Phase Space Analysis of the Accelerating Multifluid Universe
Abstract
We study in detail the phase space of a FriedmannRobertsonWalker Universe filled with various cosmological fluids which may or may not interact. We use various expressions for the equation of state, and we analyze the physical significance of the resulting fixed points. In addition we discuss the effects of the stability or an instability of some fixed points. Moreover we study an interesting phenomenological scenario for which there is an oscillating interaction between the dark energy and dark matter fluid. As we demonstrate, in the context of the model we use, at early times the interaction is negligible and it starts to grow as the cosmic time approaches the latetime era. Also the cosmological dynamical system is split into two distinct dynamical systems which have two distinct de Sitter fixed points, with the earlytime de Sitter point being unstable. This framework gives an explicit example of the unification of the earlytime with latetime acceleration. Finally, we discuss in some detail the physical interpretation of the various models we present in this work.
pacs:
04.50.Kd, 95.36.+x, 98.80.k, 98.80.Cq, 11.25.wI Introduction
In modern theoretical cosmology, the most striking event was the observation of the latetime acceleration riess that our Universe undergoes at present time. Admittedly this observation has utterly changed the way of thinking of modern cosmologists, since this latetime acceleration is a feature of our Universe that was never thought it would actually occur. Consequently, the focus for the last nearly 20 years is to model in a successful way this latetime acceleration and also to harbor the late and earlytime acceleration era in a unified theoretical framework. Towards this unified description, many proposals, especially those suggesting to modify the gravitational sector, have been introduced ever since, see the reviews reviews1 ; reviews2 ; reviews3 ; reviews4 for details. From the first moment that the latetime acceleration has been observed, it was realized that no perfect matter fluid known at that time was able to realize the latetime acceleration era, and therefore the need for alternative generalized cosmological fluids was compelling. By using generalized cosmological fluids, both the late and earlytime acceleration era can be realized, and up to date there are many theoretical proposals that use generalized fluids, for example in Refs. inhomogen1 ; nojodineos1 ; inhomogen2 ; brevik1 ; brevik2 ; fluid1 ; fluid2 imperfect fluids are used in order to describe the cosmological evolution of our Universe, and in some cases certain particular examples are used, called viscous fluids are (see Brevik:2017msy for reviews). It is notable that the imperfect fluids may describe even phantom evolution of our Universe, without using phantom scalar fields, which violate the energy conditions, see Refs. inhomogen1 ; inhomogen2 for details. Furthermore, other cosmological evolutions like bouncing cosmology, in the context of both classical and loop quantum cosmology imperfect fluids were studied in oikonomouimperfect , and also singular cosmology can be realized by imperfect fluids oikonomoufluid ; Brevik:2016kuy . Furthermore, several models which take into account bulk viscosity were discussed in Refs. ALUL ; CLLPS ; LS ; VWM ; VS ; VCFCB ; VSFZ ; V ; IS ; M ; CLP ; CCL ; LOS , and also an important class of models which assume an interaction between dark matter and dark energy fluids, can be found in Refs. ALUL ; ZP ; CCMUL .
In this paper we shall perform a detailed phasespace analysis of a FriedmannRobertsonWalker Universe, filled with different fluid components, which may or may not interact between them. The dynamical evolution of such kind of model is described by the Friedmann equations,
(1) 
(2) 
or equivalently,
(3) 
and also the energy conservation equations hold true,
(4) 
In the above, and also the equation of state (EoS) may be highly nontrivial for some fluid components. We shall appropriately choose the variables in order to capture the phase space dynamics in the most optimal way, and we shall analyze the structure of the phase space by providing an analytic treatment of the cosmological dynamical equations. Seeing the cosmological equations as a dynamical system is a particularly appealing way to investigate the phase space in many cosmological contexts, see for example Coley:1999uh ; GarciaSalcedo:2012dn ; Boehmer:2008av ; Boehmer:2014vea ; Haba:2016swv ; Bolotin:2013jpa , but also in modified gravity too, see for example Abdelwahab:2007jp ; Carloni:2007eu ; Carloni:2004kp ; Leon:2014yua ; Xu:2012jf ; Carloni:2017ucm ; Leon:2010pu ; Carloni:2007br ; Khurshudyan:2016qox ; Hohmann:2017jao . In most cases, the choice of the dynamical system variables plays a crucial choice, and in some cases the resulting cosmological dynamical system may be rendered autonomous Odintsov:2015wwp . Also it is possible to choose dimensionless variables, see for example Refs. Carloni:2007eu ; Carloni:2004kp for an gravity cosmological dynamical system, and also see Ref. STT , for a cosmological theory with higher derivatives of the scalar curvature. The dynamical systems approach for cosmological systems has many attributes, with the most important being the fact that the fixed points of the dynamical system actually provide new insights with regards to the behavior of the attractor solutions and also reveals the stability structure of the dynamical system near the attractors. It is conceivable that the choice of the variables plays an important role, as we also demonstrate by this work.
