Dynamical System Analysis of a Three Fluid Cosmological Model : An Invariant Manifold approach
aa r X i v : . [ g r- q c ] F e b Dynamical System Analysis of a Three Fluid Cosmological Model : AnInvariant Manifold approach
Subhajyoti Pal ∗ Department of Mathematics, Sister Nibedita Govt GeneralDegree College For Girls, Kolkata-700027, West Bengal, India.
Subenoy Chakraborty † Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India.
The present paper considers a three-fluid cosmological model consisting of noninteractingdark matter, dark energy and baryonic matter in the background of the Friedman-Robertson-Walker-Lemaˆıtre flat spacetime. It has been assumed that the dark mattertakes the form of dust whereas the dark energy is a quintessence (real) scalar field withexponential potential. It has been further assumed that the baryonic matter is a perfectfluid with barotropic equation of states. The field equations for this model takes the formof an autonomous dynamical system after some suitable changes of variables. Then acomplete stability analysis is done considering all possible parameter (the adiabatic indexof the baryonic matter and the parameter arising from the dark energy potential) valuesand for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperboliccritical points, the invariant manifold theory (center manifold approach) is applied. Finallyvarious topologically different phase planes and vector field diagrams are produced and thecosmological interpretation of this model is presented.
Keywords :
Non-hyperbolic point, Center manifold theory, Field equations.PACS Numbers : 98.80.-k, 05.45.-a, 02.40.Sf, 02.40.Tt
I. INTRODUCTION
Recently, a large number of observational data from various sources such as Type Ia Supernova [1,2], CMB anisotropies [4, 7], Large Scale Structures [3, 5] and Baryon Acoustic Oscillations [6]suggests that we are in a spatially flat universe which after the big bang has undergone two ∗ [email protected] † [email protected] accelerated expansion phases, one occurred before the radiation dominated era and the otherone started not too long ago. We are presently in this accelerated expansion phase.This present time accelerated expansion has been attributed to a unseen and unknown matterwith very large negative pressure called the dark energy. Analysis of different cosmological datasuggests that our universe is composed of around 70% dark energy, about 25% dark matter andthe rest accounts for baryonic matter and radiation.Very few properties of the dark energy is known [9]. As far as the mathematical modelling ofthis exotic matter is concerned, the simplest choice for it has been the Cosmological ConstantΛ [8, 11, 16]. Although these models are capable of explaining most of the observational data,all of them fails to explain the coincidence problem (why expansion is happening now and whyis it accelerated) and the fine-tuning problem (why some of the parameters take exorbitantlyhigh values where others are not). To resolve these issues, lots of dynamical dark energy modelshas been prescribed. There the dark energy has been modelled as a scalar field. Some scalarfield models like quintessence [17, 18], K -essence and Tachyonic models have attracted lots ofattention.In this paper we consider a cosmological model consisting of non-interacting dark matter,dark energy and baryonic matter in the background of the Friedman-Robertson-Walker-Lemaˆıtreflat spacetime. The dark matter has been assumed to take the form of dust whereas the darkenergy is assumed to be a scalar field with exponential potential. It has been further assumedthat the baryonic matter is a perfect fluid with barotropic equation of states. All these threefluids are assumed to be non-interacting and minimally coupled to gravity.In order to study this model qualitatively, we derive the Einstein’s field equations and theKlein-Gordan equation for the scalar field. After some