Cosmological Evolution of Interacting Dark Energy in Lorentz Violation

# Cosmological evolution of interacting dark energy in Lorentz violation

## Abstract

The cosmological evolution of an interacting scalar field model in which the scalar field interacts with dark matter, radiation, and baryon via Lorentz violation is investigated. We propose a model of interaction through the effective coupling . Using dynamical system analysis, we study the linear dynamics of an interacting model and show that the dynamics of critical points are completely controlled by two parameters. Some results can be mentioned as follows. Firstly, the sequence of radiation, the dark matter, and the scalar field dark energy exist and baryons are sub dominant. Secondly, the model also allows the possibility of having a universe in the phantom phase with constant potential. Thirdly, the effective gravitational constant varies with respect to time through . In particular, we consider a simple case where has a quadratic form and has a good agreement with the modified CDM and quintessence models. Finally, we also calculate the first post–Newtonian parameters for our model.

###### pacs:
98.80.Cq; 98.80.-k

## 1 Introduction

There has been a growing appreciation of the importance of the violations of Lorentz invariance in recent years. The intriguing possibility of the Lorentz violation is that an unknown physics at high-energy scales could lead to a spontaneous breaking of Lorentz invariance by giving an expectation value to certain non Standard Model fields that carry Lorentz indices, such as vectors, tensors, and gradients of scalar fields (1). Recently, it has been proposed a relativistic theory of gravity where gravity is mediated by a tensor, a vector, and a scalar field, thus called TeVeS gravitational theory (2). It provides modified Newtonian dynamics (MOND) and Newtonian limits in the weak field nonrelativistic limit, and is devoid of a causal propagation of perturbations. TeVeS could also explain the large-scale structure formation of the Universe without recurring to cold dark matter (3), which is composed of very massive slowly moving and weakly interacting particles. On the other hand, the Einstein–Aether theory (4) is a vector-tensor theory, and TeVeS can be written as a vector-tensor theory which is the extension of the Einstein–Aether theory (5). In the case of generalized Einstein–Aether theory (6), the effect of a general class of such theories on the solar system has been considered in Ref. (7). Moreover, as has been shown by authors in Ref. (8), the Einstein–Aether theory may lead to significant modifications of the power spectrum of tensor perturbation. The strong gravitational cases including black holes of such theories have been studied in Refs. (9).

The existence of vector fields in a scalar-vector-tensor theory of gravity also leads to its applications in modern cosmology and it might explain inflationary scenarios (10); (11) and accelerated expansion of the universe (6); (12). The accelerated expansion and crossing of the phantom divided line has been studied recently by authors in Ref. (13) Based on a dynamical vector field coupled to the gravitation and scalar fields, we have studied to some extent the cosmological implications of a scalar-vector-tensor theory of gravity (14).

Since the discovery of accelerated expansion of our Universe  (15), identifying the contents of dark energy and dark matter is one of the most important subjects in modern cosmology. The dark energy is described by an equation of state parameter , the ratio of the spatially homogeneous dark energy pressure to its energy density . A value of is required for accelerated expansion. The classification of dark energy might be due to: quintessence field (16), tachyon models (17), Chaplygin gas (18) if , cosmological constant if  (19); (20); (21); (22), or phantom field if  (23). A recent comprehensive review on dark energy is available in (24). Of course, as it has been discussed in (25); (26) the vector field is also a viable dark energy candidate and effects on the cosmic microwave background radiation and the large scale structure (27).

In the previous work (28), the attractor solutions in Lorentz violating scalar-vector-tensor theory of gravity without interaction with background matter was studied. In this framework, both the effective coupling and potential functions determine the stabilities of the fixed points. In the model, we considered the constants of slope of the effective coupling and potential functions which lead to the quadratic effective coupling with the (inverse) power-law potential. Differing from the previous work, in this work, we investigate the cosmological evolution of the scalar field dark energy and background perfect fluid by means of dynamical system. We study the cases of scalar field dark energy interacting with background perfect fluid. The interaction terms are taken to be two different forms which are mediated by the slope of the effective coupling. For more realistic model we assume that the background matter fields might be dark matter, radiation, and baryons.

