase of the early Universe to an acceleration phase of the late Universe if we choose the constants Ki(i = 1, 2, 3) to be positive. In this subsection we will show that • the quantitative predictions for the Hubble parameter H(z) are in a general agreement with the observational data, • the distance moduli for GRBs seem to show a peak at large z for the deceleration parameter q(z) in agreement with our predictions, • our unified model for the dark sector is unable to predict the present dark matter contribution Ωm0 and, therefore, it can neither explain the BAO data nor the data on the CMB-shift parameter by using the present models for them.
At present we do not know what the changes for the prediction of the Hubble parameter (180) would look like had we generalized our model to a general covariant theory presented in section 9. Therefore we have not performed, so far, any 'least squares fits to the data.

Hubble parameter
In fig 1 we compare the best out of three predictions of our model for h(z) given in [15] with observed Hubble parameters which we adopt from table 1 of [30]. In addition we show the ΛCDM-fit to these data taken from [38] In [15] we used zt = 0.71 and q0 = −0.57 taken from a ΛCDM-fit to some data given in [31]. Let us point out that these values for zt resp. q0 are in agreement with very recent model independent constraints on cosmological parameters [35], [36], [37]. For our prediction we have fixed the normalization in fig. 1  leading to k = 2.123.

Remarks on distance moduli data
The most recent supernovae (SN Ia) distance moduli data are those of the Union 2.1 compilation [39]. But we do not see any necessity to give a detailed comparison of these data with the predictions of our model (see eq. (195)) because it has been claimed very recently by O.
Farooq et al. [30] that the recent 21 Hubble parameter versus redshift data points (see fig.   1) give constraints on DE models which are almost as restrictive as those from the Union 2.1 compilation. In addition to the SN Ia data we have for larger z values up to z = 8.1 distance moduli data from Gamma-Ray Bursts (GRBs) at hand (see table II in [40]]). Using such data one obtains the tentative result that the deceleration parameter q(z) shows a maximum roughly at z ∼ 1.6 [41] in qualitative agreement with the predictions from our model (see fig.   2 in [15]).

Remarks on BAOs and the CMB-shift parameter
Baryon acoustic oscillations in the early Universe are imprinted as peaks in the late time matter power spectrum (cp. [42]). But the physical interpretation, mainly of the first peak position, depends heavily on the present baryonic and dark matter contents Ω b0 resp. Ωm0 of the Universe (see fig. 4 in [42]. This issue is usually ignored in the data analyses performed in the literature [43]. But our model, as a unified model for the dark sector of the Universe, is unable to predict Ωm0 (for more details cp. Section 6.6) and therefore we cannot at present analyse these peaks. The same argument holds for the CMB-shift parameter (cp. [44]). This issue has been discussed already in our recent paper [15].

Comparison with alternative models for dark energy
In the following we will sketch how to perform a comparison of our model with alternative models of the dark energy. Any details will be left for the future.
The Friedmann equations, derived from GR including a cosmological 'constant' Λ, are for a flat Universe (see e.g. [27]) where ρ is the total energy density of the Universe (sum of the matter (M ) and dynamical dark energy (DE) if we neglect radiation) and, if we assume that matter is described by pressure less dust, p is the (supposed negative) pressure of DE ρ = ρM + ρDE, p = pDE < 0.
Let us now assume that the first Friedmann equation (197) holds. Then the second equation (198) can be replaced, as is well known, by the continuity equation Note that if we allow a weak time-dependence of Newton's gravitational constant G the last equation (200) would have to be generalised (see [45], [46]). However, this case will not be discussed here, instead we refer the reader to [46] and the literature cited therein.
At this stage it is important to contrast here two possibilities for the description of the 'dark energy' [45]: • Some kind of fluid described by an equation of state (EoS) W (z) • A (possibly) time-dependent Λ with a fixed EoS Now our Friedmann-like equations are (see (175) and (178)) and Comparing the corresponding right hand sides with (197,198) we obtain two equations for the effective functions ρ, ρΛ and p In the first scenario we have to put ρΛ = 0 and arrive at two equations for the three functions ρM , ρDE and p. If we suppose that we have no interaction between CDM and DE  baryonic matter), ρM,0, which is not determined in our unified model for the dark sector of the Universe, i.e. ρM,0 is a free parameter in our model (for a very detailed discussion of this issue see [47]). Therefore our model, as any other unified dark fluid model, is not able to predict an EoS for DE. However, on the other hand, we can define an EoS WD for the darkon fluid as a whole Whether WD for some z-region shows a phantom-like behaviour (wD < −1) will be left for future considerations.
In the second scenario we have to put ρDE = p = 0 which corresponds to model II in [46].
We decompose ρM into its DM and B parts and take into account the local conservation of ρB which leads, in the cosmological limit, to (cp. (208)) ρB = ρB,0(1 + z) 3 .
Note that ρB,0, in contradistinction to ρM,0, is directly observable. Then (205), (206) lead to predictions for ρDM and ρΛ for a given set of constants Ki (i = 1, 2, 3). More details and their comparison with the results communicated in [46] will be left for the future.