suitable changes of variables, theseequations take the form of an autonomous dynamical system. Then we find the critical pointsand analyze the stability of each critical point. We note that in [10] the model like ours havebeen considered. But the authors did not consider the cases of non-hyperbolic critical points.Our results differ from them in various ways. Firstly, we apply some of the very rich theoriesfor the dynamical systems, namely the Invariant Manifold and the Center Manifold Theory[13–15, 20] to compute the center manifolds for all the non-hyperbolic critical points and thencontinue to do the stability analysis for them. The other one is that we have considered alltheoretically possible values of the parameters to do a complete analysis here, whereas in theirarticle, they chose only a few suitable values. Lastly, we have presented all possible topologicallydifferent phase plane diagrams here where they included only some of them in their article.The motivation to do stability analysis is that after considering all cosmological and ob-servational constraints of data, the stable critical points in our model may depict our presentuniverse as a global attractor. If they do fit with the data of present percentages of dark energy,dark matter and baryonic matter together with radiation in the universe then our model wouldsuccessfully describe the universe.The organization of this article is as follows : The section II describes the Einstein fieldequations, Klein-Gordan scalar field equation and energy conservation relations for our model.Section III describes the formation of an autonomous system. The section IV is where wepresent our work on complete stability analysis. At the end of this section, we produce thephase plane diagrams for different topological cases. Finally section V presents the cosmologicalinterpretations of our results and concludes our work. II. EQUATIONS
The homogeneous and isotropic flat Friedman-Robertson-Walker-Lemaˆıtre spacetime is the back-ground of our model. This universe is assumed to be filled up by non-interacting dark matter,dark energy and baryon. Dark matter is assumed to be dust with energy density ρ m and thedark energy is assumed to be a scalar field φ with the potential as V ( φ ). The density ρ d andthe pressure p d of the scalar field follows the following equations : ρ d = 12 ˙ φ + V ( φ ) (1)and p d = 12 ˙ φ − V ( φ ) . (2)Here ˙ denotes differentiation with respect to cosmic time t. The baryonic matter is assumed to be a perfect fluid with linear equation of state p b = ( ν − ρ b (3)where p b and ρ b are the density and the pressure of the fluid and ν is the adiabatic index of thefluid satisfying < ν ≤ . In particular ν = 1 and ν = corresponds to the dust and radiationrespectively. Here we also assume that ν = 1 All three matter are non-interacting and minimallycoupled to gravity.The Einstein field equations for this model is3 H = k ( ρ m + ρ d + ρ b ) . (4)where H is the Hubble parameter and k = 8 πG , where G is the gravitational constant, the speedof light has been scaled to 1 . The Klein-Gordan equation of the scalar field is¨ φ + 3 H ˙ φ + dVdφ = 0 (5)The energy conservation relations take the following form˙ ρ m + 3 Hρ m = 0 (6)˙ ρ d + 3 H ( ρ d + p d ) = 0 (7)˙ ρ b + 3 H ( ρ b + p b ) = 0 (8)From (4),(5),(6),(7) and (8) we derive2 ˙ H = − k ( ρ m + ρ b + p b + ˙ φ ) . (9)The equations (4),(9) and (5) are the evolution equations for this model. Next we find suitablecoordinate changes such that these evolution equations form a system of autonomous dynamicalsystem. This is done in the following section. III. THE AUTONOMOUS SYSTEM