Furthermore, to test the model in the solar system we present the post–Newtonian parameters (PN). In the PN approximation we restrict ourselves to the first post–Newtonian. The parameterized post–Newtonian (PPN) parameters are determined by expanding the modified field equations in the metric perturbation. Then, we compare the solution to the PPN formalism in first PN approximation proposed by Will and Nordtvedt (29); (30) and read off the coefficients (the PPN parameters) of post Newtonian potentials of the theory.

This paper is organized as follows. In Section 2, we set down the general formalism of the scalar field interacting with background perfect fluid in the scalar-vector-tensor theory where the Lorentz symmetry is spontaneously broken due to the unit-norm vector field. We derive the governing equations of motion for the canonical Lagrangian of the scalar field. In Section 3, we study the interaction models and the attractor solutions. The critical points of the system and their stability are presented. The cosmological implication is discussed in Section 4. In Section 5, we present the parameterized post-Newtonian parameters of the model. The final Section is devoted to the conclusions.

In what follows, the conventions that we use throughout this work are the following: Greek letters represent spacetime indices, while Latin letters stand for spatial indices and repeated indices mean Einstein’s summation. The symbol stands for terms of order . Finally, we use the metric signature .

## 2 The action and field equations

In the present section, we develop the general reconstruction scheme for the scalar-vector-tensor gravitational theory. We will consider the properties of general four-dimensional universe, i.e. the universe where the four-dimensional space-time is allowed to contain any non-gravitational degree of freedom in the framework of Lorentz violating scalar-tensor-vector theory of gravity. Let us assume that the Lorentz symmetry is spontaneously broken by imposing the expectation values of a vector field as . The action can be written as the sum of four distinct parts:

 S = Sg+Su+Sϕ+Sm , (1)

where the actions for the tensor field , the vector field , the scalar field , and the ordinary matter , respectively, are given by

 Sg = ∫d4x√−g 116πGR , (2) Su = ∫d4x√−g[−β1∇μuν∇μuν−β2(∇μuμ)2 (3) −β3∇μuν∇νuμ+λ(uμuμ+1)] , Sϕ = ∫d4x√−g [−12(∇ϕ)2−V(ϕ)] , (4) Sm = ∫d4x√−g Lm(ϕ,Ψi,gμν) . (5)

In the above () are the functions of , and is a Lagrange multiplier. In Eq. (5), we allow for an arbitrary coupling between the matter fields and the scalar field .

Varying the action (1) with respect to , we have field equations

 Rμν−12gμνR=8πGTμν , (6)

where is the Ricci tensor, is the scalar curvature, is the metric tensor, and is the energy-momentum tensor for all the fields present, . , and are the energy-momentum tensors of vector, scalar fields, and ordinary matter, respectively, given by

 T(u)μν = 2β1(∇μuτ∇νuτ−∇τuμ∇τuν)−2∇τ(u(μJτν)) (7) −2∇τ(uτJ(μν))+2∇τ(u(μJν)τ) −2uσ∇τJτσuμuν+gμνLu , T(ϕ)μν = ∇μϕ∇νϕ−12gμν[(∇ϕ)2+2V(ϕ)] , (8) T(m)μν = (ρm+pm)vμvν+pmgμν , (9)

where is the four velocity and the current tensor in Eq. (7) is given by

 Jμν=−β1∇μuν−β2δμν∇τuτ−β3∇νuμ . (10)

The energy-momentum tensor is conserved

 ∇ν(T(u)νμ+T(ϕ)νμ+T(m)νμ)=0 . (11)

In general, however, the Bianchi identity implies that each energy species in the cosmic mixture is not conserved, namely

 ∇νT(u)νμ=σ(u)μ ,  ∇νT(ϕ)νμ=σ(ϕ)μ ,  ∇νT(m)νμ=σ(m)μ . (12)

Here () is an arbitrary vector function of the space-time coordinates that determines the rate of transfer of energy, where . This is in accordance with Eq. (11). Equation (12) is the basic feature of interacting models in which there is exchange of energy between the components of the cosmic fluid. Moreover, the projection of the non conservation equation along the velocity of the whole fluid is

 Q(u)=−Q(ϕ)−Q(m) , (13)

where is a scalar.

