arXiv:1701.08667v1 [gr-qc] 30 Jan 2017

Interactive Unified Dark Energy and Dark Matter from Scalar Fields David Benisty and E.I. Guendelman email: [email protected] and [email protected] bgu.ac.il Department of Physics, Ben Gurion University of the Negev Beer-Sheva 84105, Israel

January 31, 2017 Abstract Here we generalize ideas of unified Dark Matter Dark Energy in the context of Two Measure Theories and of Dynamical space time Theories. In Two Measure Theories one uses metric independent volume elements and this allows to construct unified Dark Matter Dark Energy, where the cosmological constant appears as an integration constant associated to the eq. of motion of the measure fields. The Dynamical space time Theories generalize the Two Measure Theories by introducing a vector field whose equation of motion guarantees the conservation of a certain Energy Momentum tensor, which may be related, but in general is not the same as the gravitational Energy Momentum tensor. By demanding that this vector field be the gradient of a scalar, the Dynamical space time Theory becomes a theory of Diffusive Unified Dark Energy and Dark Matter, this is because demanding that the vector field be the gradient of a scalar introduces now a mechanism that produces non conserved energy momentum tensors instead of conserved energy momentum tensors which leads at the end to a formulation of interacting DE-DM dust models in the form of a diffusive type interacting Unified Dark Energy and Dark Matter scenario.

1

Introduction

There have been theoretical approches to gravity theories, where a fundamental constraint is implemented. For example in the Two Measures Theories [1]-[9] one works, in addition to the regular measure of integration in the ac√ tion −g, also with another measure which is also a denstiy and which is also a total derivative. In this case, one can use for constructing this measure 4 scalar fields ϕa , where a = 1, 2, 3, 4. Then we can define the density Φ = εαβγδ εabcd ∂α ϕa ∂β ϕb ∂γ ϕc ∂δ ϕd , and then we can write an action that uses both of these densities: 1

S=

Z

4

d xΦL1 +

Z

√ d4 x −gL2

(1)

As a consequence of the variation with respect to the scalar fileds ϕa , assuming that L1 and L2 are independet of the scalar fields ϕa , we obtain that: Aα a

αβγδ

Aα a ∂α L1 = 0

(2) det[Aα a]

3

where = ε εabcd ∂β ϕb ∂γ ϕc ∂δ ϕd . Since ∼ Φ as one easily see, then that for Φ 6= 0 ,(2) implies that L1 = M = Const. This result can expresed as a covariant conservation of a stress energy momentum of the µν form T(Φ) = L1 g µν , and using the 2nd order formalism, where the covariant µν derivative of gµν is zero, we obtain that ∇µ T(Φ) = 0, implies ∂α L1 = 0. This suggests generalizing the idea of the Two Measures Theory, by imposing the covariant conservation of a more nontrivial kind of energy momentum tensor, µν [10]. Therfore we consider an action of the form: which we denote as T(χ) Z Z √ √ 1 µν S = S(χ) + S(R) = d4 x −gχµ;ν T(χ) + (3) d4 x −gR 16πG µν where χµ;ν = ∂ν χµ − Γλµν χλ . If we assume T(χ) to be independent ofχµ and λ having Γµν being defined as the Christoffel Connection Coefficients, then the µν variation with respect to χµ gives a covariant conservation ∇µ T(χ) = 0. Notice the fact that the energy density is the canonically conjugated variable to χ0 , which is what we expect from a dynamical time (here represented by the dynamical time χ0 ). Some cosmological solutions of (3) have been studied in [11], in the context of spacially flat radiation like solutions, and considering a gauge field equations in curved space time. For a related approach where a set of dynamical space-time coordinates are introduced, not only in the measure of integration, but also in the lagrangian, see [12]. It is of interest to introduce also a mechanism that produces non conserved energy momentum tensors which can lead eventually to a formulation of interacting DE-DM models. To start we discuss a toy model in one dimension describing a system that allows the non conservation of a certain energy functional, which increases or decreases linearly with time, while there is another energy which is conserved.

2

A mechanical system with a constant power

In order to see the applications of the ideas, we start with a simple action of one dimensional particle in a potential V (x). We introduce a coupling between the total energy of the particle 21 mx˙ 2 + V (x) and the second derivative of some dynamical variable B: Z ¨ 1 mx˙ 2 + V (x)]dt S = B[ (4) 2 2

The equation of motion according to the dynamical variable B, give that the second devirative of the total energy is zero. In other words, the total energy of the partice is linear in time: 1 mx˙ 2 + V (x) = E(t) = P t + E0 2

(5)

where P is a constant power which given to the particle or taken from it, and E0 is the total energy of the particle at time equals zero. From the equation of motion according to coordinate x we get a close connection between the dynamical variable B and the coordinate of the particle: m¨ x

d2 B d3 B d2 B ′ + m x ˙ = V (x) dt2 dt3 dt2

(6)

with the equation (5) give: ¨˙ 2V ′ (x) B P = p − ¨ B 2m(E(t) − V (x)) 2(E(t) − V (x))

