The late Universe with non-linear interaction in the dark sector: the coincidence problem

# The late Universe with non-linear interaction in the dark sector: the coincidence problem

Mariam Bouhmadi–López,    João Morais,    and Alexander Zhuk,
###### Abstract

We study the Universe at the late stage of its evolution and deep inside the cell of uniformity. At such a scale the Universe is highly inhomogeneous and filled with discretely distributed inhomogeneities in the form of galaxies and groups of galaxies. As a matter source, we consider dark matter (DM) and dark energy (DE) with a non-linear interaction , where is a constant. We assume that DM is pressureless and DE has a constant equation of state parameter . In the considered model, the energy densities of the dark sector components present a scaling behaviour with . We investigate the possibility that the perturbations of DM and DE, which are interacting among themselves, could be coupled to the galaxies with the former being concentrated around them. To carry our analysis, we consider the theory of scalar perturbations (within the mechanical approach), and obtain the sets of parameters which do not contradict it. We conclude that two sets: and are of special interest. First, the energy densities of DM and DE on these cases are concentrated around galaxies confirming that they are coupled fluids. Second, we show that for both of them, the coincidence problem is less severe than in the standard CDM. Third, the set is within the observational constraints. Finally, we also obtain an expression for the gravitational potential in the considered model.

\affiliation

Departamento de Física, Universidade da Beira Interior, 6200 Covilhã, Portugal
Centro de Matemática e Aplicações da Universidade da Beira Interior (CMA-UBI), 6200 Covilhã, Portugal
Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain
IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain
Astronomical Observatory, Odessa National University,
Dvoryanskaya st. 2, Odessa 65082, Ukraine

## 1 Introduction

Recent observations [1] clearly indicate that our Universe is dark. The mass-energy balance of the Universe consists of 69% of dark energy (DE) and 26% of dark matter (DM). However, the nature of these constituents is still unclear. In addition, the CDM model (as well as a lot of other dark energy models) faces the coincidence problem, i.e. why is the cosmological constant at present of the same order of magnitude as the dark matter energy density? To solve these problems, different dynamical dark energy models were proposed. Among them, the models with interacting DE and DM are of great interest. A vast literature is devoted to these models, their implication in the future evolution of the universe, and their compatibility with the current observational data (see, e.g., [2, 3, 4, 5, 7, 6] and numerous references there). These interacting models can be roughly split into two classes: the models with linear and non-linear interaction, respectively. For the former models, the interaction term is proportional to either the energy density of DM, , the energy density of DE, , or their linear combinations. However, as it was noted in [8], an interaction term proportional to leads to a large-scale instability at early times111It is worth noting that such large-scale instability problem can be solved within a generalized parametrized post-Friedmann approach [9, 10]., whereas a term proportional to could give rise to a negative in the future. Therefore, models where an interaction term is described by a non-linear function of and were proposed (see, e.g., [2, 3, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]). In these papers, in most of the cases the interaction term is a product of the energy densities of DM and DE. This is motivated by the analogy with two-body chemical reactions where the interaction rate is proportional to the product of the number densities of the two particles species [2, 12]. Following this reasoning, in our paper we consider an interaction term of the form:

 Q=3Hγ¯¯¯εDE¯¯¯εDM¯¯¯εDE+¯¯¯εDM,

where is a dimensionless parameter that modules the strength of the interaction. It can be easily seen that such a model is free of the above mentioned shortcomings present in models with a linear interaction in the dark sector [8]. We also suppose that DM is pressureless and DE satisfies the equation of state (EoS) , where . It was shown in [3] that this model satisfies the scaling solution

 ξ≡¯¯¯εDM¯¯¯εDE=ξ0(a0a)−3(w+γ),ξ0=\rm const.

This solution follows from the conservation equations for DM and DE with the above described interaction term. Notice that some papers start directly from the scaling solution ansatz to investigate the coincidence problem222The coincidence problem within interacting models was also discussed in [21, 22].[23, 24, 25]. Obviously, if , the ratio is a constant and, consequently, the coincidence problem is absent (if ). To get the late-time DE dominance, we should demand that . In the case of the standard CDM model . Therefore, for any DM-DE interacting model with , the coincidence problem is less severe than in CDM. In our paper, we will provide two of such models with .

