aa r X i v : . [ a s t r o - ph ] M a y Unified model of Baryonic matter and dark components
L.P. Chimento and M´onica Forte Dpto. de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, Pabell´on I, 1428 Buenos Aires,Argentina (Dated: October 30, 2018)We investigate an interacting two-fluid cosmological model and introduce a scalar field representa-tion by means of a linear combination of the individual energy densities. Applying the integrabilitycondition to the scalar field equation we show that this ”exotic quintessence” is driven by an expo-nential potential and the two-fluid mixture can be considered as a model of three components. Thesecomponents are associated with baryonic matter, dark matter and dark energy respectively. We usethe
Simon, V erde & Jimenez (2005) determination of the redshift dependence of the Hubble param-eter to constrain the current density parameters of this model. With the best fit density parameterswe obtain the transition redshift between non accelerated and accelerated regimes z acc = 0 .
66 andthe time elapsed since the initial singularity t = 19 . Gyr . We study the perturbation evolution ofthis model and find that the energy density perturbation decreases with the cosmological time.
PACS numbers:Keywords:
I. INTRODUCTION
Astrophysical data suggests that the Universe is ac-celerating [1, 2]. This acceleration may be explained byvery different models, among them, the simplest one isthe ΛCDM [3, 4]. It assumes a cosmological constantarising from the energy density of the zero point fluctu-ations of the quantum vacuum and cold dark matter inform of pressureless dust. While it fits rather well allthe observational constraints, the small positive value ofthe energy density of the vacuum remains as an explana-tory challenge for physics today. See also Refs. [5, 6] forobjections to this interpretation. The next step is to pro-pose a dark energy component that may vary with timeand that is generally modeled by a scalar field. Most ofthese models assume that dark matter and scalar fieldcomponents evolve independently. Again, this is not thesolution because in the analysis of these models throughSNIa or WMAP data, best fit models for one set dataalone is usually ruled out by the other set at a large con-fidence limit [7]. As these conclusions are valid only forstandard models, where dark energy and dark matter aredecoupled, many papers have been devoted to interactingmodels [8]. Certain models conceive the interaction as atime-variable dark mass, evolving with an inverse powerlaw potential or an exponential potential, [9], [10], [11],[12].In this paper we clarify this point and establish anexactly solvable model with a smooth transition from amatter dominated phase to a period of accelerated expan-sion. We introduce an interacting two-fluid cosmologicalmodel and investigate the effects of imposing the integra-bility condition of the whole equation of conservation.It forces the dynamic of the model to be governed bya modified Friedmann equation with three components.One of them, associated with an exponential potential,drives an exotic scalar field, (exotic quintessence). Here,we show that the problem of an accelerating universe can be realized in a comparatively simple manner within theframework of general relativity. Finally, the perturbationevolution of the model is investigated.Our paper is organized as follows. In section II we con-sider the general interacting two-fluid cosmological modeland introduce the exotic quintessence. There, we obtainthe evolution equation of the exotic field, find their im-plicit solutions, build the modified Friedmann equationand show the asymptotic behavior of the scale factor byusing stability analysis. In section III we introduce bary-onic matter, dark matter and dark energy componentsand show that dark components satisfy separately an ef-fective equation of conservation with variable equationof state. In section IV we find confidence regions for theparameters of the model by using the Hubble functionH(z) data, the age of universe and the redshift of thetransition from non-accelerated to accelerated regime forthe best fit model. In section V we present the equationsgoverning the perturbations of the model. In section VIwe express the conclusions.