This paper is organized as follows: In section II we present some well known features of dynamical systems approach in cosmological context with generalized fluids, in section III we discuss in brief how interactions between dark matter and dark energy may be introduced and by using appropriately chosen variables we present how the dynamical systems analysis can be performed in this case. In section IV we generalize the formalism we developed in the previous sections and by using dimensionless variables, we investigate the physical consequences of having various equations of state for the fluid components of the cosmological system. In section V by using appropriately chosen dimensionless variables, we investigate in detail how the interaction of dark matter and dark energy fluids may affects the phase space structure. We study the stability and behavior of the fixed points of the dynamical system and we also discuss how the early and latetime acceleration eras are affected by the various functional forms of the interaction coupling between dark matter and dark energy. Finally the conclusions follow in the end of the paper.
Ii Standard Approach on Dynamical Systems and Cosmological Dynamics
In this section we present the simplest case of the dynamical systems approach in cosmological dynamics. We consider the simplest case in which the Universe is filled with radiation and a perfect fluid with a nontrivial EoS of the form NO1 . Such nontrivial EoS can be considered as some sort of viscous fluid or generalized EoS fluid, see Bamba:2012cp for a review on this topic. In addition, such an EoS can be considered as an effective fluid presentation of some modified gravity theory reviews2 ; reviews3 . In the case at hand, the full dynamical system takes the following form,
(5)  
(6)  
(7) 
The appearance of the term might seem unconventional, from a thermodynamic point of view, and we need to briefly describe the motivation for using such a term. This term encompasses the viscous part of the cosmological fluid, so it mainly quantifies the viscosity of the fluid. In the Universe, nd especially in the very early stages of it’s evolution, the effects of a viscous cosmological component are most likely expected to occur during the neutrino decoupling process, which occurs at the end of the lepton era Misner:1967uu . Hence, viscosity is encompassed in the very own fabric of the Universe. In addition, a strong motivation for using viscous fluid components comes from the fact that the perfect fluid approach among cosmologistshydrodynamicists is just an ideal approach, and does not describe the real world. Finally, due to the fact that early and latetime acceleration may be described by an unknown form of a cosmological fluid, it is natural to assume that the fluid has the most general form, which means that a viscous component is needed^{1}^{1}1Note that terms of the form in the cosmological relation between the effective pressure and the energy density, naturally occur in modified gravity thermodynamics Bamba:2016aoo .
Having described the motivation for the use of viscous fluid components, we can rewrite the above cosmological equations in terms of dimensionless variables. In the case at hand, there is only one independent variable due to the constraint equation (5). By using the foldings number^{2}^{2}2In this case time derivatives for some variable transform as and by introducing a new dimensionless variable defined as,
we obtain the next dynamical equation,
(8) 
where the variables , and must be expressed in terms of depending on the choices of the functions and .