We would like to stress that the quadratic interactions we have assumed is phenomenological: (i) it was introduced in Ref. [8] as a mean to remove the instabilities present at the perturbative level of some linear DE-DM interaction and (ii) is inspired on the two-body chemical reactions where the interaction rate is proportional to the densities of the two type of particles involved [2, 12]. We would like also to stress that the non-linear interaction we assume, arise in a natural way within the new generalised Chaplygin gas (NGCG) [8]. This feature might be very appealing as it might hint towards a particle physics motivations for this interaction as the standard Chaplygin gas can be obtained from d-branes physics and has a super-symmetric realisation [26]. Yet, we must say we are not aware that this is necessarily the case for a general NGCG.

In our analysis we consider the Universe at late-time and deep inside the cell of uniformity. At such scales, the Universe is highly inhomogeneous, and inhomogeneities in the form of galaxies and groups of galaxies have non-relativistic peculiar velocities. In this case, the mechanical approach is an adequate tool to study the scalar perturbations [27, 28]. Further methods related to the mechanical approach were also proposed in [29, 30, 31, 32, 33]. This approach enables us to get the gravitational potential and to consider the motions of galaxies [34]. The mechanical approach was applied to a number of DE models to study their compatibility with the theory of scalar perturbations. For example, we considered a perfect fluid with a constant equation of state parameter [35], the model with quark-gluon nuggets [36], the CPL model [37], the Chaplygin gas model [38], the non-linear model [39] and the model with a scalar field [40]. One of the main features of all these models is that perfect fluids (as well as scalar field) are considered in a very specific ”coupled” form [41]. This means that their fluctuations are concentrated around the galaxies, therefore screening their gravitational potential. Consequently, the peculiar velocities of such coupled perfect fluids are also non-relativistic. In the present paper, we investigate the possibility that non-linearly interacting DM and DE are coupled to the galaxies. The linear interacting model within the mechanical approach was considered in [42]. In what follows, we shall show that non-linearly interacting DM and DE can be coupled to the galaxies. They possess a number of interesting properties. First, the energy densities of interacting DM and DE are concentrated around the galaxies. Second, the sum of parameters is less than 1. Hence, the coincidence problem is not so severe as for the CDM model (see the definition of the ratio above). Third, the set of parameters is within the current cosmological observations [8].

Our paper is structured as follows. In section 2, we describe the background model for the non-linear interacting DM and DE. In section 3, we consider the theory of scalar perturbations for the considered model. Here, we define the sets of parameters which do not contradict the theory of scalar perturbations. In the concluding section 4, we summarise the obtained results and discuss them.

## 2 Non-linear interaction in the dark sector: background equations

and

where ,   ( is the speed of light and is the Newtonian gravitational constant) and for open, flat and closed Universes, respectively. The conformal time, , and the synchronous or cosmic time, , are connected as , where is the scale factor. As usual, for radiation we have the EoS . is the average energy density of the non-interacting non-relativistic matter with the comoving rest mass density . In general, we assume that dark matter may exist both as an interacting and a non-interacting fluid. This type of models was recently investigated, e.g. in [43, 44]. In our case, accounts for non-interacting cold dark matter and baryonic matter. The pressure for all kinds of non-relativistic matter (interacting as well as non-interacting) is assumed to vanish, i.e., . On the other hand, the equation of state for the interacting dark energy is

 ¯¯¯pDE=w¯¯¯εDE,w≠0. (2.3)

Such a perfect fluid may result under some conditions in an accelerating expansion of the Universe. This is the reason to call it ”dark energy”. While it is natural to expect that the current accelerated expansion of the universe requires a negative (in fact, in the absence of interaction the violation of the strong energy condition imposes that ), in general, we shall not restrict ourself to such values of . On the other hand, the presence of such a perfect fluid does not guarantee that the acceleration will happen for sure at present. Therefore, we still keep in Eqs. (2.1) and (2.2) the cosmological constant term . It can be easily realized that even though the presence of such a term influences the dynamical behaviour of the Universe at the background level, it will not affect the analysis of the perturbations. In the case when the interacting dark energy provides the late-time accelerated expansion of the Universe, we can drop and investigate this model from the point of its concordance with the observable data. Such models with dynamical dark energy may help to resolve the coincidence problem.