II. EXOTIC QUINTESSENCE
Cosmological models are based in the Einstein equa-tions of the gravitational field where the source in-cludes different kinds of matter known, for instance pro-tons, neutrons, photons, neutrinos, etc., as well as non-relativistic non-baryonic cold dark matter and dark en-ergy. We embark in a less ambitious project by con-sidering a model consisting of two perfect fluids withan energy momentum-tensor T ik = T (1) ik + T (2) ik . Here T ( n ) ik = ( ρ n + p n ) u i u k + p n g ik , where ρ n and p n are theenergy density and the equilibrium pressure of fluid n and u i is the four-velocity. Assuming that the two fluidsinteract between them in a spatially flat homogeneousand isotropic Friedmann-Robertson-Walker (FRW) cos-mological model, the Einstein’s equations reduce to twoalgebraically independent equations:3 H = ρ + ρ , (1)˙ ρ + ˙ ρ + 3 H ((1 + w ) ρ + (1 + w ) ρ ) = 0 , (2)where a(t) is the FRW scale factor and H ( t ) = ˙ a/a isthe Hubble expansion rate. We introduce an equationof state for each fluid component w n = p n /ρ n n = 1 , w , w are constantsand ρ , ρ >
0. This simplified model leads to a reductionof the number of fundamental parameters required to de-scribe observations. It can be considered as an advantagefrom the computational point of view. We choose unitssuch as the gravitational constant is set to 8 πG = 1 and c = 1.The whole equation of conservation (2) shows the inter-action between both fluid components allowing the mu-tual exchange of energy and momentum, meaning that,there will be no local energy-momentum conservationfor these fluids separately. Then, we assume an over-all perfect fluid description with an effective equation ofstate, w = p/ρ = − H/ H −
1, where p = p + p and ρ = ρ + ρ . So that, from Eqs. (1)-(2) we get − H = (1 + w ) ρ + (1 + w ) ρ = (1 + w ) ρ. (3)To avoid an eventual super acceleration of the universe,which could lead to a “big rip” singularity, we choose w > − w > − w ) ρ = − H > φ representationof the interacting two-fluid mixture˙ φ = (1 + w ) ρ + (1 + w ) ρ , (4)with ˙ φ = − H . The dynamic equation for the scalarfield is obtained from the equation of conservation (2)¨ φ + 32 (1 + w ) H ˙ φ + w − w ρ ˙ φ = 0 . (5)It can be integrated by setting the interaction betweenthe two fluids by ˙ ρ + A ˙ φρ = 0 , (6)where A is a new constant parameter of the model. In-tegrating Eq. (6), we find that the energy density ofthe second fluid can be associated with an exponentialpotential ρ = ρ H e − A ( φ − φ ) = V ( φ ) , (7)where ρ is a positive integration constant and H , φ are the present values of Hubble constant and scalar field.From Eqs. (4), (5) and (7) we obtain the total en-ergy density and pressure of the fluid mixture and thedynamical equation for the scalar field ρ = ˙ φ w + w − w w V, (8) p = w ˙ φ w − w − w w V, (9)¨ φ + 32 (1 + w ) H ˙ φ + w − w dVdφ = 0 . (10)with ρ + p = ˙ φ . These equations are different than theconventional ones describing quintessence, in contrast,define an exotic scalar field. When the interacting two-fluids system is related to the scalar field in the form ρ = ˙ φ / ρ = V ( φ ), with equations of state p = ρ and p = − ρ , meaning that w = 1 (stiff matter) and w = − (vacuum energy) , the exotic scalar field reducesto quintessence. Then, due to the interactions betweenthe two-fluid components the energy-momentum tensorconservation of the system, as a whole, is equivalent tothe Klein-Gordon equation. For any other interactingtwo-fluid mixture, the cosmological model contains anexotic quintessence field φ driven by an exponential po-tential.Using the integrability condition (6) in the field equa-tion (5), its first integral is given by˙ φ = AH + cH (1 + z ) w ) / , (11)and φ = φ − A ln (1 + z ) − cH Z z (1 + z ) (1+3 w ) / H dz, (12)where c is an arbitrary integration constant, z = − a /a is the redshift parameter and a is the present scalefactor. This model is finally closed when the Eq. (11)is inserted into the energy density (8) [14] . Hence, theFriedmann equation (1) reads3 H = 13(1 + w ) − A h cAH H (1 + z ) w ) / +3( w − w ) ρ + 3 c H (1 + z ) w ) i (13)As a consequence of the linear term in the expansionrate H , this equation can be seen as amodified Fried-mann equation. Its solution gives the scale factor andthe model we propose, containing exotic quintessence,could be formally solved.In order to obtain the asymptotic behavior of the scalefactor it will be useful to find the constant solutions ofthe dynamical equation for the overall equation of state˙ w = − H ( w − w ) r A w ) − (1 + w ) ! , (14)and investigate their asymptotic stability. This equa-tion has two stationary solutions w and A / − w = − A < w ), we find that w is an unstable solu-tion while A / − w = − H/ H −
1, meaning that the universe beginsto evolve from an unstable phase as it were dominatedby the first fluid w at early times, a ≈ t / w ) , andends in a stable expanding phase dominated by the ex-ponential potential, a ≈ t /A . The latter becomes anexpanding accelerated phase when the slope of the po-tential satisfy the inequality A < III. DYNAMIC OF BARYONIC AND DARKCOMPONENTS
The integrability condition (6) can be considered asan effective equation of conservation for the second fluid.This allows us to identify ρ with the energy density ofthe dark energy component and w de = − A ˙ φ/ H withits effective equation of state. Expressing the latter interm of the exotic field φ we get w de ≡ − A ˙ φ H = − A ˙ φ q
3[ ˙ φ − w V ] , (15)and the relation w = − A (1 + w de ) , (16)linking the overall and dark energy equations of state.Also, for convenience we write ρ = ρ H (1 + z ) λ with3 λ ln (1 + z ) = − A ( φ − φ ) . (17)The H linear term in the Eq. (13) is adequate to describenon-relativistic non-baryonic cold dark matter compo-nents whose energy-momentum tensor is approximatelydust-like. Finally, the baryonic matter is introduced bysetting w = 0 in the third term of the Eq. (13). Mak-ing these identifications the modified Friedmann equation(13) becomes H H = Ω dm (1+ z ) / HH +Ω de (1+ z ) λ +Ω b (1+ z ) , (18)where Ω dm = 2 cA − A , (19)Ω de = − w ρ − A , (20)Ω b = c − A , (21)are the present dark matter, dark energy and baryonicmatter density parameters respectively. As these are con-strained according to Ω dm + Ω dm + Ω b = 1, we concludethat ( c + A ) − w ρ = 3 . (22)During the accelerated epoch of the universe ¨ a >
0, theSEC is violated and ρ +3 p = ρ +(1+3 w ) ρ < − < w < − /
3. Hence, one finds that the conditions A < cA > cA > A ˙ φ > ρ is a Liapunov function and the solu-tion ρ = 0, of Eq. (6), is asymptotically stable. Also,from Eqs. (4), (7) and (11) the energy density of the firstfluid has a vanishing limit in the remote future. Hence,this general model is viable and it does not contradictbasic cosmological conjectures.From Eqs. (19) and (21) we can express the parameter A and the integration constant c in terms of the presentdensity parameters Ω dm , A = ± √ dm p Ω dm + 4Ω b , (23) c = ± √ b p Ω dm + 4Ω b (24)besides, − w ρ = 12Ω de Ω b Ω dm + 4Ω b . (25)Finally the original problem of the interacting two-fluidmixture governed by system equations (1)-(2) is equiva-lent to an effective model with a “three-fluid” mixture.So that, the effective dynamical equations of our modelread 3 H = ρ dm + ρ de + ρ b , (26)˙ ρ dm + 3 H (1 + w dm ) ρ dm = 0 , (27)˙ ρ de + 3 H (1 + w de ) ρ de = 0 , (28)˙ ρ b + 3 Hρ b = 0 , (29)where ρ dm = 3 H Ω dm (1 + z ) / H, (30) ρ de = 3 H Ω de (1 + z ) λ ) , (31) ρ b = 3 H Ω b (1 + z ) , (32)are the effective energy densities of dark and baryoniccomponents and w de = 2Ω b Ω dm + 4Ω b (cid:20) − dm H H (1 + z ) / (cid:21) , (33)is the effective equation of state of the dark energy. Also,we find the following relation w dm = 12 (cid:20) − A (1 + w de ) (cid:21) , (34)between the effective equations of state of dark compo-nents. So that, the knowledge of them determines theremaining ones including w . To obtain the above effec-tive dynamical equations of the model we have taken intoaccount that ρ de ∝ ρ . Hence, we have identified theequation of conservation (6) with (28) to express, afterusing (8), the effective w de for dark energy in terms ofthe observed density parameters, the present Hubble ex-pansion rate and the redshift parameter.In our model the stationary solutions w e = 0 at earlytimes and w l = w lde = − A / w ede = − A/ √ w lde = − A / w edm = 0, w ldm = ( − A / /
2. As the observed density parame-ter satisfies the condition Ω dm < b , then A < w ede is an unstablesolution at early times and w lde becomes asymptoticallystable at late times. Here, the evolution of the geome-try represents a universe that begins to evolve as it werematter dominated at early times and ends in an accel-erated phase dominated by the dark energy component.As the accelerated epoch begins at ¨ a = 0 or w = − / z acc = − (cid:20) Ω b Ω de ( √ − A ) (cid:21) w ( z acc ) / , (35)On the other hand, the Eq. (33) allow us to find theelapsed time t from the creation t = Ω dm b H " √ A − , (36)where we have used that the scale factor behaves as a ≈ a ( t/t ) / , at early times. IV. OBSERVATIONAL CONSTRAINTS
In Ref. [15] it was used the recently published Hub-ble function H ( z ) data [SV&J(2005)][16], extracted fromdifferential ages of passively evolving galaxies. This is in-teresting for, among other reasons, the function is not in-tegrated over, in contrast to standard candle luminositydistances or standard ruler angular diameter distancesSince the Hubble parameter depends on the differen-tial age of the Universe as a function of z in the form H ( z ) = − (1 + z ) − dz/dt , it can be directly measuredthrough a determination of dz/dt. In the procedure ofcalculating the differential ages, Simon et al. have em-ployed the new released Gemini Deep Deep Survey [17]and archival data [18], [19] to determine the 9 numeri-cal values of H ( z ) in the range 0 < z < .
8, and theirerrors [see Table 1]. These data will be inserted in ourEqs. (17) and (18) to derive restrictions on the range ofpossible values for the density parameters (19)-(21).We adopt the prior H = 72 kms − M pc − for H ( z =0). It is exactly the mean value of the results from theHubble Space Telescope key project [20] and consistentwith the one from WMAP 3-year result [21]. z H ( z ) 1 σ ( kms − Mpc − ) uncertainty ± ± . ± ± . ± . ± . ± . ± ± . Fig. 1: Observational H ( z ) with 1 σ uncertaintiesfrom SV & J (2005) and the best fit theoretical H ( z ). The parameters of the model can be determined byminimizing the function χ (Ω dm , Ω de ) = X i =1 [ H th (Ω dm , Ω de ; z i ) − H ob ( z i )] σ ( z i ) (37)where H th (Ω dm , Ω de ; z i ) = H (1 + z i ) / × Ω dm s(cid:18) − Ω dm (cid:19) + Ω de (cid:2) (1 + z i ) λ − − (cid:3) (38)is the predicted value for the Hubble parameter, obtainedfrom Eq. (18), and λ is calculated from Eqs. (12) and(17). H ob is the observed value of H at the redshift z i , σ isthe corresponding 1 σ uncertainty, and the summation isover the 9 observational H ( z i ) data points at redshift z i [22]. Also, we adopt the prior Ω b = 0 .