Let us here discuss the simplest choice of EoS, which is , which implies that and . It is easy to see that in this case, Eq. (8) takes the following form,
(9) 
where the “prime” denotes differentiation with respect to the foldings number. The equation that determines the stationary points for the above dynamical equation, is quadratic with discriminant , so the existing solutions are always real. For the case there is the only one solution, which is, there are two solutions, which are: . For the case
(10)  
(11) 
and this case is the most interesting, since it provides us plenty dynamical solutions. Now note that the physical values that the variable can take, are , but for we have , which give us the following restrictions for the free parameters:
(12)  
(13) 
Let us study the stability conditions of the existing stationary points. It is clear that for every point there is only one eigenvalue , and for the fixed points at hand, we have,
(14)  
(15) 
Therefore, we find that for any values of parameters , one of the fixed points is stable and the other one is unstable. Note that for the case we have and therefore it is compelling to investigate the corresponding center manifold, but fortunately this is not interesting case from a physical point of view. The physical significance of the stationary point is that it corresponds to a Universe with and . With regard to the fixed point , it corresponds to a Universe with some fixed relation between and .
Iii Models with Dark Matter Interacting to Dark Energy
Let us discuss at this point some models which describe interactions of dark matter with dark energy. The dynamical system in this case can be written in the following form,
(16)  
(17)  
(18)  
(19) 
where the prime denotes as previously differentiation with respect to the foldings number , and quantifies the interaction between dark energy and dark matter, and the term corresponds to the bulk viscous pressure of the dark matter fluid.
We will assume that the bulk viscous coefficient has the following form ALUL ; Brevik:2005bj ; Brevik:2007ig ,
(20) 
The motivation for using this kind of ansatz, comes from astronomical estimates on the relation between and , which was studied in Ref. Normann:2016jns , where a general form of the relation was assumed, and it was of the form . The choice we used in Eq. (20) yields a good fit between astronomical constraints and the fluid approach.
In this case we introduce the following set of dimensionless variables,
(21)  
(22)  
(23) 
where only and are independent dynamical variables. By using the set of dimensionless variables, the dynamical equations (16)(19) take the form^{3}^{3}3Note that the constraint (16) and the equation for are already took into account in this system.:
(24)  
(25) 
and by using the EoS we used in the previous section, namely, , we obtain,
(26)  
(27) 
We can easily find the stationary points for the above dynamical system, by multiplying (24) by , (25) by and by combining the resulting equations we find,
(28) 
The above equation indicates that there exist at most three stationary points, and it is clear that even in the most general case, the stationary points may be found analytically, and these are equal to,
(29) 
(30) 
Since there are a many free parameters, we investigate some particular cases, which are interesting from a physical point of view. We start off with the case , in which case the first fixed point takes the form,
(31) 
and the corresponding eigenvalues are,
(32)  
(33) 
Accordingly, the second fixed point is,
(34) 
and the corresponding eigenvalues are,
(35)  
(36) 
so the first point always unstable, whereas the second one may be stable, depending on the choice of the free parameters. However, if the first point lies in the physically allowed region, the second fixed point is rendered always unstable. Typical phase portrait with the two fixed points in the physical region, are presented in Fig.1.
Now let us consider another physically interesting case, for which . In this case, there exist three fixed points which are,
(37)  
(38)  
(39) 
A particularly interesting subcase of the above is if we further choose, , then the fixed points become,
(40)  
(41)  
(42) 
We can see that two fixed points are always unstable, whereas one of the three, the first or the second one depending on the sign of the parameter , is stable.
In Fig. 2 we plotted the phase portrait for the case , while in Fig. 3 we plotted the phase portrait for , and finally in Fig. 4 we can see the phase portrait for . As it can be seen in all figures, two of the three fixed points are unstable and one of the three is stable. Furthermore, it can be seen that by using Eq. (30), it is possible to find the case (by appropriately choosing the parameters) for which there will be some stationary point with fixed relation . This task may be solved numerically but we refrain from going into details on this.
Iv Generalized Form of Cosmological Fluids
In this section we extend the cases we presented in section II to include generalized form of the EoS. We consider the simplest scenario for which the Universe is filled with and a perfect fluid with nontrivial EoS NO1 . In this case, the dynamical system takes the following form,
(43)  
(44)  
(45)  
(46) 
By using the foldings number as independent variable and also by introducing the dimensionless variables,
the dynamical system can be cast in the following form,
(47)  
(48) 
where according to the definition, and . Moreover we can see from the system (47)(48) that the fixed point always exists, except for some very special choices of functions and .