(7)

The momentums for this toy model are: πx =

∂L = mx˙ B˙ ∂ x˙

(8)

d ∂L d ∂L (9) − = −m E(t) ˙ ¨ dt dt ∂B ∂B ∂L = E(t) (10) ΠB = ¨ ∂B Using Hamiltonian formalism (with second order derivative [13][14]) we get that the hamiltonian of the system is: πB =

2 ˙ B + BΠ ¨ B − L = πx + Bπ ˙ B = Const H = xπ ˙ x + Bπ ¨ B

(11)

Since the action in not dependent explicitly on time, the hamiltonian is conserved. So even that the total energy of the particle is not conserved, there is the conserved hamiltonian (11). A generalization of this notion to generally coordinate invariant models will give us similar phenomena.

3

The generalization as modified gravity case

Lets consider a 4 dimensional case, where there is a coupling between a scalar µν field χ, and a stress energy momentum tensor T(χ) : S(χ) =

Z

√ µν d4 x −gχ,µ;ν T(χ) 3

(12)

where , µ; ν are covariant devirative of the scalar field. When Γλµν is being defined as the Christoffel Connection Coefficients, the variation with respect to χ gives a covariant conservation of a current f µ : µν ∇µ T(χ) = f ν ; ∇ν f ν = 0

(13)

which it is the source of the stress energy momentum tensor. This correspons to the ”dynamical space time” theory (3), where the dynamical space time 4-vector χµ is replaced by a gradient of a scalar field χ. In the ”dynamical space theory” we obtain 4 equations of motion, by the variation of χµ , which correspond µν to covariantt coservaion of energy momentum tesor ∇µ T(χ) = 0. By changing the 4 vector to a gradient of a scalar ∂µ χ at the end, what we do is to change the conservation of energy momentum tensor to asymptotic conservation of energy momentum tensor (13) which corresponds to a conservation of a current ∇ν f ν = 0. In an expanding universe, the current fµ gets diluted, so then we µν recover asymptoticaly a covariant conservation law for T(χ) again. This equations (13) have close correspondence with those obtained in a ”diffusion scenario” for DE-DM exchange [16][17]. Other models of DE-EM interactions have been studied in [18][19]. Those approaches are not based on an action principle. This stress energy tensor is substantially different from stress energy tensor we µν = Rµν − 21 g µν R. Like the mechanical sysall know, which is defined as T(G) tem, where the total energy of the particle is not conserved (but there is some constant power to the system) and there is some hamiltonian which it does µν keep conservation, in the 4-D case, the stress energy momentum tensor T(χ) is ν not consereved (but there is some conserved current f , which is the source to this stress energy momentum tensor non conservation), here there is some µν consereved stress energy tensor T(G) , which comes from variation of the action according to the metric: √ −2 δ( −gLM ) µν µν T(G) = √ ; ∇µ T(G) =0 (14) −g δg µν

µν The lagrangian LM could be the modified term χ,µ;ν T(χ) , but as we will see, it µν could be added to more action terms. Using different expressions for T(χ) which depend on another variables, will give the connection between the scalar filed χ and those other variables. Notice that for the theory the shift symmetry holds, and µν µν χ → χ + Cχ ; T(χ) → T(χ) + g µν CT

(15)

will not change any equation of motion. when Cχ , CT are some arbitrary constants. This means that if the matter is coupled through its energy momentum tensor as in (12), a process of redefinition of the energy momentum tensor, will not affect the equations of motion. Of course such type of redefinition of the energy momentum tensor is exactly what is done in the process of normal ordering in Quantum Field Theory for example. 4

4

Gravity, ”k-essence” and Diffusive behavior

Our starting point is the following non-conventional gravity-scalar-field action, which will produce a diffusive type of interacting DE-DM theory: Z Z Z √ √ √ 1 µν S= d4 x −gR + −gL(φ, X) + d4 x −gχ,µ;ν T(χ) (16) 16πG with the following explanations for the different terms: R is the Ricci scalar ,and give us Einstein-Hilbert action. L(φ, X) is general-coordinate invariant Lagrangian of a single scalar field φ ,which can be of an arbitrary generic kessence type: some function of a scalar filed φ and the combination X = ∂µ φ∂ µ φ [19-23]): ∞ X L(φ, X) = An (φ)X n − V (φ) (17) N =1

As we will see, this last action will produce a diffusive interaction between DEµν DM type theory. For the anzats of T(χ) we chose using some tensor proportional to the metric, with a proportionality function Λ(φ, X): Z µν µν T(χ) = g Λ(φ, X) ⇒ S(χ) = d4 xΛ ⊔ χ (18) From the variation of the scalar field χ we get: ⊔Λ = 0, whose solution will be interpretated as a dynamicaly generated Cosmological Constant with diffusive source. We take the simple example for this generalized thoery, and for the functions L, Λ we take the first order of the Taylor expantion from (17), or L = Λ = X (A1 = 1). From the variation according to the scalar field we get a conserved µ current j;µ = 0: jα = 2(⊔χ + 1)φ,α (19) For a cosmological solution we take into account only change as funtion of time φ = φ(t). From that we get that the ’0’ component of the current jα is non zero. The last variation we should take is according to the metric (using the identities at appendix A), which gives us a conserved stress energy tensor: µν T(G) = g µν (−Λ + χ,σ Λ,σ ) + j µ φ,ν − χ,µ Λ,ν − χ,ν Λ,µ