From Eqs. (2.1) and (2.2) we get the following auxiliary equation:

The non-interacting components of matter/energy are conserved independently. This results, e.g., in a and dependence for the energy densities of non-relativistic matter and radiation, respectively. However, the interacting components should satisfy the common conservation equation:

 ¯¯¯ε′tot+3H(¯¯¯εtot+¯¯¯ptot)=0, (2.5)

where in our case and . We can split this equation into two:

 ¯¯¯ε′DM+3H(¯¯¯εDM+¯¯¯pDM)=¯¯¯ε′DM+3H¯¯¯εDM=Q, (2.6)

and

 ¯¯¯ε′DE+3H(¯¯¯εDE+¯¯¯pDE)=¯¯¯ε′DE+3(1+w)H¯¯¯εDE=−Q,w≠0. (2.7)

Here, the term describes the interaction between dark energy (DE) and dark matter (DM). If , there is a transfer of energy from DE to the interacting DM component, and vice versa for . As we have mentioned in the introduction, there are a quite large number of phenomenological expressions for . In the present paper, we investigate the case of the non-linear interaction [3, 8]:

 Q=3Hγ¯¯¯εDE¯¯¯εDM¯¯¯εtot,¯¯¯εtot=¯¯¯εDM+¯¯¯εDE, (2.8)

where is a free parameter of the model. Since we consider an expanding Universe, we have at all times, and consequently . If the interaction is switched off, i.e., if we set , it follows immediately from Eqs. (2.6) and (2.7) that:

 ¯¯¯εDM∝(a0a)3,¯¯¯εDE∝(a0a)3(1+w). (2.9)

As found in [3], the system of Eqs. (2.6) and (2.7) with the interaction (2.8) is analytically solvable. First, it was shown that the ratio for the considered model satisfies the scaling solution:

 ξ=¯¯¯εDM¯¯¯εDE=ξ0(a0a)−3(w+γ), (2.10)

where . It is worth noting that cosmological models with the scaling ansatz (2.10) were also investigated in [23, 24, 25]. In addition, we can re-write the conservation Eqs. (2.6) and (2.7) as

 ¯¯¯ε′DM+3H(1−γ1+ξ)¯¯¯εDM=0, (2.11) ¯¯¯ε′DE+3H(1+w+γξ1+ξ)¯¯¯εDE=0. (2.12)

From these equations we can identify the effective EoS parameters for the interacting DM and DE

 w(eff)DM=−γ1+ξ,w(eff)DE=w+γξ1+ξ. (2.13)

There are three possible scenarios for the background evolution, cf. equation (2.10), depending on the sign of . We now analyse each case individually.

### 2.1 w+γ>0

In this case DM becomes dominant at late-times, since and . Therefore, it is of no interest to cosmology with late-time DE dominance.

### 2.2 w+γ=0

This leads to scaling solutions (self-similar solution) where the proportion of DM to DE is constant, i.e., . The DM, DE, and total (DM+DE), energy densities can be written as functions of the scale factor as

 ¯¯¯εDM = (2.14) ¯¯¯εDE = (2.15) ¯εtot = (¯¯¯εDM,0+¯¯¯εDE,0)(a0a)3(1+w∗), (2.16)

where . Thus behaves as a perfect fluid with a constant parameter of EoS , a case which was analysed within the Mechanical Approach in [35] (however as we will see below the results are not the same at the perturbative level). In the case of , we find that (as well as its constituent parts) behaves as a cosmological constant. In order to obtain late-time acceleration we must impose .

