Nonlinear bulk viscosity and the stability of accelerated expansion in FRW spacetime

G. Acquaviva∗ and A. Beesham†

Department of Mathematical Sciences, University of ZululandPrivate Bag X1001, Kwa-Dlangezwa 3886, South Africa

(Dated: June 3, 2014)

In the context of dark energy solutions, we consider a Friedmann-Robertson-Walker spacetimefilled with a non-interacting mixture of dust and a viscous fluid, whose bulk viscosity is governedby the nonlinear model proposed in [R. Maartens and V. Mendez, Phys. Rev. D 55, 4 (1997)1937]. Through a phase space analysis of the equivalent dynamical system, existence and stabilityof critical solutions are established and the respective scale factors are computed. The results pointtowards the possibility of describing the current accelerated expansion of the Universe by means ofthe abovementioned nonlinear model for viscosity.

PACS numbers: 98.80.-k, 95.36.+x

Introduction

The discovery [1, 2] and confirmation of the presentaccelerated expansion of our Universe has opened up oneof the main modern challenges in theoretical cosmologytogether with the early inflationary mechanism: theidentification of the energy content responsible for suchbehaviour [3]. In fact, assuming the validity of Einstein’stheory of gravitation at supergalactic scales (i.e., disre-garding here modifications in the geometric sector), anaccelerated expansion can be obtained through a varietyof energy-momentum tensor choices, among which werecall the introduction of scalar fields with suitablepotentials, such as quintessence [4] or phantom fields[5]. However, these kinds of models are often subject tosevere fine tuning problems and, even though the finetuning argument in itself is not sufficient to rule out suchpossibilities, some kind of justification is expected to beprovided from particle physics. Another approach whichallows one to obtain an accelerated expansion consists ofmodifying the simple barotropic equation of state (EoS)of the fluid in favour of more exotic forms, such as theChaplygin gas model [6] and its generalizations [7]. Fora recent review of the efforts aimed at resolving thisdark energy problem, see [11] and references therein.

In this work we focus on another kind of approachwhich, to some extent, tends to minimize the appeal toexotic forms of matter: the introduction of dissipativeprocesses through the modeling of viscous effects inordinary fluids. The only dissipative contribution thatplays a role in homogeneous and isotropic scenarios isthe bulk viscous pressure Π: the effective pressure of thecosmic fluid can be then split as peff = p + Π, wherep is the equilibrium pressure. During expansion thedissipative term, contributing negatively to the effectivepressure, causes a reduction of the kinetic pressure of

∗Electronic address: acquavivag@unizulu.ac.za†Electronic address: beeshama@unizulu.ac.za

the fluid. Such an effect has already been shown to beable to produce accelerated growth of the scale factorin the context of Israel-Stewart’s linear causal theory[12, 13], its truncated causal form or the non-causalEckart theory (see [14] and references therein). On theother hand, some authors (such as [16]) pointed outthat the linear theory is not able to produce cosmicevolution in accordance with observations. The mainproblem with the linear Israel-Stewart theory is thatit relies on the assumption of small deviations fromthermodynamical equilibrium (|Π| < p), an assumptionthat is not always expected to hold, in particular whenaccelerated expansion takes place, like in early inflation,and we are dealing with dark energy. For this reason,several authors have tried to extend the theory ofviscosity away from equilibrium [17, 18], introducing forexample nonlinear effects in the dynamics of the viscouspressure Π.