(20)

For a cosmological solutions the interpretation for dark energy is for term proportional to the metric −Λ + χ,σ Λ,σ , and dark matter dust from the ’00’ component of the tensor j µ φ,ν − χ,µ Λ,ν − χ,ν Λ,µ . Lets see this solution for Fridman Robertson Walker Metric: ds2 = −dt2 + a2 (t)[

dr2 + r2 dΩ2 ] 1 − kr2

(21)

The basic combination becames L = Λ = X = ∂µ φ∂ µ φ = −φ˙ 2 . We get that the variation of the scalar field (18) will give: 2φ˙ φ¨ = 5

C2 a3

(22)

which can be integrated: φ˙ 2 = C1 + C2

Z

dt a3

(23)

The conserved current from eq (19) will give us the relation: ˙ 2φ(⊔χ + 1) =

C3 a3

which can be also integrated to give: Z Z C4 1 dt C3 χ˙ = 3 + 3 a3 dt − 3 a a 2a φ˙

(24)

(25)

which gives us the anzats for the scalar field χ. From eq (20) we get the terms for DE-DM densities: ¨ pde = −ρde ρde = φ˙ 2 − 2χ˙ φ˙ φ,

(26)

C3 ˙ ¨ pdm = 0 φ + 4χ˙ φ˙ φ, (27) a3 This lead to a Fridman equations with (26)(27) as source, and there a few solutions that be we want to discuss about. ρdm =

5

Perturbative solution

One simple case is when C2 = 0. That means that the dark enegy of this universe is constant φ˙ 2 = C1 and φ¨ = 0. The equation of motions for the dark energy and dust (26-27) are independent√on the scalar filed χ, and therfore the density of dust is that universe is C3a3C1 . This solution says there is no interaction between dark energy and dark matter. This is precisely the solution of Two Measure Theory [24-25], with the action: Z Z √ √ S= −gR + (Φ + −g)L(X, φ) (28) The conclusion from this equivalence is that for a diffusion between dark energy and dark matter dust at the late universe, the coefficient C2 is very low, and therfore we can estimate the solution by pertubation theory. The exact solution of the case of standard dark energy and dust is [24]: C3 1 a0 (t) = ( √ ) 3 sinh3/2 (αt) C1

(29)

√ where α = 23 C1 and the pertubations for the dark energy and the correction of the normalization of the dust: ∆Λ = −C2 [

C4 sinh(αt) − 2αt C3 1 t + + √ + coth(αt)] (30) 2 C1 sinh4 (αt) α sinh4 (αt) 4α sinh4 (αt) 6

2C3 C2 coth(αt) (31) 3C1 We can see from those terms, that for a deviation from the unperturbed standart solution, the behavior of dark enery and dust are opposite - for rising dark energy (for example the components are C2 < 0; C1 , C3 , C4 > 0), the dark matter amount (a3 ρdm ) goes lower. Or in case of decreasing dark energy, the amounts of dark matter goes up (and C1 , C2 , C3 , C4 > 0 ). ∆CD = −

6

Diffusive dark energy and dust

A related approach, but without using the action principle, is a diffusive dark energy and dust picture which was formulated by Calogero [16][17]. The claim is that the total energy momentum tensor which appers in Einstein equation (in µν our terminology T(G) ) is consereved. But for the dark energy and dust stress tensors there is some current which is the source of those tensors (together they conserved): µν µν ν − ∇µ T(Λ) = ∇µ T(Dust) = J ν , J;ν =0 (32)

The solution for this equation gives the folowing dependence between the density and the scale parameter: Z dt (33) ρde = C1 + C2 a3 C2 t C3 (34) ρdm = 3 − 3 a a A complete set of solutions of these differential equations (in the form of Fridman equations) is very complicated, but one phenomenologycal solution for this theory predicts a DE-DM similar ratio to the observed one [26][27]. Both approches (which are described in this paper and in Calogero’s theory) become very similar when the time derivative of the scalar field is very low χ˙ ≪ 1 or the scale parameter is very large. In that case Z dt (35) ρde = C1 + C2 a3

The dark matter dust will reduce to the term: C3 ρdm = 3 φ˙ (36) a and for those equation implies a diffusion between dark energy and dark matter dust, like Calogero has found. In this model they assumed that the dark enegy and the dust are not separatly conserved. As we can see, the term for dark energy has a precise correspondece in our model and Calogeromodel. The term for dust is quite similar only when the R integration adt3 will be well approximated by a simple product at3 . By solving this condition, we get that it is a good pertubative solution only when: a˙ (t − t0 ) ≪ 1 a 7

(37)

