Phase-space analysis of the cosmological 3-fluid problem: Families of attractors and repellers
We perform a phase-space analysis of the cosmological 3-fluid problem consisting of a barotropic fluid with an equation-of-state parameter , a pressureless dark matter fluid, plus a scalar field (representing dark energy) coupled to exponential potential . Besides the potential-kinetic-scaling solutions, which are not the unique late-time attractors whenever they exist for , we derive new attractors where both dark energy and dark matter coexist and the final density is shared in a way independent of the value of . The case of a pressureless barotropic fluid () has a one-parameter family of attractors where all components coexist. New one-parameter families of matter-dark matter saddle points and kinetic-matter repellers exist. We investigate the stability of the ten critical points by linearization and/or Lyapunov’s Theorems and a variant of the theorems formulated in this paper. A solution with two transient periods of acceleration and two transient periods of deceleration is derived.
Observations have ever confirmed the so-called transient period of acceleration (TPA) , in this regard any realistic cosmological model should include component(s) with negative pressure. In this work we consider a spatially flat Friedmann-Roberston-Walker (FRW) cosmology where the content of the universe has two components with negative pressure and a pressureless component. This is the so-called 3-fluid problem consisting of a barotropic fluid with equation of state where , a pressureless dark matter (DM) density and a dark-energy-scalar-field (DE) coupled to exponential potential with equation of state , and . We assume that the three components are noninteracting. In this model, the barotropic fluid represents visible matter if [radiation if or ordinary matter (baryons) if ]. For short, we will call this component matter (). In this cosmological 3-fluid model ; however, from a physical point of view, there is no compelling reason to constrain the values of to the interval , in this regard it has been established that the teleparallel DE  cosmological model allows for .
Existing analytical methods  have failed  to produce exact solutions to the 3-fluid problem. Apart from some trivial solutions (power law inflationary solutions where the scale factor of the universe evolves as for ), no exact solution to the 3-fluid problem seems to exist to our knowledge.
In Ref.  we restricted ourselves to ordinary matter, where , and to positive exponential potentials and resorted to numerical approach by which we derived new solutions to the 3-fluid problem. The solutions where classified hyperbolic and trigonometric according to the value of , this extends the classification made for the solutions to the cosmological 2-fluid problem  (consisting of a barotropic fluid plus a DE-scalar-field coupled to exponential potential), which were first derived in . For the whole range of , we were able to construct solutions with one TPA where the universe undergoes the deceleration-TPA-eternal deceleration expansions. No solutions with two or more TPA’s were found. However, as we shall see later, solutions with one TPA and a late-time eternal acceleration expansion do exist for the range (which were not reported in Ref. ), where the universe undergoes the deceleration-TPA-TPD-eternal acceleration expansions (TPD: For transient period of deceleration). We shall also derive solutions with two TPA’s and two TPD’s for approaching from below 2/3. Solutions with many TPA’s and TPD’s may exist too.
Phase-plane and -space analyses of autonomous differential equations lead to specific solutions that may provide the late-time attractors or the early-time repellers. Both type of solutions are interesting and provide rich insight into the evolution of the universe. Prior to the determination of the exact solutions to the cosmological 2-fluid problem - , phase-plane analyses of the 2-fluid problem were performed -  (for a general procedure see [16, 17]) and led to the discovery of potential-kinetic-scaling solutions, which are the unique late-time attractors whenever they exist for .
We shall carry a phase-space analysis of the autonomous differential equations governing the dynamics of the 3-fluid problem. Among the conclusions we reach (1) the stability of the scalar field dominated solution for , (2) the stability of the potential-kinetic-matter scaling solution for , which are quantitatively different from the corresponding results for the 2-fluid problem , and the existence of (3) new attractors (the potential-kinetic-DM scaling solution and the potential-kinetic-matter-DM scaling solution) and (4) new repellers and saddle points. In Sect. 2 we derive the autonomous differential equations of the 3-fluid problem and some other useful formulas. In Sect. 3 we discuss and extend the methods used for the stability analysis. In Sect. 4 we derive the critical points, investigate their stability and and their cosmologic implications, and construct numerically solutions with two TPA’s and two TPD’s as well as solutions with one TPA and a late-time eternal acceleration expansion. In Sect. 5 we discuss which physical scenarios are well fitted by the 3-fluid problem. We conclude in Sect. 6.