We introduce the following coordinate transformations of variables : x = r k φH , (10) y = r k p V ( φ ) H (11)and the density parameters Ω m = kρ m H , (12)Ω b = kρ b H , (13)Ω d = kρ d H . (14)These coordinate changes transform the Friedmann equation (4) and the equation (9) respec-tively as the following : Ω d = x + y , (15)Ω m + Ω b + x + y = 1 (16)and ˙ H = − H ( x + Ω m ν Ω b . (17)As Ω m and Ω b , the density parameters are non-negative real quantities, so from (16) 0 ≤ Ω m ≤ , ≤ Ω b ≤ x and y satisfies x + y ≤ . The strict equality is only possible if theenergy densities of the dark matter and the baryonic matter is zero.Differentiating the equations (10), (11) and (12) with respect to N where N = ln a ( a ( t ) isthe scale factor of the universe) and using (16), (17), (5) and (6), (7) and (8) we derive thefollowing autonomous dynamical system : dxdN = 3 x [ x − m ν − Ω m − x − y )] − r k V dVdφ y (18) dydN = y [3 x + 3 ν − Ω m − x − y ) + 3 Ω m r k V dVdφ x ] (19) d Ω m dN = − m [1 − Ω m − x − ν (1 − Ω m − x − y )] (20)We end this section by expressing the relevant cosmological parameters in terms of the abovetransformed variables as Ω d = x + y , (21) ω d = p d ρ d = x − y x + y , (22) ω eff = p d + p b ρ m + ρ d + ρ b = − x + ν (1 − Ω m − x − y ) + Ω m (23)and q = − (1 + ˙ HH ) = − [(1 − x − ν − m − ν ) + 3 ν x + y )] . (24)We note that for accelerated expansion, ω eff ≤ − and q ≤ IV. STABILITY ANALYSIS
We start working on the stability analysis of the dynamical system (18) in this section. Weassume that the potential of the scalar field representing the dark energy is exponential, ie V dVdφ = some constant. We choose q k V dVdφ = α. Then the autonomous system (18) transformsinto : dxdN = 3 x [ x − m ν − Ω m − x − y )] − αy (25) dydN = y [3 x + 3 ν − Ω m − x − y ) + 3 Ω m αx ] (26) d Ω m dN = − m [1 − Ω m − x − ν (1 − Ω m − x − y )] (27)There are ten critical points of this autonomous system. They are listed in the table below. TABLE I: Critical PointsCritical Point Name Critical Point C (0 , , C (1 , , C (0 , , C ( − , , C ( − α , √ − α , C ( − α , − √ − α , C ( − ν α , α √ ν − ν , C ( − ν α , − α √ ν − ν , C ( − α , α , (1 − α )) C ( − α , − α , (1 − α )) The value of the relevant cosmological parameters for each of the critical points are given in thefollowing table :
TABLE II: Values of the Different Cosmological Parameters at the Critical PointsCritical Points ω d ω eff Ω m Ω d Ω b qC Undefined ν − ν − C C Undefined 0 1 0 0 C C α − α − α − C α − α − α − C ν − ν − ν α − ν α ν − C ν − ν − ν α − ν α ν − C − α α C − α α For every critical point and for each of their subcases we will do two successive change ofvariables to bring them to a form with which the calculation of center manifold and dynamicsof the reduced system will be easy. For each of the cases and subcases the transformations areas follows :If X = xy Ω m , ¯ X = ¯ x ¯ y ¯Ω m and ¯¯ X = ¯¯ x ¯¯ y ¯¯Ω m , then ¯ X = X − A and ¯¯ X = P − ¯ X where A and P is some 3 × × P being non-singular. The exact form of A and P will vary from case to case and we willmention them while studying each of the cases and subcases. A. Critical Point C For critical point C , A = and the Jacobian matrix of the system (25) at this criticalpoint has the characteristic polynomial λ + (6 − ν ) λ + [ 3 ν (3 ν − ν − ν − λ − ν (3 ν − ν − . (28)which has eigen values as 3 ν − , ν − ν . The stability analysis when ν is not 0 , C .Before we write down the table, we introduce a notation. ′ represents derivative with respect to N. TABLE III: C (Reduced System) ν Center Manifold Reduced System0 ¯¯ x = O ( ¯¯Ω m ) , ¯¯ y = − α ¯¯Ω m + O ( ¯¯Ω m ) ¯¯Ω ′ m = − α ¯¯Ω m + O ( ¯¯Ω m )1 ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 02 ¯¯ x = O (¯¯ y ) , ¯¯Ω m = O (¯¯ y ) ¯¯ y ′ = 0 For all these subcases, P = . We summarize our results for C in the table below to end this subsection. TABLE IV: Summary for the critical point C Case ν Stability(RS) Stability(DS)a ν < ν = 0 Stable Stablec 0 < ν < ν = 1 Center Center-Saddlee 1 < ν < ν = 2 Center Center-Unstableg 2 < ν NA Unstable
Here RS stands for the Reduced System, DS stands for the whole system and NA stands forNot Applicable (Hyperbolic Cases).