7

Discussion, Conclusions and Prospects

In this paper we have generalized the TMT and the dynamical space time theory, which imposes the covariant conservation of an energy momentum tensor. By demanding that dynamical space time 4-vector χµ , that appears in the dynamical space time theory be a gradient ∂µ χ. We don’t obtain the covariant conservation of energy momentum tensor that is introduced in the action. Instead we obtain a current conservation. The current being the divergence of this energy momentum tensor. This current that drives the non-conservation of the energy momentum tensor, is dissipated in the case of an expanding universe. So we get an asymptotic conservation of this energy momentum tensor. This energy tensor, in not the gravitational energy tensor which appears in the right hand side of the Einstein tensor, in the gravity equations. But the non covariant conservation of the energy momentum tensor that appears in the action induces an energy momentum transfer between the dark energy and dark matter components, of the gravitational energy momentum tensor. In a way that resembles the ideas in [16][17][26][27]. But they don’t provide any action principle to support their ideas. Although the mechanism is similar, our formulation and theirs are not equivalent. We have seen that asymptotically the behavior of dark energy and dust are different - for rising dark energy (for example the components are C2 < 0; C1 , C3 , C4 > 0), the dark matter amount (a3 ρdm ) goes lower. Or in case of decreasing dark energy, the amounts of dark matter goes up (and all the constants of integration are positive). In the future we will study not only the asymptotic behavior, but the full numerical solution of the dark energy and dark matter components, starting from the early universe

8

Appendix A - identities ∂g αβ 1 = − (g αµ g βν + g αν g βµ ) ∂gµν 2 1 ∂Γτλσ = − (g µτ Γνλσ + g ντ Γµλσ ) ∂gµν 2 ∂Γτλα 1 = [g µτ (δαν δλσ + δλν δασ ) + g µν (δαµ δλσ + δλµ δασ ) − g τ σ (δαµ δλν + δλµ δαν )] ∂gµν,σ 4

αβ T(G)

−2 ∂ = √ −g

√ µν −gχµ;ν T(χ) ∂gαβ

∂ ∂ 2 +√ −g ∂xσ

A general solution for eq (13) in FRWM is: 8

√ µν −gχµ;ν T(χ) ∂gαβ,σ

a˙ C ρ˙ + 3 (ρ − p) = 3 a a when C = 0, the stress energy tensor is conserevd.

References [1] E.I. Guendelman. Mod. Phys. Lett. A14 , 1043 (1999)e-Print: gr-qc/9901017 [2] E.I. Guendelman and A.B. Kaganovich, Phys. Rev. D53 , 7020 (1996) [3] F. Gronwald, U. Muench, A. Macias, F. W. Hehl, Phys. Rev. D58 , 084021 (1998), e-Print: gr-qc/9712063. [4] E.I. Guendelman, A.B. Kaganovich, Class. Quantum Grav. 25 , 235015 (2008), e-Print: arXiv:0804.1278 [gr-qc] [5] H. Nishino, S. Rajpoot, Mod. Phys. Lett. A21 , 127 (2006), e-Print: hep-th/0404088 [6] E.I. Guendelman, A.B. Kaganovich, Ann. of Phys. 323 , 866 (2008), e-Print: arXiv:0704.1998 [gr-qc] [7] E.I. Guendelman, A.B. Kaganovich, Phys. Rev. D75 , 083505 (2007), e-Print: gr-qc/0607111 [8] E.I. Guendelman, A.B. Kaganovich, Int. J. Mod. Phys. A21 , 4373 (2006), e-Print: gr-qc/0603070 [9] Eduardo Guendelman, Ramon Herrera, Pedro Labrana, Emil Nissimov, Svetlana Pacheva , Gen.Rel.Grav. 47 (2015) arXiv:1408.5344 [gr-qc] [10] E.I. Guendelman. Annalen der Physik, 322(10):891921, Int.J.Mod.Phys. A25 (2010) 4081-4099 DOI: 10.1142/S0217751X10050317 e-Print: arXiv:0911.0178 [gr-qc]. [11] David Benisty, E.I. Guendelman. Sep 11, 2016. 6 pp. Radiation Like Scalar Field and Gauge Fields in Cosmology for a theory with Dynamical Time Published in Mod.Phys.Lett. A31 (2016) no.33, 1650188 DOI: 10.1142/S0217732316501881 arXiv:1609.03189 [12] J. Struckmeier, Phys. Rev. D91 , 085030 (2015) ( arXiv:1411.1558 [gr-qc]). [13] B.M Baker and R.F. O’Connell, Physics Letters Volume 78A, number 3, 4 August 1980 [14] (Merida, IPN), Rosario Gmez-Corts (Veracruz U.), Alberto Molgado (San Luis Potosi U., Dual C-P Inst. High Energy Phys.), Efran Rojas (Veracruz U.). Oct 21, 2013. 24 pp. Published in J.Math.Phys. 57 (2016) no.6, 062903 arXiv:1310.5750 9