2 Autonomous differential equations of the 3-fluid problem
The three components being noninteracting each fluid satisfies a conservation equation of the form where is the corresponding stress-energy tensor (). Keeping the relevant conservation equation for our analysis (corresponding to ) and using a similar notation as in , the dynamics of the three fluids in a spatially flat FRW universe, with a scale factor and a Hubble parameter , are governed by the very Eqs. (1) to (3) of  upon slightly modifying the first equation by adding the contribution attributable to DM
where . These equations are constrained by
From now on we consider only positive potentials where . The dimensionless variables
where the variable is solved by
The equation governing the motion of is independent of
In general (for all ), the constraint (9) restricted the motion to within the unit solid 2-sphere of center at the origin: . A necessary formula for the stability analysis is readily derived upon combining (6), (7) and (8) setting
Thus for , the motion happens on an ellipsoid of revolution around the axis in the phase space. The ellipsoid, which is inside the 2-sphere , does not contain all the trajectories for ; as we shall see, there are some equilibrium points inside and outside the ellipsoid; there are also other trajectories corresponding to () and to ().
The relative densities are defined by , , and obey the conservation equation . Other relevant quantities are the parameter which is constrained by and the deceleration parameter which is, by the field equations, the same as leading to
3 The critical points (CP’s) – Lyapunov’s Stability and Instability Theorems (LST and LIT)
The CP’s are the equilibrium points () in the phase space obtained upon solving the nonlinear algebraic equations , , , and . To determine the stability of the CP’s we proceed to the linearization of the system (6)- (8) setting , , () where the new variables () still obey the full nonlinear system (6)- (8). Upon linearization, the system (6)- (8) is brought to the matrix form:
where , , .
The test for stability of almost linear systems  states that [18, 20] if (1) all the eigenvalues of the nonsingular matrix () have negative real parts, then the CP is asymptotically stable but if (2) any eigenvalue has positive real part, then the CP is unstable. If some eigenvalues are zero () or have zero real parts (and still ), we will employ appropriate arguments (among which Lyapunov’s Theorems, LST - , and LIT ) for the determination of the stability of the corresponding CP as the above-mentioned test is no longer valid [18, 20]. Mathematically speaking, we shall not make use of the notion of saddle points for a saddle CP is generically unstable. However, physically, we shall distinguish between a repeller and a saddle point.
LST assumes the existence of a continuously differentiable function that is positive definite in a neighborhood of the CP and has an isolated minimum at the CP, which is the origin in the new coordinates : . If further the derivative of along a solution curve, with and , is negative definite on (except at the origin): , then the CP is asymptotically stable. We are not concerned with the case where the CP is stable - .
LIT  for 2-dimensional systems generalizes to higher dimensional systems in a straightforward way. It consists in finding a function that is continues on a domain containing the CP, which is assumed to be isolated. The Theorem assumes that and that there is at least a point in each disc in , of center CP, where . If is positive definite on (except at the origin): , then the CP is unstable.
The intuition behind LIT is as follows. Assume the above conditions are satisfied. In any disc in select a solution curve that starts at
In LIT, need not be zero at the CP since one can add any positive or negative constant to without modifying the condition of stability, and need not be positive
In cosmology both LST and LIT are very useful. One is interested to stable CP’s or attractors where different solution curves end up at regardless of their initial conditions. One is also interested to unstable solutions or repellers which represent starting or intermediate events.
The main difficulty in applying LST and LIT is that there is no method how to find . There are, however, some directions for that purpose . However, the construction of may be greatly simplified relying on the assumptions of LST and LIT concerning . This will be illustrated in the following discussion.
4 The critical points (CP’s) – Stability analysis – Cosmological implications
We have counted ten CP’s labeled from to . In the following we will provide the values of the CP’s in the form , determine their stability conditions and discuss their cosmological implications. The stability conditions are determined in terms of intervals of and/or and are derived using the “Hessian” test for stability as well as both LST and LIT. Particularly, the LST and LIT are employed to determine the stability conditions at the endpoints of the intervals of and/or , a task generally overlooked, skipped or difficult without use of the theorems [3, 13, 14, 17, 21, 22]. We summarize our results in Table 1. As was mentioned earlier, no distinction is made in the text between a saddle point and an unstable CP; This distinction appears only in Table 1.
Following the classification made for the analytic solutions to the 2-fluid problem , the solutions with are called hyperbolic and those with are called trigonometric. Due to different conventions, the value of used in , , is related to the value of used in this work by .
|& : SN|
|& : SN|
|& : SS|
|& : AS|
|& : SP|
|& : Un|
|& : SN|
|& : SS||0||1|
|& : AS|
|& : SP|
For , has at least one positive eigenvalue: . This CP is unstable. For , is singular. However, it is straightforward to show that in this case the CP is also unstable. This is achieved upon linearizating (6) and (7) in which case we obtain the eigenvalues of opposite signs.