B. Critical Point C For critical point C , A = and the Jacobian matrix of the system (25) at this criticalpoint has the characteristic polynomial λ + (3 ν − α − λ + [18 − (3 ν − α + 3) − ν ] λ − (9 ν − α + 3) = 0 . (29)which has eigen values as 3 , − ν and α + 3 . The stability analysis when ν is not 2 and α is not − C . TABLE V: C (Reduced System) ν α Center Manifold Reduced System2 − m = O ( k (¯¯ x, ¯¯ y ) k ) ¯¯ x ′ = − x ¯¯ y , ¯¯ y ′ = − x ¯¯ y = − y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 0 = 2 − y = O (¯¯ x ) , ¯¯Ω m = − ¯¯ x + O (¯¯ x ) ¯¯ x ′ = − ¯¯ x ν = 2 and α = − P is − . For ν = 2 and α = − , P = −
00 0 10 1 0 . When ν = 2 and α = − P = −
11 0 00 1 0 . We end this subsection by summarizing our results for C in a table. TABLE VI: A summary for the critical point C Case ν α
Stability(RS) Stability(DS)a < < − < − < > − > − − < − > < − > − > > − C. Critical Point C For C the A = . The Jacobian Matrix at C has the characteristic polynomial λ + (3 ν − λ − λ + [ 274 − ν − , and 3 − ν. Hence the stability analysis is almost trivial if ν = 1 . Because then it is an immediate applicationof the stability analysis of the linear cases and the Hartman-Gr¨obman theorem.Therefore we assume that ν = 1 and proceed with our stability analysis. TABLE VII: C (Reduced System) ν Center Manifold Reduced System1 ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 0 For this subcase P is . At the end, here is the summary of the results for C . TABLE VIII: The critical point C , a summaryCase ν Stability(RS) Stability(DS)a < > D. Critical Point C For critical point C , A = − and the Jacobian matrix of the system (25) at this criticalpoint has the characteristic polynomial λ + (3 ν + α − λ + [(3 ν − α − − α + 9] λ − (3 ν − α −
9) = 0 . (31)which has eigen values as 3 , − ν and 3 − α. The stability analysis when ν is not 2 and α isnot 3 (hyperbolic cases) is easy again as said before.Now we will present a table containing all the non-hyperbolic subcases and the result of theirstability analysis. Later We will write the stability results for all possible subcases for C .2 TABLE IX: C (reduced System) ν α Center Manifold Reduced System2 3 ¯¯Ω m = O ( k (¯¯ x, ¯¯ y ) k ) ¯¯ x ′ = − x ¯¯ y , ¯¯ y ′ = 3¯¯ x ¯¯ y = 3 ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 0 = 2 3 ¯¯ y = O (¯¯ x ) , ¯¯Ω m = ¯¯ x + O (¯¯ x ) ¯¯ x ′ = − ¯¯ x When ν = 2 and α = 3 the P is . For ν = 2 and α = 3 , P =
00 0 10 1 0 . When ν = 2 and α = 3 then P =
11 0 00 1 0 . We end this subsection by summarizing our results for C in a table. TABLE X: C , a stability analysisCase ν α Stability(RS) Stability(DS)a < < < < > > < > < > > > E. Critical Point C Here it is necessary that | α |≤ . A = − α √ − α for this case.The Jacobian matrix of the system (25) at this critical point has the characteristic polynomial λ + (3 ν − α + 6) λ + [18 ν − να − α + 89 α + 9] λ − (9 ν − α )(2 α − α + 81)27 = 0 . (32)This characteristic polynomial has eigenvalues as α − ν, α − α − . The stabil-ity analysis when ν = α and α = ± , ± √ (hyperbolic cases) is easy as said in the abovesubsections.Now we will present a table containing all the non-hyperbolic subcases and the result of theirstability analysis. Since for C , the total number of all possible subcases is too many, we wouldnot present the results for the complete case in a table at the end as we did in the previoussubsections. We would rather present a diagram in α − ν plane to show the stability analysisfor all possible values of