05 [23].We find a local minimum of χ for Ω dm = 0 . de = 0 . χ = 8 . H ( z ) data in Fig. 1 with error bars and the theo-retical line corresponding to the best fit parameter. The Fig. 2: Acceleration vs. redshift for the best fitmodel. Ω dm - Ω d e X Ω dm = de = OPEN MODEL CLOSED MODEL Ω dm - Ω d e Fig. 3: The 1 σ and 2 σ confidence regions, (inside andbetween the elliptic contours) for Ω dm and Ω de from SV & J (2005). The cross is the best fit model. Thestraight line corresponds to the flat cosmology makingthe separation between open and closed universes. z - - - ww dm w de Fig. 4: State parameters for dark matter, dark energyand overall fluid. - ρ b ρ dm ρ de Fig. 5: Energy densities (in units of 3 H ) vs. redshiftz Ruled out from oldest stellar ageAllowed from oldest stellar age Ρ dm H t Fig. 6: The age of the universe (in units of H − ).The solid line corresponds to our model and thedashed line represents the ΛCDM one. Fig. 2 shows that the Universe begins to accelerate on z ∼ .
66. A similar result can be obtained from ΛCDMflat cosmology when the density parameters are Ω m = 0 . Λ = 0 . dm - Ω de plane. The true values of those param-eters are inside the inner ellipse or between both ellipseswith 68 . . w dm = p dm /ρ dm , dark energy w de = p de /ρ de and overallfluid w = p/ρ . According to the previous stability analy-sis predictions we see that the asymptotic behavior of theoverall state parameter varies from w ∼ w = − A / ∼ − . z .We calculate the age of the universe (36) with the bestfit parameters and find that t = 19 . Gyr . In Fig.6 we plot the time elapsed (in units of H − ), since theinitial singularity to present days, for our model and theflat ΛCDM model, as a function of the matter density.We also show the border t = 11 Gyr coming from thebound of the oldest stellar ages. The age of the universein this coupled scenario tends to be much higher whencompared with the ACDM case [25]
V. LINEAR PERTURBATIONS
Cosmological models with two interacting fluids havebeen investigated with the purpose of describe the evolu-tion of dark components. There the energy momentumtensor of the interacting components is not separatelyconserved. Usually these cosmological model are pre-sented with interacting matter species have a non con-stant equation of state parameter [26] or with DE havinga constant equation of state parameter coupled to DM[27],[28]. However, in the model we are investigating theinteraction between the two fluids is setting by Eq. (6).For this choice the field equation can be integrated, gen-eralizing the case of quintessence driven by the exponen-tial potential. Then, it will be interesting to investigatethe evolution of the density perturbation.In the synchronous gauge the line element is given by: ds = a ( τ )[ − dτ + ( δ ij + h ij ) dx i dx j ] , (39)where the commoving coordinate are related to theproper time t and position r by dτ = dt/a , d x = d r /a and h ij is the metric perturbation. The scalar mode of h ij is described by the two fields h ( k , τ ) and η ( k , τ ) inthe Fourier space, h ij ( x , τ ) = Z d ke i k · x (cid:20) ˆ k i ˆ k j h + ( ˆ k i ˆ k j − δ ij ) η (cid:21) . (40)with k = k ˆ k . The Einstein equations to linear order