In ref [20] a single viscous fluid was analysed inthe linear causal theory, which was shown to lead tostable accelerated expansion. In that paper it was alsosuggested that it would be interesting to introduceanother fluid without viscosity and to study the effectsof both. In the following we take into account thenonlinear model proposed in [19] and study the phasespace of the theory in FRW spacetime in the presenceof both dust and a generic viscous fluid governed by anEoS parameter γ. As we shall see later in our case, theviscous pressure in the case of accelerated expansion ofthe scale factor is larger than the equilibrium pressure,thus justifying the need for a nonlinear extension. It hasto be kept in mind also that the condition underlying thehydrodynamic description requires the mean interactiontime tc between fluid particles to be less than thecosmological time scale, i.e. tc θ−1, where θ is theexpansion scalar. In the case of accelerated expansionwe have to phenomenologically assume the existence ofa long-range interaction, perhaps mediated by gravity,which ensures such a description. In the context of adynamical system description, both local (critical pointsand their stability) and global features (characteristic

2

trajectories and basins of attraction) are analyzed. Oneof the aims of our work is to see, under fairly generalconditions, whether one could get accelerated expansionat late times. Secondly, in [16] it was found in thecontext of Eckart theory under reasonable assumptionsthat, whilst obtaining late time acceleration, one cannotobtain critical points corresponding to radiation andmatter dominated eras, which are necessary in anyreasonable model of the Universe. So a further motiva-tion for this work is to see if, in addition to late timeacceleration, one could also obtain radiation and matterdominated transient stages.

There have been several authors who have studied vis-cous fluids as a source of dark energy, mostly using Eckarttheory, e.g. [21]. Even in this non-causal theory it is pos-sible [22] to construct a reasonable cosmological scenariostarting with an initial singularity, followed by decelera-tion at early times and then a smooth transition to anacceleration at late times. Fabris et al. [23] investigateda single fluid viscous model in Eckart theory, finding thatit is possible to have a matter dominated era followed bylate time acceleration. The evolution of density pertur-bations was studied and it was found that it is possible toreproduce the observed power spectrum. This was laterextended [24] by a dynamical system analysis of a viscousfluid plus dust, finding similar conclusions and good fitwith observations. In fact, the idea of bulk viscosity giv-ing rise to late time acceleration without a cosmologicalconstant was known [25–28] even before the observationaldiscovery via supernovae. Gagnon and Lesgourgues [29]studied a bulk viscous model and showed that linear per-turbations are well behaved for reasonable values of theparameters of the model, and that no apparent fine tun-ing is needed. Several authors [30, 31] found that cos-mology based on Eckart theory is indistinguishable fromthe usual ΛCDM model. Piattella et al. [32] also foundthat for reasonable parameters, the truncated causal the-ory is in close agreement with the usual ΛCDM model.By studying observational constraints from supernovae,baryon acoustic oscillations and the cosmic microwavebackground, Velten et al. [33] have motivated for thepresence of bulk viscosity in some form or the other.

I. RELEVANT EQUATIONS

Given the Friedmann-Robertson-Walker (FRW) met-ric

ds2 = −dt2 + a2(t)(dr2 + r2 dΩ2

),

the expansion scalar is defined as θ = 3a/a. We let8πG = 1. We consider the Universe to be filled with anon-interacting mixture of dust, with energy density ρd(pd = 0), and a viscous fluid with energy density ρv andeffective pressure p = pv(ρv) + Π. Here, pv is the equi-librium part of the viscous pressure, whereas Π is thenon-equilibrium part, which obeys an evolution equation

to be given later. Then, from Einstein’s field equations(EFEs), we obtain the Raychaudhuri equation

θ = −1

3θ2 − 1

2(ρd + ρv + 3pv + 3Π) , (1)

and the Friedmann constraint equation

ρd + ρv −1

3θ2 = 0 (2)

Conservation of energy-momentum provides the otherrelevant evolution equations,

ρv = −θ (ρv + pv + Π) (3a)

ρd = −θ ρd . (3b)

We consider a barotropic EoS for the viscous fluid:

pv = (γ − 1) ρv (4)

Using the EoS eq.(4) and the constraint eq.(2), eq.(1)and eq.(3a) become

θ = −1

2θ2 − 3

2

[(γ − 1) ρv + Π

](5a)

ρv = −θ (γ ρv + Π) (5b)