Due to the fact that the ratio , it seems that there is no energy density transfer between DM and DE. However, this is not so. To show it, let us consider the case , which implies . In this case , therefore,

 γ<0⇒1+w∗>1,1+w∗=1+w1+ξ0<1+w, (2.17)

and the comparison of Eqs. (2.9), (2.14), and (2.15) demonstrates that DM (DE) decreases with the growth of the scale factor faster (slower) in the presence of interaction. Obviously, this happens due to the energy density transfer from DM to DE which ensures the constancy of the ratio . Similarly, we can show that in the case , we obtain , and there is an energy density transfer from DE to DM.

### 2.3 w+γ<0

In this case DE becomes dominant at late-time, since and . Therefore the solutions of equations (2.11) and (2.12) have the following limiting behaviours for DM

 ¯¯¯εDM(ξ≫1)≃(a0a)3and¯¯¯εDM(ξ≪1)≃(a0a)3(1−γ), (2.18)

and for DE

 ¯¯¯εDE(ξ≫1)≃(a0a)3(1+w+γ)and¯¯¯εDE(ξ≪1)≃(a0a)3(1+w). (2.19)

We see that at very late-time the interaction affects the evolution of DM, which behaves as a perfect fluid with an effective parameter of state , while essentially leaving DE unaffected. If DE is to be responsible for the current acceleration of the Universe, then .

The behaviour in (2.18) and (2.19) is precisely what is observed in the formal solutions of equations (2.11) and (2.12) (see [3] for a derivation of these solutions)

 ¯¯¯εDM = (2.20) = ¯¯¯εDM,0(ξ0ξ)1−γw+γ[1+ξ1+ξ0]−γw+γ,
 ¯¯¯εDE = (2.21) = ¯¯¯εDE,0(ξ0ξ)1+ww+γ[1+ξ1+ξ0]−γw+γ.

Eqs. (2.18) - (2.21) show that, in spite of the fact that DE dominates at the late stage of the Universe evolution, the energy density transfer exists in both directions depending on the sign of . Indeed, for positive , the energy density of DM decays slower than due to the energy density transfer from DE to DM and vise versa in the case of .

Notice that since goes to zero in the future, as , we can write the following Maclaurin series

 [1+ξ1+ξ0]−γw+γ=(11+ξ0)−γw+γ+∞∑i=0αiξi. (2.22)

From the definition of the coefficients of a Maclaurin series, we obtain the expression

 αi ≡ (2.23) = 1i![−γw+γ][−γw+γ−1]...[−γw+γ−(i−1)] = 1i!(−1w+γ)iγ[γ+(w+γ)]...[γ+(i−1)(w+γ)].

Therefore, the coefficients are given by

 α0 = 1, (2.24) αi>0 = (−1)ii!i∏j=1γ+(j−1)(w+γ)w+γ=(−1)ii!(γw+γ)i. (2.25)

In Eq. (2.25), we have introduced the Pochhammer symbol, , to designate the ascending factorial [45, 46]. We can now write the energy density of DM and DE as

 ¯¯¯εDM = ¯¯¯εDE,0(11+ξ0)γ|w+γ|(a0a)3(1+w)+∞∑i=0αiξi+1, (2.26) ¯¯¯εDE = ¯¯¯εDE,0(11+ξ0)γ|w+γ|(a0a)3(1+w)+∞∑i=0αiξi, (2.27)

where we took into account that .

## 3 Cosmological perturbations in the late Universe: mechanical approach

As we have already mentioned in the introduction, we consider the Universe at its late stage of evolution and deep inside the cell of uniformity. At such scales the Universe is highly inhomogeneous. The inhomogeneities in the form of galaxies, groups and clusters of galaxies have already formed. To describe the dynamical behaviour of the inhomogeneities, the discrete cosmology (mechanical) approach was proposed in [27, 28, 34] (see also the related methods developed in [29, 30, 31, 32, 33]). The above mentioned inhomogeneities affect the background model, that is, the background metric as well as the background matter/energy in the model. The scalar perturbations of a Friedmann-Lemaître-Robertson-Walker metric in the Newtonian gauge read:

 ds2≈a2[(1+2Φ)dη2−(1−2Φ)γαβdxαdxβ]. (3.1)