ink-space, expressed in terms of h and η , are given by thefollowing four equations [29]: k η − a ′ a h ′ = 4 πGa δT , (41) k η ′ = 4 πGa ( ρ + p ) θ, (42) h ′′ + 2 a ′ a h ′ − k η = − πGa δT ii , (43) h ′′ + 6 η ′′ + 2 a ′ a ( h ′ + 6 η ′ ) − k η = − πGa ( ρ + p ) σ. (44)Here, the quantities θ and σ are defined as ( ρ + p ) θ = ik j δT j , ( ρ + p ) σ = − ( k i k j − δ ij / ij and Σ ij = T ij − δ ij T kk / T ij .In addition, θ is the divergence of the fluid velocity θ = ik j v j and ′ means d/dτ .Let us consider a fluid moving with a small coordinatevelocity v i = dx i /dτ , then, v i can be treated as a pertur-bation of the same order as energy density, pressure andmetric perturbations. Hence, to linear order in the per-turbations, the energy-momentum tensor, with vanishinganisotropic shear perturbation Σ ij , is given by T = − ( ρ + δρ ) , (45) T i = ( ρ + p ) v i = − T i , (46) T ij = ( p + δp ) δ ij . (47)For a fluid with equation of state p = wρ , the per-turbed part of energy-momentum conservation equations T µν ; µ = 0 in the k-space leads to the equations δ ′ = − (1 + w )( θ + h ′ − H ( δpδρ − w ) δ, (48) θ ′ = −H (1 − w ) θ − w ′ w θ + δp/δρ w k δ, (49)where δ = δρ/ρ and H = a ′ /a = aH = ˙ a . Besides, usingequations (41), (43), (45) and (47) we arrive at h ′′ + H h ′ + 3 H (cid:18) δpδρ (cid:19) δ = 0 . (50) We have showed that our interacting two-fluid modelcan be associated with an overall perfect fluid descrip-tion based in an effective equation of state w = ( w ρ + w ρ ) / ( ρ + ρ ). Hence, we investigate the asymptoticregimes at early and late times assuming nearly constantequations of state w ≈ w e = 0 and w ≈ w l = − A / w ≈
0, theeffective fluid perturbations evolve similar to those of or-dinary dust with ˙ θ = θ = 0, and from Eqs. (48-50) weobtain ¨ δ + 2 H ˙ δ − H δ = 0 (51)and δ = c t − + c t / , where c and c are arbitraryintegration constants. In this dust dominated era theperturbation grows as δ ≈ a showing an initial unsta-ble phase and compatible with the observation that theprimordial universe would have tiny perturbations whichseed the formation of structures in the later universe.At late times, we are interested to find the evolutionof the linear scalar perturbations for any mode k . Tothis end we write the second order differential equationfor the density perturbation δ and the first order differ-ential equation for the divergence of the fluid velocity θ ,evaluating them on the asymptotically stable equation ofstate w ≈ w l . In this case, from Eqs. (48-50) we get: δ ′′ + H δ ′ + (cid:20) w l k −
32 (1 + w l )(1 + 3 w l ) H (cid:21) δ +3 w l (1 + w l ) H θ = 0 , (52) θ ′ = −H (1 − w l ) θ + w l w l k δ. (53)Taking into account that in the late time regime thescale factor behaves as a ∝ t / w l ) we can calculatethe conformal time τ , a and H = a ′ /aτ ∝ t (1+3 w l ) / w l ) (54) a ∝ τ / (1+3 w l ) (55) H = 2(1 + 3 w l ) τ . (56)From Eqs. (52) and (53) the perturbation evolution be-comes mode dependent with the k / H term, and for lowenergy modes their solutions can be obtained assuming apower law dependence of the perturbations with the scalefactor, δ ∝ a n and θ ∝ a s . In this case the approximatesolutions for w l = − A / − .