[15] Canonical and phantom scalar fields as an interaction of two perfect fluids Mauricio Cataldo (Biobio U.), Fabiola Arevalo, Patricio Mella (Concepcion U.). Jan 2013. 9 pp. Published in Astrophys.Space Sci. 344 (2013) 495-503 [16] S. Calogero, A kinetic theory of difusion in general relativity with cosmological scalar field,JCAP 1111 (2011) 016, [arXiv:1107.4973]. [17] S. Calogero, Cosmological models with fluid matter undergoing velocity difusion, J. Geom. Phys. 62 (2012) 22082213 [arXiv:1202.4888]. [18] Cosmological dynamics with non-linear interactions Fabiola Arevalo, Anna Paula Ramos Bacalhau, Winfried Zimdahl. Dec 2011. 21 pp [19] T. Chiba, T.Okabe and M. Yamaguchi, Phys. Rev. D62, 023511 (2000). (arXiv:astro-ph/9912463) [20] C. Armendariz-Picon, V. Mukhanov and P. Stein- hardt, Phys. Rev. Lett. 85, 4438 (2000). (arXiv:astroph/ 0004134) [21] C. Armendariz-Picon, V. Mukhanov and P. Steinhardt, Phys. Rev. D63, 103510 (2001). (arXiv:astro-ph/0006373) [22] T. Chiba, Phys. Rev. D66, 063514 (2002). (arXiv:astroph/ 0206298) [23] Eduardo Guendelman, Douglas Singleton, Nattapong Yongram. A two measure model of dark energy and dark matter May 2012. 6 pp. Published in JCAP 1211 (2012) 044 DOI: 10.1088/1475-7516/2012/11/044 arXiv:1205.1056 [24] Eduardo Guendelman, Emil Nissimov, Svetlana Pacheva. Unified Dark Energy and Dust Dark Matter Dual to Quadratic Purely Kinetic K-Essence. Nov 22, 2015. 11 pp. Published in Eur.Phys.J. C76 (2016) no.2, 90 DOI: 10.1140/epjc/s10052-016-3938-7. arXiv:1511.07071 [25] M.V. Sazhin, O.S. Sazhina, U. Chadayammuri September 13, 2011 The Scale Factor in the Universe with Dark Energy arXiv:1109.2258v1 [astroph.CO] 10 Sep 2011 [26] Zbigniew Haba, Aleksander Stachowski, Marek Szydowski. Mar 13, 2016. 23 pp. Dynamics of the diffusive DM-DE interaction Dynamical system approach Published in JCAP 1607 (2016) no.07, 024 DOI: 10.1088/14757516/2016/07/024 arXiv:1603.07620 [27] Marek Szydlowski, Aleksander Stachowski Does the diffusion DM-DE interaction model solve cosmological puzzles? arXiv:1605.02325v1

10

Interactive Unified Dark Energy and Dark Matter from Scalar Fields David Benisty and E.I. Guendelman email: [email protected] and [email protected] bgu.ac.il Department of Physics, Ben Gurion University of the Negev Beer-Sheva 84105, Israel

January 31, 2017 Abstract Here we generalize ideas of unified Dark Matter Dark Energy in the context of Two Measure Theories and of Dynamical space time Theories. In Two Measure Theories one uses metric independent volume elements and this allows to construct unified Dark Matter Dark Energy, where the cosmological constant appears as an integration constant associated to the eq. of motion of the measure fields. The Dynamical space time Theories generalize the Two Measure Theories by introducing a vector field whose equation of motion guarantees the conservation of a certain Energy Momentum tensor, which may be related, but in general is not the same as the gravitational Energy Momentum tensor. By demanding that this vector field be the gradient of a scalar, the Dynamical space time Theory becomes a theory of Diffusive Unified Dark Energy and Dark Matter, this is because demanding that the vector field be the gradient of a scalar introduces now a mechanism that produces non conserved energy momentum tensors instead of conserved energy momentum tensors which leads at the end to a formulation of interacting DE-DM dust models in the form of a diffusive type interacting Unified Dark Energy and Dark Matter scenario.

1

Introduction

There have been theoretical approches to gravity theories, where a fundamental constraint is implemented. For example in the Two Measures Theories [1]-[9] one works, in addition to the regular measure of integration in the ac√ tion −g, also with another measure which is also a denstiy and which is also a total derivative. In this case, one can use for constructing this measure 4 scalar fields ϕa , where a = 1, 2, 3, 4. Then we can define the density Φ = εαβγδ εabcd ∂α ϕa ∂β ϕb ∂γ ϕc ∂δ ϕd , and then we can write an action that uses both of these densities: 1

S=

Z

4

d xΦL1 +

Z

√ d4 x −gL2

(1)

As a consequence of the variation with respect to the scalar fileds ϕa , assuming that L1 and L2 are independet of the scalar fields ϕa , we obtain that: Aα a

αβγδ

Aα a ∂α L1 = 0

(2) det[Aα a]