Cosmologically, the only solution curves that may reach emanate from with and ; otherwise, some solution curves (only those emanating from ) may just get close to it, but do not cross it, as shown in Fig. 1. At this CP, all densities vanish for a dominant DM component , is an indeterminate, and the universe undergoes a decelerated expansion with .
The matrix has the eigenvalues where for , respectively, so they are generically unstable. They are also unstable in the special case where is singular
This CP exits for (the case leads to the previous case). The eigenvalues of are: . In the case , the CP is asymptotically stable for and unstable for [this includes the cases ( and ) and ( and )]. If , it is asymptotically stable in the cases ( and ) and ( and ) upon linearizing [(7) and (8)] and [(6) and (7)], respectively.
There remains the case and where we expect the CP [in this case ] to be asymptotically stable. We apply LST and select of the form: , which is positive definite if and . The CP is an isolated minimum of with . The directional derivative along the solution curves, with and , is evaluated using the r.h.s’s of (6), (7) and (8) after converting to new coordinates ():
which is negative definite in the vicinity of the CP. We choose to be any neighborhood of the CP (including the CP), where , and not including other points of the surface . This way we make negative definite
For , this CP is an attractor with a dominant scalar field component , and a decelerated expansion .
From the set of the eigenvalues of , , the CP is generically unstable if .
Now, consider the case where is singular (). In the special case , the CP is unstable for the linearization of (7) and (8) leads to , . The same conclusion is achieved from . For the CP is also unstable by LIT or upon linearizing (6) and (7) which results in the eigenvalues of opposite signs. The case is stable since we have .
Thus, is a matter dominant attractor () if , a saddle point if , and a repeller if . With the parameter remains undetermined, the state of the universe at undergoes a decelerated expansion if or an accelerated expansion if . This is a novel point because one may have a TPA without necessary having (at the same time) a minimum kinetic energy and a maximum potential energy as in the case of the 2-fluid problem [8, 6]. In fact, at both kinetic and potential energies are zero. Thus if, for , a solution curve approaches the saddle point then deviates to a CP (this would be the CP ), the TPA there (at ) may last longer than the TPA occurring away from saddle points. This is because a saddle point behaves partly as an attractor and partly as a repeller. This is in fact the case in Fig. 3 that is a plot, for and , of twice the deceleration parameter, , and the kinetic and potential relative densities, (dashed line) and (continuous line). The parameter crosses the axis at: , , , and . This solution has thus two TPA’s and two TPD’s: The first and second TPA’s are observed in the intervals and , respectively, and the first and second TPD’s are observed in the intervals and , respectively. The first TPA starts at , which is the moment where and [, , ], that is the corresponding point on the solution curve is near . The graph of continues to oscillate for below the line , this is a sign that solutions with many TPA’s and TPD’s may exist if careful choice of the parameters is carried out. Fig. 4 is a similar plot with different inputs and . It is a solution with one TPA and one TPD which depicts a case with a TPA starting at the moment where is minimum and is maximum.
, , .
This unstable CP generalizes in that may assume any value between 0 and 1; it also generalizes .
As is singular, it is not possible to draw any conclusion concerning stability by linearization of the system (6)- (8) for this CP. With , Eq. (12) becomes . A solution curve that starts near the CP has (the CP is on the sphere and all solution curves are inside the sphere). Since , we have and thus the solution curve moves in the direction of decreasing and never returns back to the CP where , which is then unstable. This gives another application of a variant of LIT (see footnote 4).
Since is not constrained, is a new one-parameter family of kinetic-matter repellers. With a stiff equation of state , the initial density is shared between the barotropic fluid and DE, , and (decelerated expansion). Solution curves starting from , which is represented by a semicircle in Fig. 5, reach .
The corresponding solution for the 2-fluid problem  is a potential-kinetic scaling solution the stability of which does not depend on the value of . For the 3-fluid problem we rather have a potential-kinetic-matter scaling solution, its stability depends on as we shall see later soon.
The eigenvalues depend on both (): where we define . This CP exists for only. If , it is asymptotically stable for and . Furthermore, this CP is (a) a stable node for for all
If , this CP is asymptotically stable for and since the linearization of (6) and (7) provides two negative eigenvalues: with (a stable node for and a stable spiral for ). For the remaining cases where is singular ( or ), the CP is unstable for and since near it we establish: . For , the eigenvalues which result from the linearization of (6) and (7) are proportional to and , ensuring asymptotic stability for and instability for . For the case and , where , we recover the CP which has been shown to be asymptotically stable.