Using Eq. (6), the Friedmann and Raychaudhuri equations can be written as

 3H2=8πG(ρu+ρϕ+ρm) , (14)

and

 2˙H=−8πG(ρu+ρϕ+ρm+pu+pϕ+pm) , (15)

where

 ρu=−3βH2,pu=−ρu+2(β˙H+˙βH), (16) ρϕ=12˙ϕ2+V ,pϕ=−ρϕ+˙ϕ2 . (17)

Here, we have defined .

Substituting Eqs. (16) and (17) into Eqs.(14) and (15), respectively, we obtain

 3(β+18πG)H2=12˙ϕ2+V+ρm (18)

and

 2(β+18πG)˙H=−2˙βH−˙ϕ2−2(ρm+pm) . (19)

Let us define the effective coupling as follows

 ¯β ≡ β+18πG , (20)

then Eqs. (18) and (19) can be simplified as

 H2=13¯β(12˙ϕ2+V+ρm) , (21) ˙HH=−˙¯β¯β−12˙ϕ2H¯β−γmρmH¯β . (22)

Here, we have defined , where is the ordinary matter barotropic parameter, which is related to the equation of state parameter through the relationship . Similarly, we also defined the scalar field barotropic parameter, and . Then the effective equation of state for the total cosmic fluid is

 γ(e)=1+pu+pϕ+pmρu+ρϕ+ρm , (23)

which, again, is related to the equation of state parameter through . The condition for an accelerated universe is . When , the universe is in quintessence phase while it is in phantom phase when .

From Eq. (16) we obtain

 ˙ρu+3H(ρu+pu)=3H2˙¯β . (24)

In order to preserve the conservation of total energy equation , where and are the total energy density and the pressure, respectively, one can write the conservation of scalar field and matter field:

 ˙ρϕ+3H(ρϕ+pϕ)=−3H2˙¯β+Qm , (25) ˙ρm+3H(ρm+pm)=Qm . (26)

The interaction term can be interpreted as a transfer from one energy component to another energy component of the cosmic fluid. These interactions are completely associated with Lorentz violation. In our case, the scalar field decays into the matter field and the vector field. The conservation of scalar field, Eq. (25), is equivalent to a dynamical equation for the scalar field ,

 Qm=−˙ϕ(¨ϕ+3H˙ϕ+V,ϕ+3H2¯β,ϕ) . (27)

The above equation reduces to Refs. (11); (28) for . Equations (21), (22), and (27) represent the basic set of equations of the model of interacting components of the cosmic fluid in the framework of Lorentz violating scalar-vector-tensor theory of gravity. In what follows we shall apply a dynamical system to analyze the cosmological dynamics of this set of equations.

## 3 Interacting model

Some models that allow interaction between the scalar field and the matter field have been proposed as a solution to the cosmic coincidence problem. These models are compatible with observational data but so far there has been no evidence on the existence of this interaction. A solution will be achieved if the dynamical system presents scaling solutions which are characterized by a constant dark matter to dark energy ratio. Even more important are those scaling solutions that are also attractors and have the accelerated solution. In this way, the coincidence problem gets substantially alleviated because, regardless of the initial conditions, the system evolves towards a final state where the ratio of dark matter to dark energy remains constant.

The explicit form of Eq. (13) can be expressed in the form

 Qϕ+Qm=−˙¯β¯β(ρϕ+ρm) . (28)

We assume the interaction term as follows

 Qm=˙¯β¯βρϕ=−¯β,ϕ¯βρm˙ϕ . (29)

The interaction term (29) means that the scalar field can exchange energy with the background matter, through the interaction between them. In this case the exchange energy is mediated by the slope of the effective coupling.