In the late Universe, the inhomogeneities (e.g. galaxies and groups of galaxies) have non-relativistic peculiar velocities. In the mechanical approach [27, 28], we also consider a special form of matter sources (playing, e.g., the role of DM and DE) which are coupled to these inhomogeneities [40, 41]. By this we mean that the fluctuations of the energy density of such matter sources are concentrated around the inhomogeneities. Therefore, the peculiar velocities of these fluctuations are also non-relativistic, and in the system of equations for the scalar perturbations we can drop the terms containing the peculiar velocities as compared with their energy density and pressure fluctuations. Then, the gravitational potential takes the form [27, 28, 35, 36, 37, 38] :

 Φ(r,η)=φ(r)c2a, (3.2)

where the function (the ”comoving” gravitational potential) depends only on the comoving spatial coordinates333It is worth noting that in general the radius-vectors of the inhomogeneities (see e.g. Eq. (4.4) below) depend on time. However, the peculiar velocities are so small (highly non-relativistic at present) that effectively at each moment in time we can calculate the potential just by knowing the position of the inhomogeneities. In this sense the mechanical approach is like a quasi-static approximation, i.e., the particles move so slowly that, at each moment in time, we can compute all relevant quantities just as if they were at rest. As it was shown in [28], the peculiar velocities of the inhomogeneities behaves as . Hence, the larger is the scale factor, , i.e. at a later time of the evolution of the Universe, the better this approach works. It is as well worth noting that this quasi-static approximation is considered with respect to the comoving space. Therefore, the scale factor of the Universe still has a dynamical behaviour. Summarizing, in the mechanical approach, the gravitational potentials of the inhomogeneities are defined by their positions and not through their velocities. This approximation is similar to the one employed in astrophysics (see section 106 in [47]). The main difference is that here we take into account the expansion of the Universe (and we additionally consider other perfect fluids in the coupled form). After determining the gravitational potential of the system, we can use it to describe the relative motion of the inhomogeneities (e.g. galaxies) [34]. . The gravitational potential satisfies the following system of linearised Einstein equations:

Here, is the Laplace operator defined with respect to the spatial metric , and the fluctuation of the energy density of the non-interacting non-relativistic matter (at the linear approximation with respect to ) reads [27]

 δT00=δρcc2a3+3¯ρcc2Φa3=δρcc2a3+3¯¯¯ρcφa4, (3.5)

where the fluctuation of the comoving rest mass density is given as .

With the help of (2.4), Eq. (3.4) reads (make mention to how was replaced)

where we also took into account that . From this EoS and within the mechanical approach, where the peculiar velocities are assumed to be non-relativistic, we can easily get that (we refer the reader to the appendix of [37] for a derivation of this result). The important point is that, according to Eq. (3.5), the accuracy of our approach is 444See e.g. the footnote 2 in [37] where we justified this approximation.. This means that in our perturbed equations, in the late Universe we should drop all the terms which behave as , with . For example, in the above equation (3.6) the term should be omitted. Following this reasoning, we get for the fluctuations of interacting dark energy

We can easily verify that the same expression for follows from the total conservation equation [37]:

 ∑i[δpXi+(¯εXi+¯pXi)Φ]=0, (3.8)

where the summation should be taken over all matter/energy constituents of the model. Keeping in mind that the contribution of the -term disappears from this equation, we must consider in our case the non-interacting non-relativistic matter and radiation, and the interacting dark sector components.