With regard to the evolution of the viscous pressure vari-able Π, we make reference to the nonlinear model pro-posed in [19]:

τ Π = −ζ θ −Π

(1 + Π

τ∗ζ

)−1

− 1

2Π τ

[θ +

τ

τ− ζ

ζ− T

T

](6)

This equation1 has been derived by assuming a nonlinearrelationship between the thermodynamic “force”χ andthe thermodynamic “flux” Π of the form

Π = − ζ χ

1 + τ∗ χ

where ζ is the bulk viscosity and τ∗ is the characteristictime for nonlinear effects (the linear Israel-Stewart the-ory is recovered for τ∗ = 0). The quantities τ and T ineq.(6) are the linear relaxational time and the local equi-librium temperature, respectively. The quantity in roundbrackets is strictly positive in order to ensure positivityand non-divergence of the entropy production rate. Wewill consider the following relations for the parametersinvolved (see [19]):

1 We refer to [19] for more details on the derivation and motiva-tions.

3

• bulk viscosity: ζ = ζ0 θ, with ζ0 > 0

• linear relaxational time: τ = ζ/(γ v2 ρv)

• nonlinear characteristic time: τ∗ = k2 τ

• temperature (barotropic fluid): T = T0 ρ(γ−1)/γ ,

from the integrability condition of the Gibbs rela-tion

Here v is the dissipative contribution to the speed ofsound V , whose complete expression is V 2 = c2s + v2.

From the condition V 2 ≤ 1 and from the definition ofadiabatic sound speed c2s ≡ δpv/δρv = γ−1, we have thebound

v2 ≤ 2− γ, with 1 ≤ γ ≤ 2 (7)

The explicit form of the evolution equation is then givenby

Π = −γ v2 ρv θ −γ v2

ζ0

Π ρvθ

(1 +

k2

γ v2

Π

ρv

)−1

− 1

2Π

[θ −

(2γ − 1

γ

)ρvρv

](8)

II. DYNAMICAL SYSTEM

In order to reduce the dynamical equations to an au-tonomous system, we define the expansion-normalizedvariables Ω = 3ρv/θ

2 and Π = 3Π/θ2, together with thenew time variable dt/dτ = 3/θ, whose associated deriva-tive will be denoted by a prime. The system eq.(5) interms of the normalized variables is

θ′

θ= −3

2

[1 + (γ − 1) Ω + Π

](9a)

3ρ′vθ2

= −3(γ Ω + Π

). (9b)

From the definition of Ω, we obtain

Ω′ =3 ρ′vθ2− 2 Ω

θ′

θ. (10)

Substituting eqs.(9) in this last equation, we get the evo-lution equation for Ω

Ω′ = 3 (Ω− 1)[(γ − 1) Ω + Π

](11)

We now introduce the evolution equation for Π. Differ-entiating Π with respect to τ and making use of eq.(9a),we get

Π′ = −3γ v2 Ω

1 +Π

3 ζ0

(1 +

k2

γ v2

Π

Ω

)−1− 3

Π2

Ω

[2γ − 1

2γ− Ω

]− 3 Π(γ − 1) (1− Ω) (12)

Thus the dynamical system is described by eqs.(11) and(12). The search for critical points amounts to looking

for values Ωc , Πc which solve the system Ω′ = Π′ = 0.

III. PHASE SPACE ANALYSIS

First of all, we stress that positivity of entropy pro-duction rate requires a restriction of the phase space tothe region

Π > −γ v2

k2Ω (13)

Note that if k2 v2, this condition narrows the possiblenegative values of Π toward zero. For finite k, in the limitv → 0, only positive values of bulk pressure are allowed.On the contrary, for k2 v2 the bounds on bulk pressureare less restrictive. A good tradeoff would be to considerk2 . v2, which, through the fact that v2 ≤ 2− γ and thedefinition τ∗ = k2 τ , means that the characteristic timefor nonlinear effects τ∗ does not exceed the characteristictime for linear relaxational effects τ .