Modeling dark matter halos by a steady state darkon fluid
We start with the darkon fluid EOMs in the Eulerian formulation as given in section 3. By taking their time-independent form (steady state equations) we will derive a nonlinear ordinary differential equation (ODE) for the gravitational potential in the spherically symmetric case.
The solutions of this ODE may serve as a model for the dark matter (DM) halos of galaxies. In particular we consider DM-dominated galaxies in which case the halo dominates the galactic gravitational potential and therefore determines the rotational velocity of stars (rotation curves (RCs)).
In subsection 7.1 we derive the nonlinear ODE for the gravitational potential in the spherically symmetric case and the behaviour of their solutions for large distances from the centre of the halo. Then, in subsection 7.2, we present the results of the numerical solutions of this ODE and compare them with the main characteristics of the observational data for the RCs of DM-dominated galaxies.

Equation for the gravitational potential
From the darkon fluid EOMs (73) we obtain immediately the following steady state equations: Steady state equation for the gravitational field where the auxiliary field θi is given by the solution of Euler equation: Continuity equation: Next we consider the halos as spherically symmetric objects. Then the vector fields appearing in (212-215) take the following form: where φ is the gravitational potential. By using (216) in the steady state equations we obtain from (212) where θ, obtained by the integration of (213), is given by For convenience we have chosen to write the integration constant as − γ 2α . Euler's eq. (214) leads to the Bernoulli eq. 1 2 and the solution of the continuity eq. gives where α is an integration constant.
As in the Lagrangian formulation the vector field θ is a conserved quantity we may wonder whether it is possible to connect the integration constants α and γ appearing here and K1 appearing in the cosmological context (see (123)).
We find, in our case, This expression has a different ξ dependence from that of the r.h.s. of (123). So, unfortunately, there is no relation between α and γ on one hand and K1 on the other.
We note that the solution (220) produces a singularity at the origin on the r.h.s. of the continuity eq. (215). This singularity might be interpreted as the central black hole of the galaxy (see [48] for the existence of a central black hole in LSB-galaxies). But a black hole describes a sink for any kind of matter. So we get α < 0 and therefore Inserting all these expressions into (217) we obtain a ODE 1 which may be understood as the Poisson eq. for a self consistently determined effective DM mass density ρ Next we switch over to a dimensionless potential φ by the substitution From the foregoing equations we easily conclude that γ has the dimension m 5 s −5 as it should be.
So, altogether, we end up with a nonlinear, non-autonomous ODE for φ in which φ is nonpositive and satisfies the natural boundary conditions We note that the ODE (226) and the first boundary condition (φ(∞) = 0) are invariant w.r.t. to the scale transformation φ(r) →φ(r) ≡ φ(λr), λ > 0.
Thus the second boundary condition is necessary to make the solution unique.
To simplify (226) we introduce, instead of φ the function ψ = −2rφ which then has to satisfy the ODE whose positive solutions have to be determined. To determine the required solutions of (229) let us consider first (229) with the boundary conditions ψ(1) = η, η > 0 and ψ = β.
In this case we can use the following lemma, which is a particular case of a more general theorem derived by S. Taliaferro (see theorem 2.1 in [49]) Lemma [49]. Let ψ(r) be a positive solution of (229) satisfying (230) for fixed η > 0 and an arbitrary real number β. Then there exists r2, satisfying 1 < r2 < ∞ such that limr→r 2 ψ(r) = 0 and ψ(r) cannot be continued past r2 as a twice continuously differentiable solution of (229).
To use this lemma to the study of properties of required solutions of (226) it is advantageous to perform first the transformation Then (229) with the boundary conditions (230) gets transformed into the following au- Using the relation (234) may be rewritten as an integral equation for φ(r): which is cast in such a form that the first line contains the usual integral of the Poisson equation, the second line a constant and the third one a term being proportional to 1 r . But the latter is in conflict with the original ODE (226). The reason is that the transformation (235) becomes singular at r = 0. Therefore the coefficient of 1 r has to vanish which fixes the value of b at But the lemma given above holds for any b, and so in particular also for b = bc. Thus, due to the relation (235), it holds also for φ(r) ie. limr→r 2 φ(r) = 0 and so φ(r) cannot be continued past r2.
Next we return to looking at solutions ϕ of (232). We know from the results given above, that • we have to find a positive solution ϕ at least for 0 < x < x2 with ϕ(x2) = 0.
• we have to vary b until it reaches a value bc which corresponds to leading to To relate the solutions of (232) to the RCs v(r) of a DM-dominated galaxy we recall that v(r) is determined from the equality of the centripetal acceleration and the gravitational force acting on a star in circular motion [50] v 2 (r) where the factor γ 2 5 arises due to the dimensionlessness of φ after the substitution (225). Then, using (235) we can determine v in terms of the derivative of ϕ