Equations (25) and (26), respectively, become

 ˙ρϕ+3H(ρϕ+pϕ)=−¯β,ϕ¯βρϕ˙ϕ , (30) ˙ρm+3H(ρm+pm)=−¯β,ϕ¯βρm˙ϕ . (31)

For a more realistic model we assume that the matter fields might be a combination of dark matter, , radiation, , and baryons, : . We also assume that the barotropic equation of state for the radiation field and that the baryons are non-relativistic particles so that holds. Hence, the equations for the energy densities of radiation and baryons are

 ˙ρr+4Hρr=0 ,˙ρb+3Hρb=0 , (32)

respectively, and we find the well-known relationships: and , is a scale factor. For the scalar field and the dark matter we have

 ˙ρϕ+3Hγ(e)ϕρϕ=0 ,˙ρc+3Hγ(e)cρc=0 , (33)

where and are the effective barotropic equation of state for scalar field and dark matter, respectively,

 γ(e)ϕ=γϕ+˙¯β3H¯β ,γ(e)c=1+˙¯β3H¯β(1+ρr+ρbρc) . (34)

Notice that for we have , and both and with Lorentz violation will dilute slower then that without Lorentz violation or const. Thus will determine both the effective equations of state and .

### 3.1 Dynamical analysis

In order to study the dynamics of the model, we shall introduce the following dimensionless variables (14); (28):

 x2≡˙ϕ26¯βH2 ,y2≡V3H2¯β , (35) λ1≡−¯β,ϕ√¯β ,λ2≡−√βV,ϕV , (36) Γ1≡¯β¯β,ϕϕ¯β2,ϕ ,Γ2≡VV,ϕϕV2,ϕ+12¯β,ϕ/¯βV,ϕ/V , (37)

and, accordingly, the governing equations of motion could be reexpressed as the following system of equations:

 H′ = −32H(1+x2−y2+13z2−√6λ1x) , (38) x′ = −x(3+H′H) (39) +√32(λ1+λ2)y2+2√32λ1x2 , y′ = −y(H′H−√32(λ1−λ2)x) , (40) z′ = −z(2+H′H−√32λ1x) , (41) u′ = −u(32+H′H−√32λ1x) , (42)

where

 z=√ρr3¯βH2 ,u=√ρb3¯βH2 . (43)

A prime denotes a derivative with respect to the natural logarithm of the scale factor, . Equation (21) gives the following constraint equation:

 Ωc=ρc3¯βH2=1−x2−y2−z2−u2 , (44)

where , , and . Notice that , are the effective cosmological density parameters which are associated with the Lorentz violation.

In general, the parameters , , and are variables dependent on and completely associated with the Lorentz violation. In order to construct viable Lorentz violation model, we require that the effective coupling and the potential function should satisfy the conditions and , respectively. In this paper, we want to discuss the phase space, then we need certain constraints on the effective coupling and potential function. Note that for const., , the scalar field dynamics in the Lorentz violating scalar-vector-tensor theories is then reduced to the scalar field dynamics in the conventional one. But, the effective gravitational constant is rescaled by Eq. (20). In this case, the cosmological attractor solutions can be studied through a scalar exponential potential of the form where const. This exponential potential gives rise to scaling solutions for the scalar field (31). In this paper we consider the case in which and are constant parameters. For example, a constant is given by an effective coupling and we have . A constant can only be obtained as a combination of and , one finds

 V(ϕ)=V0(¯β(ϕ))s , (45)

where is a constant parameter. In general, one can write the potential as a function of effective coupling, .