3

where = ε εabcd ∂β ϕb ∂γ ϕc ∂δ ϕd . Since ∼ Φ as one easily see, then that for Φ 6= 0 ,(2) implies that L1 = M = Const. This result can expresed as a covariant conservation of a stress energy momentum of the µν form T(Φ) = L1 g µν , and using the 2nd order formalism, where the covariant µν derivative of gµν is zero, we obtain that ∇µ T(Φ) = 0, implies ∂α L1 = 0. This suggests generalizing the idea of the Two Measures Theory, by imposing the covariant conservation of a more nontrivial kind of energy momentum tensor, µν [10]. Therfore we consider an action of the form: which we denote as T(χ) Z Z √ √ 1 µν S = S(χ) + S(R) = d4 x −gχµ;ν T(χ) + (3) d4 x −gR 16πG µν where χµ;ν = ∂ν χµ − Γλµν χλ . If we assume T(χ) to be independent ofχµ and λ having Γµν being defined as the Christoffel Connection Coefficients, then the µν variation with respect to χµ gives a covariant conservation ∇µ T(χ) = 0. Notice the fact that the energy density is the canonically conjugated variable to χ0 , which is what we expect from a dynamical time (here represented by the dynamical time χ0 ). Some cosmological solutions of (3) have been studied in [11], in the context of spacially flat radiation like solutions, and considering a gauge field equations in curved space time. For a related approach where a set of dynamical space-time coordinates are introduced, not only in the measure of integration, but also in the lagrangian, see [12]. It is of interest to introduce also a mechanism that produces non conserved energy momentum tensors which can lead eventually to a formulation of interacting DE-DM models. To start we discuss a toy model in one dimension describing a system that allows the non conservation of a certain energy functional, which increases or decreases linearly with time, while there is another energy which is conserved.

2

A mechanical system with a constant power

In order to see the applications of the ideas, we start with a simple action of one dimensional particle in a potential V (x). We introduce a coupling between the total energy of the particle 21 mx˙ 2 + V (x) and the second derivative of some dynamical variable B: Z ¨ 1 mx˙ 2 + V (x)]dt S = B[ (4) 2 2

The equation of motion according to the dynamical variable B, give that the second devirative of the total energy is zero. In other words, the total energy of the partice is linear in time: 1 mx˙ 2 + V (x) = E(t) = P t + E0 2

(5)

where P is a constant power which given to the particle or taken from it, and E0 is the total energy of the particle at time equals zero. From the equation of motion according to coordinate x we get a close connection between the dynamical variable B and the coordinate of the particle: m¨ x

d2 B d3 B d2 B ′ + m x ˙ = V (x) dt2 dt3 dt2

(6)

with the equation (5) give: ¨˙ 2V ′ (x) B P = p − ¨ B 2m(E(t) − V (x)) 2(E(t) − V (x))

(7)

The momentums for this toy model are: πx =

∂L = mx˙ B˙ ∂ x˙

(8)

d ∂L d ∂L (9) − = −m E(t) ˙ ¨ dt dt ∂B ∂B ∂L = E(t) (10) ΠB = ¨ ∂B Using Hamiltonian formalism (with second order derivative [13][14]) we get that the hamiltonian of the system is: πB =

2 ˙ B + BΠ ¨ B − L = πx + Bπ ˙ B = Const H = xπ ˙ x + Bπ ¨ B

(11)

Since the action in not dependent explicitly on time, the hamiltonian is conserved. So even that the total energy of the particle is not conserved, there is the conserved hamiltonian (11). A generalization of this notion to generally coordinate invariant models will give us similar phenomena.

3

The generalization as modified gravity case

Lets consider a 4 dimensional case, where there is a coupling between a scalar µν field χ, and a stress energy momentum tensor T(χ) : S(χ) =

Z

√ µν d4 x −gχ,µ;ν T(χ) 3

(12)

where , µ; ν are covariant devirative of the scalar field. When Γλµν is being defined as the Christoffel Connection Coefficients, the variation with respect to χ gives a covariant conservation of a current f µ : µν ∇µ T(χ) = f ν ; ∇ν f ν = 0

(13)

which it is the source of the stress energy momentum tensor. This correspons to the ”dynamical space time” theory (3), where the dynamical space time 4-vector χµ is replaced by a gradient of a scalar field χ. In the ”dynamical space theory” we obtain 4 equations of motion, by the variation of χµ , which correspond µν to covariantt coservaion of energy momentum tesor ∇µ T(χ) = 0. By changing the 4 vector to a gradient of a scalar ∂µ χ at the end, what we do is to change the conservation of energy momentum tensor to asymptotic conservation of energy momentum tensor (13) which corresponds to a conservation of a current ∇ν f ν = 0. In an expanding universe, the current fµ gets diluted, so then we µν recover asymptoticaly a covariant conservation law for T(χ) again. This equations (13) have close correspondence with those obtained in a ”diffusion scenario” for DE-DM exchange [16][17]. Other models of DE-EM interactions have been studied in [18][19]. Those approaches are not based on an action principle. This stress energy tensor is substantially different from stress energy tensor we µν = Rµν − 21 g µν R. Like the mechanical sysall know, which is defined as T(G) tem, where the total energy of the particle is not conserved (but there is some constant power to the system) and there is some hamiltonian which it does µν keep conservation, in the 4-D case, the stress energy momentum tensor T(χ) is ν not consereved (but there is some conserved current f , which is the source to this stress energy momentum tensor non conservation), here there is some µν consereved stress energy tensor T(G) , which comes from variation of the action according to the metric: √ −2 δ( −gLM ) µν µν T(G) = √ ; ∇µ T(G) =0 (14) −g δg µν

