The late Universe with nonlinear interaction in the dark sector: the coincidence problem
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 nonlinear 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.
Departamento de Física, Universidade da Beira Interior, 6200 Covilhã, Portugal
Centro de Matemática e Aplicações da Universidade da Beira Interior (CMAUBI), 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
mbl@ubi.pt (On leave of absence from UPV/EHU and IKERBASQUE) \emailAddjviegas001@ikasle.ehu.eus \emailAddai.zhuk2@gmail.com
1 Introduction
Recent observations [1] clearly indicate that our Universe is dark. The massenergy 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 nonlinear 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 largescale instability at early times^{1}^{1}1It is worth noting that such largescale instability problem can be solved within a generalized parametrized postFriedmann 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 nonlinear 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 twobody 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:
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
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 problem^{2}^{2}2The 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 latetime DE dominance, we should demand that . In the case of the standard CDM model . Therefore, for any DMDE 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 DEDM interaction and (ii) is inspired on the twobody 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 nonlinear 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 dbranes physics and has a supersymmetric 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 latetime 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 nonrelativistic 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 quarkgluon nuggets [36], the CPL model [37], the Chaplygin gas model [38], the nonlinear 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 nonrelativistic. In the present paper, we investigate the possibility that nonlinearly 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 nonlinearly 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 nonlinear 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 Nonlinear interaction in the dark sector: background equations
We start with the Friedmann and Raychaudhuri equations for the homogeneous background which read
(2.1) 
and
(2.2) 
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 noninteracting nonrelativistic matter with the comoving rest mass density . In general, we assume that dark matter may exist both as an interacting and a noninteracting fluid. This type of models was recently investigated, e.g. in [43, 44]. In our case, accounts for noninteracting cold dark matter and baryonic matter. The pressure for all kinds of nonrelativistic matter (interacting as well as noninteracting) is assumed to vanish, i.e., . On the other hand, the equation of state for the interacting dark energy is
(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 latetime 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:
(2.4) 
The noninteracting components of matter/energy are conserved independently. This results, e.g., in a and dependence for the energy densities of nonrelativistic matter and radiation, respectively. However, the interacting components should satisfy the common conservation equation:
(2.5) 
where in our case and . We can split this equation into two:
(2.6) 
and
(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 nonlinear interaction [3, 8]:
(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:
(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:
(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 rewrite the conservation Eqs. (2.6) and (2.7) as
(2.11)  
(2.12) 
From these equations we can identify the effective EoS parameters for the interacting DM and DE
(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
In this case DM becomes dominant at latetimes, since and . Therefore, it is of no interest to cosmology with latetime DE dominance.
2.2
This leads to scaling solutions (selfsimilar 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
(2.14)  
(2.15)  
(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 latetime 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,
(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
In this case DE becomes dominant at latetime, since and . Therefore the solutions of equations (2.11) and (2.12) have the following limiting behaviours for DM
(2.18) 
and for DE
(2.19) 
We see that at very latetime 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)
(2.20)  
(2.21)  
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
(2.22) 
From the definition of the coefficients of a Maclaurin series, we obtain the expression
(2.23)  
Therefore, the coefficients are given by
(2.24)  
(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
(2.26)  
(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 FriedmannLemaîtreRobertsonWalker metric in the Newtonian gauge read:
(3.1) 
In the late Universe, the inhomogeneities (e.g. galaxies and groups of galaxies) have nonrelativistic 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 nonrelativistic, 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] :
(3.2) 
where the function (the ”comoving” gravitational potential) depends only on the comoving spatial coordinates^{3}^{3}3It is worth noting that in general the radiusvectors of the inhomogeneities (see e.g. Eq. (4.4) below) depend on time. However, the peculiar velocities are so small (highly nonrelativistic 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 quasistatic 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 quasistatic 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:
(3.3)  
(3.4) 
Here, is the Laplace operator defined with respect to the spatial metric , and the fluctuation of the energy density of the noninteracting nonrelativistic matter (at the linear approximation with respect to ) reads [27]
(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)
(3.6) 
where we also took into account that . From this EoS and within the mechanical approach, where the peculiar velocities are assumed to be nonrelativistic, 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 ^{4}^{4}4See 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
(3.7) 
We can easily verify that the same expression for follows from the total conservation equation [37]:
(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 noninteracting nonrelativistic matter and radiation, and the interacting dark sector components.
Next, it is convenient to split into two parts:
(3.9) 
where
(3.10) 
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
(3.11) 
If we differentiate equation (3.11), we obtain
(3.12) 
Substituting the conservation equations (2.5) and (2.7) in Eq. (3.12), we find after some algebra
(3.13) 
Notice that if we turn off the interaction term, we obtain the following solution to
(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])
(3.15) 
When we turn off the interaction term the solution of this equation is
(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].^{5}^{5}5The case and is also mathematically consistent but this is just the case of a Universe filled by noninteracting 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):
(3.17) 
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
(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.
We are now ready to go back to the component of the perturbed Einstein equations (3.4), which we can rewrite as
(3.20) 
By substituting (3.11) and (3.19) in this equation, we find
(3.21) 
The lefthandside (l.h.s.) of this equation does not depend on time. Therefore, the same should be true within the adopted accuracy for the righthandside (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 selfconsistent. 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 noninteracting 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 noninteracting nonrelativistic 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
Let us first analyse the case of , which corresponds to a selfsimilar 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
(3.22) 
where
(3.23) 
and
(3.24) 
Here we took into account the dependence of the fluctuation of radiation on the scale factor:
(3.25) 
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 decelerating^{6}^{6}6The 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
(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
(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 :

, and ;

and may be nonzero;

and ;

, , and . This case reduces to the cosmological constant with no interaction^{7}^{7}7This 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]).;

and . This case leads to phantom behaviour and is defined by the relation (3.26).
3.2
In this case, the expressions for and are given by (2.27). Substituting them on r.h.s. of Eq. (3), we obtain
(3.28)  
Here, we have introduced the coefficients
(3.29)  
(3.30)  
(3.31) 
and
(3.32) 
where
(3.33)  
(3.34)  
(3.35)  
(3.36)  
(3.37) 
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
(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
(3.39)  