α ’s and ν ’s. TABLE XI: C (Reduced System) ν α Center Manifold Reduced System2 3 id to subcase (e) of C id to subcase (e) of C = 2 3 id to subcase (b) and (h) of C id to subcase (b) and (h) of C − C id to subcase (e) of C = 2 − C id to subcase (b) and (h) of C √ ¯¯Ω m = √ ¯¯ x + O ( k (¯¯ x, ¯¯ y ) k ) ¯¯ x ′ = − √ x , ¯¯ y ′ = − √ x ¯¯ y = 1 √ ¯¯ x = r ¯¯ y + O (¯¯ y ) , ¯¯Ω m = s ¯¯ y + O (¯¯ y ) ¯¯ y ′ = − y − √ ¯¯Ω m = √ ¯¯ x + O ( k (¯¯ x, ¯¯ y ) k ) ¯¯ x ′ = 3 √ x , ¯¯ y ′ = 3 √ x ¯¯ y = 1 − √ ¯¯ x = r ¯¯ y + O (¯¯ y ) , ¯¯Ω m = s ¯¯ y + O (¯¯ y ) ¯¯ y ′ = − y α α ¯¯ y = t ¯¯ x + O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = − u ¯¯ x + O (¯¯ x )0 0 ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = O (¯¯ x ) Where ”id” means ”identical” and r = √ ν − ν − , s = √ − ν )4(2 ν − , t = √ − α (9 α − α )4 α − α +405 α − and u = − α √ − α α − . We also note that in the subcase before the last one in the table, the orderedpair ( ν, α ) is not permitted to take the values (2 , , (2 , − , (1 , √ ) and (1 , − √ ) . ν = 1 and α = √ the P is . For ν = 1 and α = √ , P = √ − νν − . For ν = 1 and α = − √ the P is −
10 0 10 1 0 and when ν = 1 and α = − √ , P = − − √ − − νν − . Also in the final two subcases P = − α (9 − α ) / α − α +81 √ − α α α and P = −
10 1 0 respectively.Since we will find that the stability diagram for C and C is identical, it will be presented atthe end of the next subsection. We now proceed with the next subsection. F. Critical Point C Here it is also necessary that | α |≤ .A = − α − √ − α in this case.The Jacobian matrix of the system (25) at this critical point has the characteristic polynomial λ + (3 ν − α + 6) λ + [18 ν − να − α + 89 α + 9] λ + (9 ν − α )(2 α − α + 81)27 = 0 . (33)5This characteristic polynomial has eigenvalues as α − ν, α − α − . The stabilityanalysis when ν = α and α = ± , ± √ (hyperbolic cases) is again easy.Now we will present a table containing all the non-hyperbolic subcases and the result of theirstability analysis. For C too the total number of all possible subcases is many, so we would notpresent the results for the complete case in a table. We would rather present a diagram in α − ν plane to show the stability analysis for all possible values of α ’s and ν ’s as we did for C . TABLE XII: C (Reduced System) ν α Center Manifold Reduced System2 3 id to subcase (e) of C id to subcase (e) of C = 2 3 id to subcase (b) and (h) of C id to subcase (b) and (h) of C − C id to subcase (e) of C = 2 − C id to subcase (b) and (h) of C √ ¯¯Ω m = − √ ¯¯ x + O ( k (¯¯ x, ¯¯ y ) k ) ¯¯ x ′ = − √ x , ¯¯ y ′ = − √ x ¯¯ y = 1 √ ¯¯ x = − r ¯¯ y + O (¯¯ y ) , ¯¯Ω m = − s ¯¯ y + O (¯¯ y ) ¯¯ y ′ = − y − √ ¯¯Ω m = − √ ¯¯ x + O ( k (¯¯ x, ¯¯ y ) k ) ¯¯ x ′ = 3 √ x , ¯¯ y ′ = 3 √ x ¯¯ y = 1 − √ ¯¯ x = − r ¯¯ y + O (¯¯ y ) , ¯¯Ω m = − s ¯¯ y + O (¯¯ y ) ¯¯ y ′ = − y α α ¯¯ y = − t ¯¯ x + O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = u ¯¯ x + O (¯¯ x )0 0 ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = O (¯¯ x ) Where r, s, t and u take usual values as defined in the subsection of C . As like in the previoussubsection here is also in the subcase before the last one in the table, the ordered pair ( ν, α ) isnot permitted to take the values (2 , , (2 , − , (1 , √ ) and (1 , − √ ) . When ν = 1 and α = √ , P = −
10 0 10 1 0 . For ν = 1 and α = √ , P = − √ − − νν − . For ν = 1 and α = − √ , P = ν = 1 and α = − √ , P = − √ − νν − . Also in the final two subcases P = α (9 − α ) / α − α +81 − √ − α α α and P = −