It is interesting to note that actually due to the constraint equation (43), in the case at hand, there is only one independent variable, but by introducing the variable and by taking into account the equation (44), allows us to obtain more information about the evolution of the dynamical system. For instance, one of the most hard tasks in such kind of dynamical systems analysis, is to interpret correctly the physical meaning of the stationary points. This approach was firstly proposed in Ref. BH . Let us introduce an additional parameter, which will be very helpful for this interpretation, which is the effective equation of state, which we denote as , and it is defined as follows:
(49) 
The parameter can be expressed in terms of the dimensionless variables (43)(44) as follows,
(50) 
By specifying the EoS, it is possible to obtain various physically interesting evolution scenarios, so in the rest of this section we shall specify the EoS and we study in detail the dynamical evolution stemming from the choice of EoS.
iv.1 A Simplified Form of EoS
Let us discuss the simplest case of EoS, which is , which in turn implies , . It is easy to verify that in this case, the equations (47)(48) take the following form,
(51)  
(52) 
Thus we have the following stationary points for the dynamical system above,

First fixed point

Second fixed point .
For the first fixed point we need to note that, this point lies in the physical region only if . The corresponding eigenvalues are , and . Using Eq. (50) we find that at this point we have .
With regard to the second fixed point, the eigenvalues are , and . Correspondingly we find that at this point . Thus we can see that the first fixed point always unstable and also that the second fixed point may be stable only if one (or both) of parameters , are strictly negative. For instance if we require that the first point lies in the physical region, we find that the second point is stable for . In Figs. (56) we plot the typical behavior of the phase trajectories.
Particularly, Fig. 5 corresponds to , and . The left fixed point corresponds to the effective EoS parameter and for the right fixed point, we have . Clearly the left fixed point represents radiation, while the right one corresponds to some phantom evolution. In Fig. 6, the phase portrait corresponds to the following choices for the parameters, , . The left point corresponds to an effective EoS parameter , while the right point corresponds to , which describes a form of collisional matter Oikonomou:2014lua .
iv.2 More Complicated Forms of the EoS
Now let us study more complicated forms of the EoS, and we choose it to be of the form . This EoS is known to lead the cosmological system to finitetime singularities, as this was demonstrated in Ref. NO1 . For this EoS, the equations (47)(48) take the form,
(53)  
(54) 
Correspondingly, the effective EoS in this case reads,
(55) 
It is clear that in the most general case, this system may be solved only numerically, so let study some appropriately chosen cases which admit analytical solutions. Consider first the case for which , , in which case, the dynamical system (53)(54) takes the following form,
(56)  
(57) 
In this case there are two stationary points: the first is , ^{4}^{4}4Note here that in all these three cases existence of the point , is not an obvious solution, but it can be confirmed by numerical investigations. and the second is , . For the first point, we have ^{5}^{5}5Infinite values of looks like the Ruzmaikin solution at . and for the second one we have .
We can see that the second point corresponds to some new nontrivial de Sitter state with and . The eigenvalues of the second fixed point are , and . The typical behavior of the phase trajectories corresponding to this case, can be found in Fig.7, for , and . As it can be seen, there exist trajectories which start from the first fixed point , (note that zero values of correspond to infinite values of , which indicates a singularity) and end up to the second fixed point, with nonsingular and nonzero values of . In effect, the second fixed point may be interpreted as a latetime acceleration de Sitter point of the cosmological dynamical system.
Now let us consider the case for which , and , in which case, the dynamical system (53)(54) takes the form,
(58)  
(59) 
In this case there are two stationary points, namely, , and also , and the situation is very similar to the previous case. For the first fixed point we have and for the second one . The eigenvalues of the second point are , and . Note also that for both these cases, and must have opposite sign, in order for the second fixed point to lie in the physical region. We also need to note that for this case there are additional fixed points, which are difficult to find analytically, but the most physically interesting cases of fixed points are the ones we just presented. The phase space behavior corresponding to this case can be found in Fig. 8, for , , , and as it can be seen, the behavior of the trajectories is similar to the previous case.