Important quantities in the characterization of criticalpoints will be the deceleration parameter q = −1 − θ′/θand the effective EoS parameter γeff = −2θ′/3θ, which

4

are given respectively by

q =1

2

[1 + 3(γ − 1) Ω + 3Π

](14)

γeff = 1 + (γ − 1) Ω + Π (15)

The region of the phase space where accelerated expan-sion occurs is found by imposing q < 0 in eq.(14):

Π < −1

3− (γ − 1)Ω

In this region the effective EoS parameter γeff < 2/3.Comparing eq.(13) and eq.(14) for q < 0, one finds thataccelerated expansion is possible in the physical phasespace only if

v2

k2>

1 + 3 Ω (γ − 1)

3 γ Ω(16)

Considering eq.(11), letting Ω′ = 0, we identify twoconditions:

Ωc = 1 (17a)

(γ − 1) Ωc + Πc = 0 (17b)

In order to locate the critical points one has to sub-stitute these conditions into the equation Π′ = 0. Wewill carry out the analysis specifying the character ofthe viscous fluid through the choice of γ. Moreover,given the considerations above, from now on we will set0 < k2 = v2 ≤ 2 − γ. We note that this choice excludesthe case of stiff matter (γ = 2) from the analysis, becausethe bound on the dissipative speed of sound would givev2 = 0.

A. Dust: γ = 1

The first condition Ωc = 1 identifies an invariant set,in which equation Π′ = 0 is given by

3

2Π3c +

3

2Π2c −

v2

ζ0(1 + 3 ζ0) Πc − 3v2 = 0

In the range of parameters considered, this cubic equa-tion has 3 real roots and only one of them is positive.Furthermore, it can be shown that the most negative rootlies always in the region of negative entropy productionrate, so that we are concerned only with the other tworoots Π+ and Π−, where ± specifies their sign. The sys-tem has thus 2 critical points in the invariant set Ωc = 1and we call them P+

d and P−d . Given the dynamical sys-tem in the general form

Ω′ = f(

Ω, Π)

(18a)

Π′ = g(

Ω, Π)

(18b)

the stability of the critical points is analyzed through theeigenvalues of the matrix

A =

(∂f∂Ω

∂f

∂Π∂g∂Ω

∂g

∂Π

)|P±i

(19)

If both eigenvalues have negative (respectively positive)real part then the point is a source (respectively a sink).A saddle point is found if the real parts of the eigenvalueshave opposite signs. We find that ∂Πf |P±

d≡ 0, so that

the eigenvalues are given by

λ1 ≡∂f

∂Ω|P±d

= 3Π± (20)

λ2 ≡∂g

∂Π|P±d

= 3Π± − v2

ζ0 (1 + Π±)2(21)

For the point P+d , given that Π+ > 0, the sign of both

eigenvalues is always positive, so it is a source point.As regards P−d , instead the sign of the eigenvalues isnegative, thus identifying a stable sink.

Considering now the condition eq.(17b), which in

the case of dust simplifies to Πc = 0, it has to benoted that it is not an invariant set, but it representsa line of points in the phase space where the flow is“momentarily at rest” in the Ω direction (a set of turningpoints). We can find a critical point on this line by

imposing also the requirement Π′ = 0, whose solutionis Ωc = 0. However, the stability analysis for the pointP 0d = 0, 0 is hampered by its location, i.e., it lies on

the entropy production rate divergence line, where thesystem is not well defined. In order to have an idea ofthe behaviour around this point, we resort to numericalplots. The global situation is depicted in Fig. 1: thegreen trajectory is a representative flow going from P+

d

towards P−d , while the red trajectory is the invariantset Ω = 1. The white region has been cut out from theanalysis because it corresponds to models with negativeentropy production rate, whereas on its boundary, therate diverges. Note that this boundary is repulsive,in the sense that no trajectory in its neighbourhoodis attracted towards it: this feature keeps the modelssafe from divergences in entropy production rate. Inthe green region q < 0, so the expansion is accelerated.As can be seen, the point P 0

d turns out to be a saddle,while P−d is the global attractor for every choice ofinitial conditions in the physical region and it describesa cosmological expanding model dominated by viscousmatter. In the portrait on the top, the choice of theparameters leads to an accelerated expansion, whilein the portrait on the bottom the stable point lies inthe deceleration region. In Table I a summary of thestability analysis is presented.