10 1 0 respectively.We finish this subsection by producing the stability diagram for both C and C in α − ν planein the following : −3 −2 −1 0 1 2 3−2−1.5−1−0.500.511.52 α ν FIG. 1: Stability Diagram ( C , C ) Here the cyan shaded area ie the area bounded by α = √ , α = − √ and ν = α representsthe α − ν pairs for which (25) is stable. The yellow shaded region represents parameter valuesfor which (25) is saddle. Lastly the origin is center-stable here. G. Critical Point C In this case, it is necessary that 0 ≤ ν ≤ α = 0 . A takes the form − ν α √ ν (2 − ν )2 α . The Jacobian matrix of the system (25) at C has the following characteristic polynomial: λ + (6 − ν λ + [ 9( ν − ν − α )4 α ] λ − ν (9 ν − α )( ν − ν + 2)4 α = 0 . (34)It has eigenvalues as 3( ν − , α ( να − α + p (2 − ν )(36 ν − να + 2 α )) and α ( να − α − p (2 − ν )(36 ν − να + 2 α )) . The stability analysis when ν = 1 , ν = α and ν = 2 , ν = 0(hyperbolic cases) are easy by application of linear stability analysis and reduction of non-linearcase to linear case under some specific conditions.So we present a table containing all the non-hyperbolic subcases. For C also, we will present adiagram in α − ν plane to show the stability analysis for all possible values of α ’s and ν ’s at theend. TABLE XIII: C (Reduced System) ν α Center Manifold Reduced System1 √ id to subcase (e) of C id to subcase (e) of C − √ id to subcase (g) of C id to subcase (g) of C = 1 ν = α , α > C id to subcase (i) of C = 1 ν = α , α < C id to subcase (i) of C ν = α ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 02 α ¯¯ x = O ( k (¯¯ y, ¯¯Ω m ) k ) ¯¯ y ′ = α ¯¯ y ¯¯Ω m , ¯¯Ω ′ m = − v ¯¯ y α id to subcase (a) of C id to subcase (a) of C where v = α − α . For the last two subcases P is w − w and α respectively with w = α √ − α − α +92 α − . Again we will find that the stability diagram for C is exactly same as C . Hence it will bepresented at the end of the next subsection.8
H. Critical Point C In this case also, it is necessary that 0 ≤ ν ≤ α = 0 and A takes the form − ν α − √ ν (2 − ν )2 α . The Jacobian matrix of the system (25) at C has the characteristic polynomial: λ + (6 − ν λ + [ 9( ν − ν − α )4 α ] λ − ν (9 ν − α )( ν − ν + 2)4 α = 0 . (35)This polynomial has eigenvalues as 3( ν − , α ( να − α + p (2 − ν )(36 ν − να + 2 α )) and α ( να − α − p (2 − ν )(36 ν − να + 2 α )) . The stability analysis when ν = 1 , ν = α and ν = 2 , ν = 0 (hyperbolic cases) are easy again.So we present a table containing all the non-hyperbolic subcases for C . For this case also, atthe end we will present a diagram in α − ν plane to show the stability analysis for all possiblevalues of α ’s and ν ’s. TABLE XIV: C (Reduced System) ν α Center Manifold Reduced System1 √ id to subcase (e) of C id to subcase (e) of C − √ id to subcase (g) of C id to subcase (g) of C = 1 ν = α , α > C id to subcase (i) of C = 1 ν = α , α < C id to subcase (i) of C ν = α ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 02 α ¯¯ x = O ( k (¯¯ y, ¯¯Ω m ) k ) ¯¯ y ′ = α ¯¯ y ¯¯Ω m , ¯¯Ω ′ m = − v ¯¯ y α id to subcase (a) of C id to subcase (a) of C Where ” v ” takes it’s value as defined for C . For the last two subcases P is q j and α respectively with j = α √ − α + α − α − and q = − α √ − α + α − α − . This subsection concludes with the stability diagram in the α − ν plane for both the C and C :9 −3 −2 −1 0 1 2 300.20.40.60.811.21.41.61.82 α ν FIG. 2: Stability Diagram ( C , C ) Here 0 ≤ ν ≤ α = 0 . The cyan-shaded area ie the area bounded by ν = 0 , ν = 1 and ν = α represents stability. The yellow region represents saddle system (25). The ’o’ markedlines are those parameter values for which the system (25) is center-stable. I. Critical Point C In this case, it is necessary that α = 0 . For this case, A = − ν α ν α α − α . The Jacobian matrix of the system (25) at C has the characteristic polynomial as following : λ + (3 ν −