### 3.2 Attractor solutions

The critical points are obtained by imposing the conditions . Substituting linear perturbation , , and about the critical points into Eqs. (39)–(42), we obtain, up to first-order in the perturbation, the equation of motion

 ddα⎛⎜ ⎜ ⎜⎝δxδyδzδu⎞⎟ ⎟ ⎟⎠=M⎛⎜ ⎜ ⎜⎝δxδyδzδu⎞⎟ ⎟ ⎟⎠ . (46)

Notice from (39)–(42) that the dynamical equations are invariant under the change of sign , and in consequence we don´t have to include the points with in our analyzes. The properties of the critical points are summarized in Table 1. There are eight critical points at all and two of them lead to attractor solutions, depending on the values of the parameters and . The scalar field dominated solution, point in Table 2, are characterized by , and the effective equations of state are given by

 γ(e)ϕ=13(λ1+λ2)2 ,γ(e)=−13(λ21−λ22) . (47)

The solution of this point exists for and the universe is accelerated for . From eq. (47) one can see that the de Sitter epoch corresponds to . The scalar field is dark energy when . In this case the effective coupling and the potential function are quadratic in , . The inflationary solution of this model has been studied in Ref. (11). Figure 1 shows that the sequence of radiation, dark matter and scalar field dark energy. The baryon is sub–dominant in this case. The parameters correspond to and . The scalar field equation of state parameter is nearly a constant, during the radiation and matter epochs because the fields are almost frozen for which . At the transition era from matter domination to the scalar field dark energy domination, and begin to grow because the kinetic energies of the fields become important. However, the universe enters the de Sitter phase during which the field rolls up the potential. More interesting of this attractor solution is of the constant potential, . The universe is in phantom phase in this case because is crossing and it is accelerated for .

The second attractor solution is the scalar field scaling solution, point in Table 2. The solution of this point exists for , corresponding to energy density parameter . The effective equations of state are given by

 γϕ=γm=1 ,γ(e)ϕ=γ(e)m=λ2(λ1+λ2) , (48) γ(e)=1−2λ1λ1+λ2 . (49)

The universe is accelerated for . In the case of the effective coupling and the potential function are quadratic in , i.e. , the universe is always accelerated. For the constant potential, , the scalar field behaves as a cosmological constant while the universe is in phantom phase. Figure 2 shows the sequence of radiation, dark matter and scalar field dark energy. The baryon is sub dominant in this case. The parameters correspond to (top panel), and (bottom panel).

## 4 A comparison of the model using supernova data

From the above detail analysis, we may investigate the cosmological consequences of a Lorentz violating scalar-vector-tensor theory which incorporates time variations in the gravitational constant. It was raised by Dirac who introduced the large number hypothesis (32), and has recently become a subject of intensive experimental and theoretical studies (33). The effective gravitational constant, , is obtained from the Friedmann equation,

 G(e)=18π¯β=G1+8πGβ , (50)

where is the parameter in the action (1). Therefore the time variation of can be written as

 ˙G(e)G(e)=−˙¯β¯β , (51)

and the effective gravitational constant is determined by the effective coupling . For the quadratic effective coupling, , the effective gravitational constant is inversely proportional to , . Recently using the data provided by the pulsating white dwarf star G117-B15A the astereoseismological bound on is found (34) to be .

In the present model the time variation in the gravitational constant is given by

 ˙G(e)G(e)=3λ1(λ1+λ2)H , (52)

in the scaling solution and

 ˙G(e)G(e)=λ1(λ1+λ2)H , (53)

in the scalar field dominated solution, where the evolution of the Hubble parameter is given by Eq. (38). For instance, in the case of power law expansion of the universe with , the time variation of leads to

 ˙G(e)G(e)∝3λ1(λ1+λ2)t−1 , (54)

in the scaling solution. Assuming the present age of the Universe as 14 Gyr, it is straightforward to derive from Eq. (54) the estimate for the case of constant potential. Our model also allows the negative value of . Let us focus on the scaling solution. If we find

 ˙G(e)G(e)=±√2λ1H . (55)

A negative implies a time-decreasing , while a positive means is growing with time. From Eq. (55), it is clear that the Lorentz violation leads to time variation of the gravitational constant.