5

Pd+

Pd-

Pd0

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

W

P

~

Pd+

Pd-

Pd0

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

W

P

~

FIG. 1: Phase plane evolution of the system with γ = 1,ζ0 = 1 and: v2 = k2 = 1 (top); v2 = k2 = 1/25 (bottom).Red flow: invariant set Ω = 1. White region: negativity ofentropy production rate. Green region: accelelerated expan-sion. Dashed plot: the line Π = 0.

Point P 0d P+

d P−d

Character saddle source sink

TABLE I: Stability of critical points for γ = 1.

B. Radiation: γ = 4/3

Just like before, we start by imposing the conditioneq.(17a) in equation Π′ = 0, obtaining the third orderequation

27

32Π3c +

9

8Π2c −

v2

ζ0

(4

3+ 3 ζ0

)Πc − 4v2 = 0

Among the three real roots, we again retain only the twothat lie in the physical phase space. Hence we find twocritical points, P+

r = 1, Π+c and P−r = 1, Π−c , where

Π+c (resp. Π−c ) is the positive (resp. negative) root.

Imposing in Π′ = 0 the second condition eq.(17b),

which now reads Πc = −Ωc/3, we find the critical pointP 0r = 0, 0 and another critical point

P ∗r =

27 ζ04 v2

(v2 − 1

32

), − 9 ζ0

4 v2

(v2 − 1

32

)(22)

whose presence and influence in the physical phase spaceis subject to the condition Ω∗c ≤ 1, that is

∀ v > 0 if 0 < ζ0 < ζ0 (23a)

0 < v ≤ v if ζ0 > ζ0 (23b)

where v ≡√

ζ0

32(ζ0 − ζ0

) and ζ0 ≡ 4/27

The dynamics and stability properties in the phasespace are similar to the case of dust only when P ∗r is notpresent, but an important difference arises when consid-ering the stability of the critical point P−r in conjunctionwith the appearance of P ∗r .

Let us analyze the stability properties in more detail.As in the previous case, we defer the qualitative stabilityanalysis of P 0

r to the numerical plot Fig. 2: this criticalpoint is a saddle. For the points P±r , the eigenvalues ofthe stability matrix eq.(19) are

1 + 3 Π±c ,9 Π±c

4− 64 v2

3 ζ0 (4 + 3 Π±c )2(24)

The critical point P+r is always a source. Critical point

P−r turns out to be a sink for Π−c < −1/3, which oc-curs outside the domain of parameters given by (23),i.e., when P ∗r is absent; if instead the parameters sat-isfy conditions (23), then P−r is a saddle and the “newlyborn” critical point P ∗r turns out to be a sink, havingeigenvalues

−4(1 + 64 v2

)±√

16 v2 (5− 64 v2)2

+ 81 (1− 32 v2)2ζ0

32

6

Pr+

Pr-

Pr0

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

W

P

~

Pr-

º Pr*

Pr+

Pr0

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

W

P~

Pr+

Pr*

Pr0

Pr-

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

W

P

~

FIG. 2: Phase plane evolution of the system with γ = 4/3 and: v =√

2/3, ζ0 = 1 (left); v = v, ζ0 = ζ0 + 1/10 (center);v = v−1/10, ζ0 = ζ0 +1/25 (right). Red flow: invariant set Ω = 1. White region: negativity of entropy production rate. Green

region: accelelerated expansion. Dashed plot: the line Π = −Ω/3.