µν The lagrangian LM could be the modified term χ,µ;ν T(χ) , but as we will see, it µν could be added to more action terms. Using different expressions for T(χ) which depend on another variables, will give the connection between the scalar filed χ and those other variables. Notice that for the theory the shift symmetry holds, and µν µν χ → χ + Cχ ; T(χ) → T(χ) + g µν CT

(15)

will not change any equation of motion. when Cχ , CT are some arbitrary constants. This means that if the matter is coupled through its energy momentum tensor as in (12), a process of redefinition of the energy momentum tensor, will not affect the equations of motion. Of course such type of redefinition of the energy momentum tensor is exactly what is done in the process of normal ordering in Quantum Field Theory for example. 4

4

Gravity, ”k-essence” and Diffusive behavior

Our starting point is the following non-conventional gravity-scalar-field action, which will produce a diffusive type of interacting DE-DM theory: Z Z Z √ √ √ 1 µν S= d4 x −gR + −gL(φ, X) + d4 x −gχ,µ;ν T(χ) (16) 16πG with the following explanations for the different terms: R is the Ricci scalar ,and give us Einstein-Hilbert action. L(φ, X) is general-coordinate invariant Lagrangian of a single scalar field φ ,which can be of an arbitrary generic kessence type: some function of a scalar filed φ and the combination X = ∂µ φ∂ µ φ [19-23]): ∞ X L(φ, X) = An (φ)X n − V (φ) (17) N =1

As we will see, this last action will produce a diffusive interaction between DEµν DM type theory. For the anzats of T(χ) we chose using some tensor proportional to the metric, with a proportionality function Λ(φ, X): Z µν µν T(χ) = g Λ(φ, X) ⇒ S(χ) = d4 xΛ ⊔ χ (18) From the variation of the scalar field χ we get: ⊔Λ = 0, whose solution will be interpretated as a dynamicaly generated Cosmological Constant with diffusive source. We take the simple example for this generalized thoery, and for the functions L, Λ we take the first order of the Taylor expantion from (17), or L = Λ = X (A1 = 1). From the variation according to the scalar field we get a conserved µ current j;µ = 0: jα = 2(⊔χ + 1)φ,α (19) For a cosmological solution we take into account only change as funtion of time φ = φ(t). From that we get that the ’0’ component of the current jα is non zero. The last variation we should take is according to the metric (using the identities at appendix A), which gives us a conserved stress energy tensor: µν T(G) = g µν (−Λ + χ,σ Λ,σ ) + j µ φ,ν − χ,µ Λ,ν − χ,ν Λ,µ

(20)

For a cosmological solutions the interpretation for dark energy is for term proportional to the metric −Λ + χ,σ Λ,σ , and dark matter dust from the ’00’ component of the tensor j µ φ,ν − χ,µ Λ,ν − χ,ν Λ,µ . Lets see this solution for Fridman Robertson Walker Metric: ds2 = −dt2 + a2 (t)[

dr2 + r2 dΩ2 ] 1 − kr2

(21)

The basic combination becames L = Λ = X = ∂µ φ∂ µ φ = −φ˙ 2 . We get that the variation of the scalar field (18) will give: 2φ˙ φ¨ = 5

C2 a3

(22)

which can be integrated: φ˙ 2 = C1 + C2

Z

dt a3

(23)

The conserved current from eq (19) will give us the relation: ˙ 2φ(⊔χ + 1) =

C3 a3

which can be also integrated to give: Z Z C4 1 dt C3 χ˙ = 3 + 3 a3 dt − 3 a a 2a φ˙

(24)

(25)

which gives us the anzats for the scalar field χ. From eq (20) we get the terms for DE-DM densities: ¨ pde = −ρde ρde = φ˙ 2 − 2χ˙ φ˙ φ,

(26)

C3 ˙ ¨ pdm = 0 φ + 4χ˙ φ˙ φ, (27) a3 This lead to a Fridman equations with (26)(27) as source, and there a few solutions that be we want to discuss about. ρdm =

5

Perturbative solution

One simple case is when C2 = 0. That means that the dark enegy of this universe is constant φ˙ 2 = C1 and φ¨ = 0. The equation of motions for the dark energy and dust (26-27) are independent√on the scalar filed χ, and therfore the density of dust is that universe is C3a3C1 . This solution says there is no interaction between dark energy and dark matter. This is precisely the solution of Two Measure Theory [24-25], with the action: Z Z √ √ S= −gR + (Φ + −g)L(X, φ) (28) The conclusion from this equivalence is that for a diffusion between dark energy and dark matter dust at the late universe, the coefficient C2 is very low, and therfore we can estimate the solution by pertubation theory. The exact solution of the case of standard dark energy and dust is [24]: C3 1 a0 (t) = ( √ ) 3 sinh3/2 (αt) C1