Predictions versus observations
To begin with we would like to discuss how to fix, at least in principle, scales for distances and velocities. Due to the dilation symmetry of our model (see (228) More detailed discussion has been postponed until we have generalised our model to a general covariant one as suggested in section 9.
For b > bc the graph of ϕ(x) crosses the x-axis at some x0 < 0 where x0 is an increasing function of b. We believe, although this is numerically hard to verify, that x0 → −∞ for b decreasing to bc. For x < x0 the solution becomes complex valued. This behaviour is illustrated in fig.2 (b = −0.03) and fig.3 (b = bc).
On the other hand for b < bc the graph of ϕ(x) starts to rise rapidly at some x < 0 and goes to infinity. This behaviour is clear from fig.4 (b = −0.032).
The behaviour of the graphs of ϕ and ϕ for b ≥ bc resp. b < bc as shown in figs.  Hence we have to compare qualitatively the RCs calculated for b = bc with observational data. For that we plot in figs. 6-7 the rotational velocities v(r) determined from eq. (241). In fig 6. we show the behaviour of v(r) at small radii with the extrinsic parameters γ 1 5 , resp. R fitted, for fixed a = 10 to the RC of the dwarf galaxy UGC 8490 (data taken from [56]). Fig   7. shows the global behaviour of v(r) in dimensionless units.
The graphs for the darkon fluid contribution to the RCs given in figs. 6-7 show the following qualitative characteristics when compared to observational data on DM haloes (for a very recent review see [51]) • They start at a radius r1 ∼ 0 and rapidly increase reaching v(r) ∼ 76kms −1 at r ∼ 4kpc.
• Beyond this increase our model predicts a rather flat, but still very slowly increasing behaviour for the RCs until extremely large radii. This is basically in agreement with all observations according to which the RCs of a galaxy stay approximately constant after attaining a maximum at about 5 kpc [52]. This statement is true, in particular, for the DM-dominated galaxies: the LSB-galaxies (cp. the RCs reproduced in [53] and in figs. 3,4 of [54]) and dwarf galaxies (cp. [55] and the RCs reproduced in [56]).
The behaviour of ϕ(x) within this flat-like region can be well approximated by (see fig.3) where c ∼ 126.5 has been fixed by f (0) = 10. Then we obtain f (0) ∼ −0.03162, which is a good approximation for bc (242). f (x) is the solution of the ODE and so is an approximate solution of the ODE (232) if ϕ ∼ const.
• They terminate at an extremely large radius roughly given by r2 ∼ e 130 with a diverging RC as r → r2. Of course, this contradicts all observations. But the size of r2 is clearly orders of magnitude larger than the size of the visible part of the Universe: In the FRW model the size R0 of the Universe is given by R0 = 0.5ct0 [57], where t0 ∼ 13.7 Gyr is the age of the Universe [50]. Thus we obtain R0 ∼ 2.1 Gpc. The choice R = 4.10 14 pc in fig. 6 gives r2 ∼ 10 71 pc, which is about 62 orders of magnitude larger than R0.
We conclude that this highly unphysical behaviour of v(r) as r → r2 is possibly a consequence of considering only an isolated halo. We expect an improvement of this situation if we embed the halo in the time-dependent darkon fluid (to be done by an appropriate generalization of the considerations in the next section 8).
8 Influence of the cosmic expansion on binary systems In the following we consider two astrophysical objects, one of large mass M (e.g. the sun) and another one of small mass m M (e.g. a planet) embedded in the time dependent darkon fluid which, asymptotically behaves like the cosmological solution (see section 6.2). We would like to assess the change of Newton's force of attraction between the two bodies due to the presence of the darkon fluid. To be specific we approximate this binary system by a point-like test particle of mass m which moves in the time-dependent gravitational potential created by a point-like mass M and the overall darkon fluid considered in the rest system of the heavy mass M .
Within GR the corresponding problem is to find, as a solution of Einstein's equation, an interpolation between the Schwarzschild and the FLRW metric. Then the motion of the test particle is the geodesic motion in this metric. A solution to this problem has been given in a very recent paper [58] (see also the recent review [59] and the literature quoted therein, which covers also the history of this problem starting with the paper by Mc Vittie in 1933 [60]). In the Newtonian limit one obtains for the EOM of the test particle (see [58]) xi =ä a xi − GM r 3 xi.
9 Outlook for a general covariant theory The Poincare algebra treats spatial and time variables on the same footing. Then its enlargement by dilations has necessarily a dynamical exponent z = 1 (cp. [61]). But in our model we have anisotropic scaling with z = 5 3 . Therefore, an enlargement of the Milne gauge group (24) to arbitrary space-time diffeomorphisms (Diff) as the symmetry group for a generalized darkon fluid model seems to be forbidden. But the dilation symmetry with z = 5 3 was an emergent symmetry and not an input of our model. So we should look for a general covariant generalization of our model without imposing any form of scale symmetry.
To do this we have first to restore the Milne gauge invariance for the Eulerian action (56) by undoing the gauge fixing (41) leading to S = 1 4πG dt d 3 x 4πGnqi(Dtu i − g i ) − θ(∂tn + ∂ k (nu k )) + nαDtβ + S f ield (257) with the spatial metric given by ηij = δij and S f ield is determined by the expression (31).
Now we propose the following action as the Diff-generalization of (257) where we have used the following notation (in units with c = 1): g is the determinant of the covariant space-time metric gµν with the invariant line element and λ denotes the covariant derivative λ u ν ≡ ∂ λ u ν + Γ ν λσ u σ .
Here Γ ν λσ is the corresponding Levi-Civita connection, n is the scalar particle density and u λ the four-velocity. θ, α and β are scalar fields. The vector fieldqµ will be deduced, as given below, from the Milne-scalar qi. Finally the last term in (258), which is the Diff-generalisation of the last term in (257), is the Einstein-Hilbert action given by where R is the Ricci-scalar.
To construct the vector fieldqµ we proceed in two steps: • By means of a Lorentz transformation we promote the Milne-scalar qi (i = 1, 2, 3) to qα (α = 0,1,2,3), which is a scalar field w.r.t. Diff but a contravariant space-like vector field, labeled by α, in tangent space.
The tetrads E α µ transform like a contravariant vector w.r.t. Diff by changing the Greek index and like a four-vector w.r.t. Lorentz transformations by changing the underlined Greek index. The tetrads are related to the metric tensor by So the limit gµν → ηµν corresponds to E α µ → δ α µ and so toqµ → qµ . Therefore we obtain, in the limiting case, for the individual terms in (258) qν u λ λ u ν → qi(Dtu i − g i ), in agreement with (257).
What about possible scaling properties of the action (258)? The generators of Diff do not form a closed algebra together with the generators of anisotropic scaling transformations  [64], to find scale invariant solutions in the ultraviolet or in the infrared regime. Any further consequences of this new ansatz (258) for a general covariant darkon fluid have still to be worked out. In particular we are interested in the cosmological implications. But due to the larger number of gauge fields in the action (instead of the three gravitational fields g i we have now ten fields of the metric tensor gµν ) we will obtain cosmological EOMs other than the ones studied so far.
In particular we will obtain a prediction for the Hubble parameter h(z) which would probably be different from (180). Therefore nothing can be said at the moment about the asymptotic behaviour of h(z).