Concluding this section, let us briefly discuss another choice of parameters for which it is possible to find analytically the fixed points, and this occurs for the choice , , in which case, the dynamical system (53)(54) takes the form,
(60)  
(61) 
In this case, the stationary points are the following, , and , . For the first fixed point we have and for the second one
V Dark Matter Interacting with Dark Energy: Some nontrivial Models
Let us now discuss some nontrivial models which describe interactions between the dark matter and dark energy fluids. The dynamical system in this case may be written in the following form,
(62)  
(63)  
(64)  
(65)  
(66) 
where we modified the interaction between dark energy and dark matter by using the multiplier ^{6}^{6}6Note that case corresponds to the standard interaction we presented in a previous section., and also the term , which corresponds to the bulk viscous pressure of the dark matter fluid. We will assume that the bulk viscous coefficient has the following form,
(67) 
We define the set of dimensionless variables as follows,
(68)  
(69)  
(70)  
(71) 
where only and are dynamical independent variables. In the new variables system, Eqs. (62)(66) take the form^{7}^{7}7Note that the constraint (62) and the equation for have already been taken into account for this system.:
(72)  
(73)  
(74) 
and using that the EoS has the form we used in the previous section, namely, , , we obtain,
(75)  
(76)  
(77) 
The effective EoS in this case (for arbitrary functions and ) reads,
(78) 
which in the case that , , becomes,
(79) 
Now we consider some special cases, and we start with the case . This case corresponds to the usual interaction term. Since the first two equations are identical to (26)(27), we have the solutions (29), (30), where we need to add .
Also, since the first two equations do not depend on , the first and second eigenvalues will be totally identical to the ones we obtained in section III, and one additional eigenvalue appears for every stationary point, which is,
(80) 
Also note that the stability or instability of the point with respect to this additional coordinate , implies stability in the past or in the future correspondingly. It mean that a stable with respect to coordinates , point which have will be stable in the past if it is stable with respect to coordinate , or equivalently will be stable for infinite values of . In addition, it will stable in the future, if it is unstable with respect to coordinate , or equivalently it will be stable for small (zero) values of .
The effective EoS (79) for the fixed point (29) is equal to for any values of parameters, which clearly describes a radiation dominated era. However, the effective EoS for the fixed points (30), has a more complicate structure, which is given below,
(81) 
Let us now further analyze the case at hand, by specifying the values of the free parameters, so we start with the parameter , and assume for the moment that . In this case the fixed point (31) has an additional eigenvalue and the corresponding effective EoS becomes , which describes radiation. Moreover, the fixed point (34) has the additional eigenvalue and the corresponding effective EoS for this point is . Thus in Fig. 1 the left fixed point corresponds to radiation with , and the right point has , and for both the fixed points, has infinite values. In Table 1 we have gathered all the fixed points which correspond to the case , .
1a  

1b  

As it can be seen in Table 1, the fixed point may describe latetime acceleration. Indeed, if we set , with , we obtain , and and as we already noted, this means that this point is stable in the future (for small values of ). Moreover, by changing the parameter , we can provide any interesting relation between and . For instance if put we obtain for this point .
Let us discuss some alternative choices for the parameters, so consider the case , in which case the fixed point (37) has an additional eigenvalue, which we denote , and it is equal to, and the corresponding effective EoS is . In addition, the fixed point (38) has the additional eigenvalue with , which describes a matter dominated state. Finally, the fixed point (39) has with . In Table 2 we have gathered all the fixed points for the case .
2a  

2b  
2c  

As it can be seen in Table 2, the fixed point is always unstable, if it lies in the physical region, that is, when . The fixed point may be stable in the past and in the future, depending on the values of the parameters. Finally, the fixed point may be stable only if . So in this case, no fixed point describes latetime acceleration, however, the fixed points and may describe a radiation dominated era and matter dominated era respectively.
Consider now the case , , in which case, the fixed point (40) has the additional eigenvalue and the corresponding effective EoS is . Accordingly, the fixed point (41) has with and finally the fixed point (42) has with . Thus in Fig.2 we have for the fixed point , for the fixed point and for the fixed point . Correspondingly, in Fig. 3 we have for the fixed point , for the fixed point and for the fixed point . Note that all the aforementioned fixed points correspond to states which have infinite values of .
3a 