Next, it is convenient to split into two parts:

where

and accounts for a possible contribution of DE to the radiation fluctuations [37]. In the CDM model, the fluctuation of radiation exactly coincides with the expression (3.10) [28]. Then, Eq. (3.7) takes the form

If we differentiate equation (3.11), we obtain

Substituting the conservation equations (2.5) and (2.7) in Eq. (3.12), we find after some algebra

 δε′DE=−4HδεDE+3H(1+w)¯¯¯εDEφc2a+Qφc2a. (3.13)

Notice that if we turn off the interaction term, we obtain the following solution to

 δεDE=c1a4−1+wc2w¯¯¯εDE,0a3+3w0a4+3wφ, (3.14)

where is an integration constant. This result is in accordance with the analysis done in [35]. On the other hand, the perturbed conservation equation for reads (see e.g. the Appendix in [37])

 δε′DE+3H(1+w)δεDE+(1+w)¯¯¯εDE(−3Φ′)=−δQ. (3.15)

When we turn off the interaction term the solution of this equation is

 δεDE=c2a3+3w−1+wc2(−1/3)¯¯¯εDE,0a3+3w0a4+3wφ, (3.16)

where is an integration constant. Notice that Eqs. (3.14) and (3.16) are consistent only in the cases with . These are the physically acceptable values of found in [35].555The case and is also mathematically consistent but this is just the case of a Universe filled by non-interacting dust and radiation - unsuitable to describe the current Universe. Indeed it is of no interest to the interacting model we are analysing.

Using Eqs. (3.11) and (3.13), we obtain from (3.15):

The results obtained so far in this section are independent of the form of and . Let us now consider the exact form of , as given by (2.8). Perturbing this interaction term, we find

 δQ=3Hγ[(11+ξ)2δεDM+(ξ1+ξ)2δεDE]. (3.18)

Notice that, while the expression in (2.8) is not obtained from a covariant formulation of the interaction, which means that there is no a priori prescription on how to obtain the perturbation , in our approach the implementation of the perturbed interaction term derived in Eq. (3.18) was done consistently with the constraints derived from the perturbed conservation equations for the total matter.

Equation (3.18) enables us to get

Here we have used (3.11) and (3.17) to eliminate and .

We are now ready to go back to the component of the perturbed Einstein equations (3.4), which we can re-write as

By substituting (3.11) and (3.19) in this equation, we find

The left-hand-side (l.h.s.) of this equation does not depend on time. Therefore, the same should be true within the adopted accuracy for the right-hand-side (r.h.s.) of Eq. (3). In general, there are two possibilities to achieve this: either the terms in r.h.s. compensate each other, or each of them do not depend on time separately. Some terms may vanish because according to our accuracy, we define fluctuations of the energy densities and pressure up to terms , inclusively. Bearing in mind that r.h.s. of (3.20) has been multiplied by , on r.h.s. of (3) we must drop all terms , with . Now, we should analyse under which conditions r.h.s. of (3) is self-consistent. It makes sense to consider separately the cases of and .

Before continuing we compare the results obtained so far, in particular Eq. (3), with those obtained in the mechanical approach for a non-interacting DE fluid with constant EoS parameter [37]. Intuitively, to recover the results of [37] we should set as this switches off the interaction both at the background, , and perturbative, , levels. However, in Eq. (3) we find terms inversely proportional to that diverge as we take the limit . This comes from the fact that Eq. (3.19), as derived from Eq. (3.18), is not defined when , i.e., in the absence of interaction we cannot derive from Eq. (3.19) a relation between the perturbations of DM and DE, and consequently between , , and . A way around this problem is to notice that in the absence of interaction, the interacting DM component should vanish, since all non-interacting non-relativistic matter is already included in the matter distribution . Therefore, when we switch off the interaction we should set to zero , , and . Consequently in Eq. (3) we should drop all the terms coming from the term in Eq. (3.20). To better identify such terms, let us set and in Eq. (3.19). The two remaining terms of r.h.s. of the equation are those that we should drop from Eq. (3) after setting in that equation. After the comparison with the equation obtained in [37] we find that we do recover the result for the case of no interaction.