32 ) λ + [ 18 να − α ] λ − α − ν − α = 0 . (36)This polynomial has eigenvalues as 3(1 − ν ) , − α +3 √ − α α and − α − √ − α α . The stabilityanalysis when ν = 1 and α = − √ , √ (hyperbolic cases) are easy.Therefore we present a table containing all the non-hyperbolic subcases for C . In the end wewill present a diagram in α − ν plane to show the complete stability analysis as usual.0 TABLE XV: C (Reduced System) ν α Center Manifold Reduced System1 √ id to subcase (a) of C id to subcase (a) of C − √ id to subcase (b) of C id to subcase (b) of C = 1 √ id to subcase (f) of C id to subcase (f) of C = 1 − √ id to subcase (h) of C id to subcase (h) of C α = − √ , √ ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 0 For the last subcase, P = α + α ( α −√ − α )12(2 α − α + α ( α + √ − α )12(2 α − − α + α ( α −√ − α )12(2 α − − α + α ( α + √ − α )12(2 α − . We find again that the stability diagram for C and C is totally same. Hence it will bepresented at the end of the next subsection of C . J. Critical Point C This is the last subcase. In this case too it is necessary that α = 0 . Also, A = − ν α ν α α − α . The Jacobian matrix of the system (25) at C has the characteristic polynomial λ + (3 ν −
32 ) λ + [ 18 να − α ] λ + 27(2 α − ν − α = 0 . (37)which has eigenvalues as 3(1 − ν ) , − α +3 √ − α α and − α − √ − α α . The stability analysis when ν = 1 and α = − √ , √ (hyperbolic cases) are easy again.Therefore we will present a table containing all the non-hyperbolic subcases for this criticalpoint. At the end we will present the stability diagram in α − ν plane.1 TABLE XVI: C (Reduced System) ν α Center Manifold Reduced System1 √ id to subcase (a) of C id to subcase (a) of C − √ id to subcase (b) of C id to subcase (b) of C = 1 √ id to subcase (f) of C id to subcase (f) of C = 1 − √ id to subcase (h) of C id to subcase (h) of C α = − √ , √ ¯¯ y = O (¯¯ x ) , ¯¯Ω m = O (¯¯ x ) ¯¯ x ′ = 0 For the last subcase, P = α + α ( α −√ − α )12(2 α − α + α ( α + √ − α )12(2 α − α − α ( α −√ − α )12(2 α − α − α ( α + √ − α )12(2 α − . Lastly, we provide the stability diagram in the α − ν plane for C and C and end this section : −3 −2 −1 0 1 2 3−2−1.5−1−0.500.511.52 α ν FIG. 3: Stability Diagram ( C , C ) Here it is necessary that α = 0 . The two squares in the top corners bounded by the lines ν = 1 , α = √ and ν = 1 , α = − √ respectively are cyan-shaded. They represent the parametervalues for which the system (25) is stable. The rest of the region represents saddle system.The’o’ marked line represents center-stability.At the end of this section, we will present the phase plane diagrams of the autonomous system(25) for various values of the parameter ν and α in figures (4)-(7) The last diagram (8) depictsthe vector fields of the autonomous system (25).2 −1 0 1−1012 x y C , α =0, ν =0 −1 0 1−101 x y C , α =1, ν =0−1 0 1−101 x y C , α =0, ν =1 −1 0 1−1012 x y C , α =1, ν =1−1 0 1 2−1012 x y C , α =0, ν =2 −1 0 1−1012 x y C , α =0.75, ν =2.12 FIG. 4: C , C y C , α =−3, ν =0 0.5 1 1.5−0.500.51 x y C , α =−3, ν =1.50.5 1 1.5−0.500.5 x y C , α =3, ν =−2.5 0.5 