The fact that —to ensure asymptotic stability of the CP— results in , while in the case of the 2-fluid problem , still given by the same formula, may have both signs. and depend on both (): , . With , the state of the universe approaching this stable point may undergo a decelerated expansion if or an accelerated expansion if . is a saddle point for .
Here again the eigenvalues depend on both parameters (): . This CP exists for only and it is asymptotically stable for and . The CP is (a) a stable node if (with ) or (b) a stable spiral if (with ). If , it is asymptotically stable for and as the linearization of (6) and (7) leads to the same eigenvalues . For the case and we recover the CP which has been shown to be asymptotically stable.
The relevant parameters are which depends only on , , , and .
For the 2-fluid problem, is the unique attractor for (for all ) . But depends on , this means that the end-behavior of the solution depends on the nature of the barotropic fluid. We have seen that, for the 3-fluid problem, is no longer stable for but is, which is a new attractor and does not depend on . Thus, no matter the barotropic fluid equation of state is, the universe’s evolution ends up at the same state provided . As we shall se below, for (), there is a line (a one-parameter family) of attractors all represented by the CP .
, , .
This unstable CP generalizes and in that may assume any value constrained by , with and remains undetermined. Since is not constrained, is a new one-parameter family of saddle points where only matter and DM are the nonvanishing components. is represented by a vertical line in the phase diagram, which is the line of Fig. 2.
Since is a free parameter, is a new one-parameter family of attractors in which all components coexist: It is a potential-kinetic-matter-DM scaling solution where , , , and . and are both arbitrary and smaller than . According to the last and most accurate observations , (corresponding to ), thus .
5 Fitting the 3-fluid model
As stated in the Introduction, any realistic model should include at least one component with a negative pressure to account for a TPA . For that purpose, many theoretical models have been suggested the simplest of which is the so-called CDM where the component with negative pressure is a vacuum energy (the cosmological constant). This model results in a constant DE equation of state, , and the coincidence problem. The next generation of models, which consider two noninteracting fluids [7, 8, 13, 14, 24], have introduced a scalar field (quintessence) to generalize the CDM model. They have emerged to tackle the coincidence problem and to provide a variable DE equation of state. Models where ordinary matter and DE interact have also emerged [17, 25].
While there is no observational evidence of the existence of any interaction between the two dark components, some authors, however, arguing that the amounts of DE and DM are comparable at the present age of the universe, have anticipated that and formulated 2- and 3-fluid problems with DE-DM [21, 22, 26] or DE-matter-radiation  interaction terms. Of course, these models reduce to the 3-fluid problem with no interaction terms if appropriate constraints are further imposed. In Ref. , the authors considered a DE-DM interaction with baryons uncoupled and radiation redshifted away or neglected. Thus, their model applies to the epoch beyond the matter-radiation decoupling, which corresponds to a redshift , and it reduces to the 3-fluid problem upon setting the DE-DM interaction coupling constant  (with this constraint, the model corresponds to ours with , as we shall see below in this section). In contrast, the authors of Ref. , considering again a flat FRW, included radiation in, but dropped baryons from, their DE-DM-radiation model to allow for a deeper investigation of the universe’s dynamics during the radiation dominant era. They considered a non-minimally and non-constant coupling, inspired from Scalar Tensor Theories (STT), the value of which depends on the trace of the energy-momentum tensor of the background component. Since radiation is traceless, it remains decoupled from the DE-DM system they investigated. The authors presented a general procedure for dynamical analysis of the STT inspired DE-DM interactions. Their model reduces to the 3-fluid problem if their DE-DM interaction coupling function  and their DM parameter (with these constraints, the model corresponds to ours with , as we shall see below in this section).
In most of the above-mentioned models, the potential functions associated with DE and/or the interaction terms have been derived or chosen, relying partly on some physical assumptions, so that the problems remain analytically tractable (even though no nontrivial exact analytic solutions have been found so far), among which we find the 3-fluid problem we are considering here with no interaction terms. However, by neglecting all types of interactions, particularly that of visible matter, we restrict the application of the 3-fluid model to beyond the epoch of matter-radiation decoupling ( ). Thus, for , the model fits well the following three physical scenarios based solely on the value of .
. In this case the components of the universe are regrouped in a way that the barotropic fluid represents radiation, the DM and baryons together make up the pressureless component with a total relative density