(29)

√ where α = 23 C1 and the pertubations for the dark energy and the correction of the normalization of the dust: ∆Λ = −C2 [

C4 sinh(αt) − 2αt C3 1 t + + √ + coth(αt)] (30) 2 C1 sinh4 (αt) α sinh4 (αt) 4α sinh4 (αt) 6

2C3 C2 coth(αt) (31) 3C1 We can see from those terms, that for a deviation from the unperturbed standart solution, the behavior of dark enery and dust are opposite - for rising dark energy (for example the components are C2 < 0; C1 , C3 , C4 > 0), the dark matter amount (a3 ρdm ) goes lower. Or in case of decreasing dark energy, the amounts of dark matter goes up (and C1 , C2 , C3 , C4 > 0 ). ∆CD = −

6

Diffusive dark energy and dust

A related approach, but without using the action principle, is a diffusive dark energy and dust picture which was formulated by Calogero [16][17]. The claim is that the total energy momentum tensor which appers in Einstein equation (in µν our terminology T(G) ) is consereved. But for the dark energy and dust stress tensors there is some current which is the source of those tensors (together they conserved): µν µν ν − ∇µ T(Λ) = ∇µ T(Dust) = J ν , J;ν =0 (32)

The solution for this equation gives the folowing dependence between the density and the scale parameter: Z dt (33) ρde = C1 + C2 a3 C2 t C3 (34) ρdm = 3 − 3 a a A complete set of solutions of these differential equations (in the form of Fridman equations) is very complicated, but one phenomenologycal solution for this theory predicts a DE-DM similar ratio to the observed one [26][27]. Both approches (which are described in this paper and in Calogero’s theory) become very similar when the time derivative of the scalar field is very low χ˙ ≪ 1 or the scale parameter is very large. In that case Z dt (35) ρde = C1 + C2 a3

The dark matter dust will reduce to the term: C3 ρdm = 3 φ˙ (36) a and for those equation implies a diffusion between dark energy and dark matter dust, like Calogero has found. In this model they assumed that the dark enegy and the dust are not separatly conserved. As we can see, the term for dark energy has a precise correspondece in our model and Calogeromodel. The term for dust is quite similar only when the R integration adt3 will be well approximated by a simple product at3 . By solving this condition, we get that it is a good pertubative solution only when: a˙ (t − t0 ) ≪ 1 a 7

(37)

7

Discussion, Conclusions and Prospects

In this paper we have generalized the TMT and the dynamical space time theory, which imposes the covariant conservation of an energy momentum tensor. By demanding that dynamical space time 4-vector χµ , that appears in the dynamical space time theory be a gradient ∂µ χ. We don’t obtain the covariant conservation of energy momentum tensor that is introduced in the action. Instead we obtain a current conservation. The current being the divergence of this energy momentum tensor. This current that drives the non-conservation of the energy momentum tensor, is dissipated in the case of an expanding universe. So we get an asymptotic conservation of this energy momentum tensor. This energy tensor, in not the gravitational energy tensor which appears in the right hand side of the Einstein tensor, in the gravity equations. But the non covariant conservation of the energy momentum tensor that appears in the action induces an energy momentum transfer between the dark energy and dark matter components, of the gravitational energy momentum tensor. In a way that resembles the ideas in [16][17][26][27]. But they don’t provide any action principle to support their ideas. Although the mechanism is similar, our formulation and theirs are not equivalent. We have seen that asymptotically the behavior of dark energy and dust are different - for rising dark energy (for example the components are C2 < 0; C1 , C3 , C4 > 0), the dark matter amount (a3 ρdm ) goes lower. Or in case of decreasing dark energy, the amounts of dark matter goes up (and all the constants of integration are positive). In the future we will study not only the asymptotic behavior, but the full numerical solution of the dark energy and dark matter components, starting from the early universe

8

Appendix A - identities ∂g αβ 1 = − (g αµ g βν + g αν g βµ ) ∂gµν 2 1 ∂Γτλσ = − (g µτ Γνλσ + g ντ Γµλσ ) ∂gµν 2 ∂Γτλα 1 = [g µτ (δαν δλσ + δλν δασ ) + g µν (δαµ δλσ + δλµ δασ ) − g τ σ (δαµ δλν + δλµ δαν )] ∂gµν,σ 4

αβ T(G)

−2 ∂ = √ −g

√ µν −gχµ;ν T(χ) ∂gαβ

∂ ∂ 2 +√ −g ∂xσ

A general solution for eq (13) in FRWM is: 8

√ µν −gχµ;ν T(χ) ∂gαβ,σ

a˙ C ρ˙ + 3 (ρ − p) = 3 a a when C = 0, the stress energy tensor is conserevd.

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