Conclusions and outlook
In the present paper we have reviewed our recent results on the darkon-fluid model which is, as outlined in section 9, a first building block for a new general covariant theory describing the dark sector of the Universe. The model involves new physics by using nonrelativistic massless particles with a nonstandard coupling to the gravitational field. The model, which contains no free parameters in its Lagrangian, predicts qualitatively correct values of the late time cosmic acceleration as well as the flat behaviour of galactic rotation curves. These successes encourage us to continue our work. We have to work out the astrophysical implications of the covariant theory and have to compare them with the predictions of the present model and with observational data.
So the main open topics of research are: • Derive the EOMs of the covariant theory.
• Restrict the EOMs to the cosmological regime and compare their predictions with the present model. Perform a best fit to the Hubble parameter resp. distance moduli data.
• Enlarge the covariant model by adding baryonic matter.
• Look for the solutions of the spherically symmetric but time dependent EOMs which attain for large distances their cosmological limits.
• Apply the results to modeling of halos, solar system tests.
• Quantization of the Hamiltonian dynamics for the darkon fluid as a whole or for the cosmological dynamics.
• Answer the question whether the present model is the nonrelativistic limit of the covariant theory in a strict sense.
• Discuss higher order derivative terms within the covariant theory.
These problems are currently under investigation. We hope to be able to present our results soon.