1 1.5−0.500.5 x y C , α =2.5, ν =20.5 1 1.5 2−0.500.5 x y C , α =−3.5, ν =2 0.5 1 1.5−0.500.51 x y C , α =−0.75, ν =2.12 FIG. 5: C , C −0.5 0 0.5−0.500.5 x y α =0, ν =0.5 −0.5 0 0.5 1−0.500.51 x y α =−0.5, ν =1.5−0.5 0 0.5 1−0.500.5 x y α =0.5, ν =1 −0.5 0 0.5 1−0.500.5 x y α =0, ν =1 FIG. 6: C −1 0 1−1012 x y C , α =3, ν =1.5 −1 0 1−101 x y C , α =−2.12, ν =1.25−1 0 1 2−101 x y C , α =2.12, ν =1.25 −1 0 1−1012 x y C , α =−1.41, ν =1 FIG. 7: C , C , C , C −2 −1 0 1 2−2−1012−1.5−1−0.500.51 xy z FIG. 8: Vector field diagram of (25)
V. COSMOLOGICAL INTERPRETATIONS AND CONCLUSION
The present work deals with a cosmological model consisting of three non-interacting fluidsnamely the baryonic matter in the form of perfect fluid ( p b = ( ν − ρ b ), dark matter in the formof dust and dark energy as a scalar field respectively. This cosmological model has been studiedin the framework of dynamical system analysis by forming the evolution equations (Einstein’sfield equations) into an autonomous system with suitable transformation of the variables. Intable I, it has been shown that there are 10 equilibrium points ( C - C ) of the autonomoussystem. The values of the relevant cosmological parameters at the equilibrium points have alsobeen presented in table II.The equilibrium point C is completely dominated by the baryonic matter and as expected instandard cosmology, the model will be in decelerating phase if the baryonic fluid is normal (ie,non-exotic: ν > ) and it will be in the accelerating era for exotic nature of the matter (ie, ν < ).The equilibrium points C and C have identical nature and both of them are not interestingfrom the cosmological viewpoint as there is deceleration (of unusual magnitude) when matteris completely dominated by DE. The point C describes a known result of standard cosmology:In the dust era there is deceleration with the value of the deceleration parameter . The equilibrium point C and C correspond to cosmological era dominated by the darkenergy(DE). The condition ( α <
3) restricts the DE to be exotic and there is acceleration. Bychoosing α appropriately it is possible to match the model with the recent observations.5Both the equilibrium points C and C in the phase space have identical cosmological behaviors.For real points in the phase space as well as for realistic baryonic fluid, ν should be restricted as0 < ν < . The cosmological model is dominated by both the DE and the baryonic matter. Infact, the cosmic phase described by these phase space points (ie C and C ) has dominance ofbaryon over DE if < ν < < ν < then DE dominatesthe cosmic evolution and there is acceleration.The scaling solutions represented by the equilibrium points C and C are dominated by boththe dark matter (DM) and DE. But they are not of much physical interest as they alwayscorrespond to dust era of evolution.Thus, from the dynamical system analysis, the equilibrium points describe various cosmo-logical eras and some of them are interesting from the cosmological viewpoints with recentobserved data. Therefore, one may conclude that from complicated cosmological models onemay get cosmological inferences without solving the evolution equations, rather by analyzingwith dynamical system approach and the present work is an example of this inference. VI. REFERENCES [1] S. J. Perlmutter et al. [Supernova Cosmology Project Collaboration], “Measurements of Omega andLambda from 42 high redshift supernovae”,
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