In the following, we study the expansion history of the universe using the 194 SnIa data (35); (36). We simplify our model by considering an interaction between dark matter and the scalar field dark energy given by Eqs. (30) and (31). The evolution of the dark matter and scalar field dark energy are given by

 ρi(z)=ρi0e3∫z01+ω(e)i(z′)1+z′dz′ ,(i=m,ϕ) , (56)

where is the redshift. Using the above relation, the Hubble parameter as a function of the redshift can be written as

 H2(z) = Missing or unrecognized delimiter for \left (57) +(1−Ωm0)(1+z)3(1+ωϕ(z))] ,

where the subscript describes the current value of the variable. Notice that the evolution of the Hubble parameter is deviated by the factor of compared to the standard one. If the functions and are given, we can find the evolution of the Hubble parameter. In this section, we consider an ansatz for the effective coupling,

 ¯β=¯β0(1+ζz2) , (58)

where is a constant.

Let us first consider the modified Cold Dark Matter (CDM) model. We have

 H2(z;ζ,Ωm0)=(H01+ζz2)2[Ωm0(1+z)3+(1−Ωm0] . (59)

Equation (59) has two free parameters and which are determined by minimizing

 χ2=∑i[μobs(zi)−μ(zi)]2σi , (60)

where is the extinction-corrected distance modulus,

 μ(z)=5log10(dL(z)1Mpc)+25 , (61)

and is the total uncertainty in the SnIa data. The luminosity distance is given by

 dL(z)=c(1+z)H0∫z0dz′H(z′) . (62)

Fitting the model to 194 SnIa data, we get , , and . For comparison, we also fit the cosmological constant model to the 194 SnIa data and find , and .

In the next model we replace the cosmological constant energy density by a scalar field dark energy with constant equation of state parameter ( constant). We set here . We evaluate and minimize with respect to and . We find

 χ2min=χ2(ζ=−0.29,ωϕ=−1.13)=195.71 . (63)

Figure  3 shows a comparison of the observed 194 SnIa Hubble free luminosity distances along the predicted curves in the context of Lorentz violating scalar-vector-tensor theory. We see that the effect of Lorentz violation appears at . We define the reduced form of Hubble parameter compared to the standard case as

 H2red=H2LV−H2stdH2std , (64)

where

 H2std(z) = H20[Ωm0(1+z)3 (65) +(1−Ωm0)(1+z)3(1+ωϕ(z))] .

Thus the reduced form of Hubble parameter, due to the effect of Lorentz violation, is

 H2red(z)=(¯β0¯β(z))2−1 . (66)

## 5 Parameterized Post-Newtonian

In order to confront the predictions of a given gravity theory with experiment in the solar system, it is necessary to compute its PPN parameters. The post-Newtonian approximation is based on the assumptions of weak gravitational fields and slow motions. It provides a way to estimate general relativistic effects in the fully nonlinear evolution stage of the large scale cosmic structures. The procedure of parameterizing our model is following to that of Ref (37), in which the authors has derived the PPN parameters in the frame of Einstein aether theory.

In the weak field approximation, we choose a system of coordinates in which the metric can be perturbatively expanded around Minkowski spacetime. The decomposition is as follows

 gμν=ημν+hμν , (67)

where is the Minkowski metric and is the metric perturbations and we take .

The equations governing the perturbation in the model Eq. (1) are found by computing the Einstein field equations in the perturbative limit. The full field equations are given by

 Rμν=8πG(T(m)αβ+T(ϕ)αβ+T(u)αβ)(δαμδβν−12gμνgαβ). (68)

We allow for an arbitrary coupling between the matter fields and the scalar field . We assume that the scalar field is coupled to a barotropic perfect fluid with a coupling function given by

 q(ϕ)=−1ρm√−gδSmδϕ , (69)

Therefore, the equation of motion for the scalar field is

 □ϕ−dVdϕ−3∑i=1dβidϕKi=q(ϕ)ρm , (70)

where

 K1=∇μuν∇μuν, K2=(∇μuμ)2, K3=∇μuν∇νuμ. (71)

In the previous discussion we have considered the coupling between the scalar field and the matter field is given by the effective coupling,