Point Character

if (23) hold if (23) do not hold

P 0r saddle saddle

P+r source source

P−r saddle sink

P ∗r sink (n/a)

TABLE II: Stability of critical points for γ = 4/3.

which are both real and negative in the ranges given by(23). A summary of the stability properties is given inTable II.

The dynamical system is plotted in Fig. 2 for threerepresentative sets of values of the parameters. In theplot on the left the global attractor P−r represents anaccelerated expanding model dominated by viscous radi-ation. In the central plot the point P−r ≡ P ∗r is again theglobal attractor, but the model is decelerating. The ploton the right displays the peculiarity with respect to thedust model: the point P−r is now an unstable saddle and

the point P ∗r , located along the (dashed) line Π = −Ω/3,is the new global attractor. This point represents a decel-erated expanding model with contributions coming bothfrom viscous radiation and non-viscous dust.

IV. SCALE FACTORS

Some general considerations on the form of the scalefactors corresponding to the critical points can be made

by integrating the cosmic-time version of eq.(9a), i.e.

θ = −1

2

[1 + (γ − 1) Ω + Π

]θ2 , (25)

and then solving θ ≡ 3 a/a for a(t). Power law scale

factors are obtained whenever 1 + (γ − 1) Ω + Π 6= 0,

because then θ 6= 0. In this case the result correspondingto a generic critical point is

a(t) = a0 (t− t0)2

3[1+(γ−1) Ωc+Πc] (26)

It is easy to show that the choice k2 = v2 made in theprevious sections prevents the occurence of models withexponential scale factors. In order for the exponentiallyexpanding models (identified by the condition 1 + (γ −1) Ω + Π = 0 in eq.(25)) to be present in the physicalregion of phase space (bounded by eq.(13)), the followinginequality must hold:

(1− γ) Ω− 1 > −γ v2

k2Ω ,

For v2 = k2 the inequality does not hold in the physicalphase space. On the other hand, only if we let v2 > k2,can the inequality be satisfied (consistently with the anal-ysis leading to eq.(39) in [19]), specifically in the phase-space region[

1− γ(

1− v2

k2

)]−1

< Ω ≤ 1 ∧ −γ < Π <2

3− γ

It may be pointed out that since we get exponentially ex-panding solutions only for a certain range of parameters,our analysis is not completely immune from fine tuning.Moreover in this case, from eq.(26), it is clear that for

1 + (γ − 1) Ω + Π < 0, i.e., below the “exponential be-haviour” line, we can find cosmological contracting mod-els. In Fig. 3 a qualitative representation of the situationis plotted.

7

0.0 0.2 0.4 0.6 0.8 1.0 W

0

P

~

0.0 0.2 0.4 0.6 0.8 1.0 W

0

P

~

FIG. 3: Qualitative phase space portrait for (a) v2 = k2 and(b) v2 > k2. The physical phase space is above the thick line.Green region: accelerated expansion. Dashed line: exponen-tial behaviour. Red region: contracting models.

Finally, if we let instead v2 < k2, the possibility of hav-ing accelerated expansion will narrow down, disappearing

from the physical phase space when v2 < γ−2/3γ k2.

With regard to the critical points analyzed in the pre-vious section, in Table III are listed the respective poly-nomial scale factors. The presence of the viscous pressureaffects the power law by slowing down the dynamics inP+ and accelerating it in P− with respect to the purenon-viscous fluid case. The behaviour in the points P 0

and P ∗ (actually on the whole line (γ − 1) Ω + Π = 0,dashed in Figs. 1 and 2) is always dust-like.

V. DISCUSSION

From the analysis carried out, it is clear that stable so-lutions exist for non-interacting two-fluids models in thepresence of nonlinear bulk viscous effects and that someof these solutions display an accelerated expansion. Ifthe viscous fluid is of dust type it will ultimately domi-nate the non-viscous component; if the viscous fluid is ofradiation type, its dominance upon the non-viscous dustdepends on the parameters involved.