### 3.1 w+γ=0

Let us first analyse the case of , which corresponds to a self-similar solution of DM and DE. As we have seen before, in this case we have , , , and and are described by Eqs. (2.14) and (2.15), respectively. If we substitute these expressions in Eq. (3) we find that r.h.s. of this equation has the structure

 r.h.s=A1a0a+A2(a0a)1+3w∗, (3.22)

where

and

 A2≡1+w∗+3ξ0(w∗)23(w∗)2(1+ξ0)¯¯¯εDE,0κa20φ2. (3.24)

Here we took into account the dependence of the fluctuation of radiation on the scale factor:

where is the distribution of the fluctuations at the present time. The two terms on r.h.s. of equation (3.22) cannot compensate each other as they have different dependencies on : the first term goes as while the second goes as (we are assuming that as implies ). Therefore, in order for r.h.s. to be time independent we need to impose, for each term separately, that it either vanishes or is independent of time.

Regarding the first term we have to impose that which results in either , or in , i.e., and (and, consequently, ) behave as radiation.

Regarding the second term we have three possibilities. The first is that , in which case the term can be dropped altogether within the accuracy of the mechanical approach. As a consequence, we find that interacting DM and DE do not contribute to the gravitational potential. The second possibility is that , in which case the term is independent of time and the late Universe is neither accelerating nor decelerating666The acceleration of the Universe takes place if the following combination is positive: . Obviously, this happens for (we remind that both and are positive).. Finally, the third possibility is that , which leads to

 1+w∗+3ξ0(w∗)2=0⇒ξ0=−1+w∗3w∗2. (3.26)

Notice that since is positive by definition, if the previous equation holds we must have , i.e., phantom behaviour of dark energy. For a fixed value of we get

 w∗=−1±√1−12ξ06ξ0,0<ξ0≤112. (3.27)

It can be easily seen that for if we choose the upper sign in (3.27). The case of implies that , i.e. , and we arrive to the case of a cosmological constant with no interaction.

Finally, we list the acceptable cases found for :

1. , and ;

2. and may be non-zero;

3. and ;

4. , , and . This case reduces to the cosmological constant with no interaction777This reduces to CDM model in absence of interactions on the dark sector. Given that this case has been previously analysed in [27], we will disregard its analysis. The same criterium will be applied whenever we recover CDM models (with constant) in absence of interactions on the dark sector, which were studied in [35] (see also the appendix in [37]).;

5. and . This case leads to phantom behaviour and is defined by the relation (3.26).

### 3.2 w+γ<0

In this case, the expressions for and are given by (2.27). Substituting them on r.h.s. of Eq. (3), we obtain

 r.h.s = κc2a3018wγδεrad2,0(2∑i=0Biξi0(a0a)1+3i|w+γ|) (3.28) + (11+ξ0)−γw+γκa20¯¯¯εDE,06wγ(+∞∑i=0Diξi0(a0a)1+3w+3i|w+γ|)φ.

Here, we have introduced the coefficients

 B0 ≡ (3γ+1)(3w−1), (3.29) B1 ≡ 2(3w−1), (3.30) B2 ≡ 3(w+γ)−1, (3.31)

and

 Di ≡ i∑j=0Cjαi−j, (3.32)

where

 C0 ≡ −[3(w+γ)+1](1+w), (3.33) C1 ≡ −3(1+w)(1+γ)−2w(1+3w), (3.34) C2 ≡ −3+3(w+γ)−w(1+3w), (3.35) C3 ≡ −1+3(w+γ), (3.36) Cj ≡ 0,j≥4, (3.37)

and the coefficients are defined in Eqs. (2.24) and (2.25).

Before continuing, let us investigate which terms we can drop from Eq. (3.28) within the accuracy of the mechanical approach.

With respect to the three terms that are related to the coefficients , we find that we need only to keep the term proportional to , which goes like , as this is already in the boundary of the accuracy. The other two terms are outside the accuracy of the approach and can be dropped.

For the terms that are related to the coefficients , we find that the maximum order that needs to be considered is defined by the biggest integer that satisfies the inequality

 1+3w+3iDmax|w+γ|≤1,⇒iDmax=E(−w|w+γ|), (3.38)

where is the floor function that returns the largest integer smaller than [46]. The value of depends then on the ratio . Taking into account the previous considerations we can reduce (3.28) to

 r.h.s = κc2a3018wγδεrad2,0B0(a0a) (3.39) + (11+ξ0)−γw+γκa20¯¯¯εDE,06