We highlight the fact that, in the case of viscous ra-diation, a trajectory in the phase space starting from aneighbourhood of P+

r with appropriate initial conditionscan pass through the following stages: i) a radiation-

dominated era (source P+r ), ii) a matter-dominated tran-

sient era (saddle P 0r ), where structure formation can oc-

cur and iii) a final, everlasting era characterized by accel-erated (either polynomial or exponential) expansion fora non-zero-measure set of parameters (sink P−r ). This isin contrast to the findings of [16] in which Eckart theorywas used and where it was found that, although late timeacceleration could be obtained, radiation and matter so-lutions were not present. The intervention of all the threestages of evolution in the dynamical system is the mainachievement of the present analysis.

From eq.(7) it is clear that the speed of sound ofthe viscous dark energy could attain quite small valuesand hence conceivably lead to clustering. This would inturn cause an inhomogeneous expansion of the Universewhich could, in principle, be detected as an anisotropyin the measurements of supernovae’s brightness-redshiftrelation [35]. Similar studies in Eckart theory [24] showgood agreement with the supernovae results. Data fromdistance-redshift as well as baryon acoustic oscillationsmeasurements place fairly tight constraints which anyreasonable cosmological model has to satisfy, e.g. on theequation of state and the fractional contribution fromdark energy. For example, without claiming here a rigor-ous analysis, if in the present model we let the effectiveEoS parameter attain the observed value γeff ' 0 [34]at the stable critical point P−i (i = d, r), then the ad-missible region of phase space for the stable point to befound is restricted by the system of eqs.(14)(15): in thespecific case of exactly vanishing γeff , it is the line ofexponential growth 1 + (γ − 1)Ω + Π = 0 which, as wehave seen in Sec.IV, can be considered physically viableonly for v2 > k2. In this sense, the fine tuning is afeature that still cannot be taken care of in this model.A thorough comparison with observational data was notthe main aim of this work. Rather, our analysis fulfillsthe previous lack of a more complete dynamical systemanalysis for the proposal of [19] (in our case extended totwo fluids) in order to establish the existence and stabil-ity of viable cosmological models. Future investigationscould be aimed to generalize some aspects of the model(e.g the choice of the parameter ζ) and to constrain themodel with observational results.

In conclusion, the kind of evolution that we have foundshares some similarities with the current accepted modelfor cosmic evolution, even though, as we pointed out, itstill requires some degree of fine tuning. Apart from thesimilarities, we also stress that, at this level, the modela) does not include an explanation for primordial infla-tion and b) obviously shares with ΛCDM the ignoranceabout the future behaviour of the Universe, in the sensethat in both models the accelerating phase (if present)lasts forever; the everlasting acceleration, in turn, is im-possible to disprove without knowing a priori the exactmatter-energy content. In fact, it has to be kept in mindthat such type of analysis is always phenomenological innature.

8

Point: P 0 P+ P− P ∗

γ = 1 a0 (t− t0)2/3 a0 (t− t0)2

3(1+Π+c ) a0 (t− t0)

2

3(1+Π−c ) (n/a)

γ = 4/3 a0 (t− t0)2/3 a0 (t− t0)2

4/3+Π+c a0 (t− t0)

2

4/3+Π−c a0 (t− t0)2/3

TABLE III: Scale factors corresponding to the critical points for 1 + (γ − 1) Ωc + Πc 6= 0 (polynomial behaviour).

Acknowledgments

The authors are thankful to the Astrophysics andCosmology Research Unit (ACRU) at the University ofKwaZulu-Natal for the kind hospitality, in particularSunil Maharaj. We also thank Rituparno Goswami and

Radouane Gannouji for useful discussions. GA thanksthe University of Zululand for the award of a Postdoc-toral Fellowship. Finally, the authors are grateful to ananonymous referee for constructive comments which ledto an improvement in the manuscript.

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