63 are given by θ ≈ θ a . (57) δ ≈ δ a . + δ a . + θ a . , (58)where θ , δ and δ are integration constants while θ isa function of θ and w l . This shows that the couplingto θ in Eq. (52) can be neglected for all scales we areinterested. Finally, expressing the Eq. (52) in term ofconformal time we get δ ′′ + 21 + 3 w l δ ′ τ + (cid:20) wk − w l w l τ (cid:21) δ = 0 . (59)The general solution of the latter equation in terms ofthe Bessel functions is δ = τ b h c J ν ( k √ w l τ ) + c J − ν ( k √ w l τ ) i , (60)with b = − w l w l ) , ν = ± w l w l ) . (61)At late times, it can be approximated by the two firstterms of the Eq. (58) showing that the energy densityperturbation decreases for large cosmological times formodes satisfying the condition k / H ≪
1. For highenergy modes, k / H ≫
1, the perturbation δ ≈ a . , (62)decreases but slowly that the low energy modes. Thisresults can be understood writing the Eq. (52) as theequation of motion of a dissipative mechanical system byusing the analogy with the classical potential problem ddτ (cid:20) δ ′ V ( δ ) (cid:21) = − D ( δ, δ ′ ) , (63)where V ( δ ) = w l k (cid:18) − H H (cid:19) δ , (64) D ( δ, δ ′ ) = 32 (1 + w l )(1 + 3 w l ) HH ′ δ + H δ ′ , (65) H = 2 w l k w l )(1 + 3 w l ) . (66)The potential V has an extreme at δ = 0, it is max-imum for H < H or a minimum for H > H . Onthe other hand, assuming that the perturbation dependson the scalar factor in the form δ ∝ a n , we find that D ≈ . H δ >
0. Hence, for any mode k the pertur-bation begins to grow at early times for H < H , whileat late times for H > H , the function inside the squarebracket in Eq. (63) is a Liapunov function and the per-turbation decreases asymptotically reaching δ = 0 in thelimit t → ∞ . VI. CONCLUSIONS
We have shown an interacting two-fluid cosmologicalmodel that allows us to reproduce the accelerated behav-ior of our universe and its probable age . The model gives rise to an exotic scalar field dubbed exotic quintessencewhich reduces to quintessence when one fluid is associ-ated with stiff matter and the other with vacuum energy.Setting the interaction between the two fluids by Eq. (6),the field equation is integrated, generalizing the case ofquintessence driven by the exponential potential and, theequation governing the scale factor (13) looks like a mod-ified Friedmann equation.We have obtained the evolution equation for the over-all equation of state of the model and showed the asymp-totic behavior of the scale factor i.e., the universe be-gins from an unstable phase dominated by the first fluid, a ≈ t / w ) , and ends in a stable expanding phasedominated by the exponential potential, a ≈ t /A . Thelatter becomes accelerated when the exponential poten-tial slope satisfy A <
2. Setting w = 0, the scale factorinterpolates between pressureless matter and dark energyphases.Using the Hubble function H ( z ) data from Table 1we minimize the χ function (37), obtaining the bestfit densities parameters Ω dm = 0 . +0 . − . and Ω de =0 . +0 . − . with a reduced χ = 1 .
24. These results areconsistent with those found in the literature, see for in-stance Ref. [30] for null coupling (Ω dm = 0 . +0 . − . ) orwith the result obtained in Ref. [31], (Ω dm = 0 . +0 . − . )through mean relative peculiar velocity measurements forpairs of galaxies. With our best densities parameters andthe priors for H = 72 kms − M pc − and Ω b = 0 .
05, weobtain the theoretical H ( z ) function, plotted togetherwith the SVJ(2005) experimental data in Fig. 1. In Fig.2 we plot the acceleration of the model as a function of theredshift and find the transition from the non acceleratedphase to the accelerated one around z acc = 0 .
66 [24]. Ourvalue agrees with the result obtained by a nearly modelindependent characterization of dark energy properties asa function of redshift, ( z acc = 0 . +0 . − . ,[32]). The prob-lem of why an accelerated expansion should occur nowin the long history of the universe seems to be naturallydressed in our model. Considering the age of the uni-verse, we take into account that the age of the oldest stel-lar objects have been constrained for instance, by usinga distance-independent method [33], ( t = 13 . ± Gyr for Globular clusters in the Milky Way) and the whitedwarfs cooling sequence method [34] ( t = 12 . ± . Gyr for the globular cluster M4). Then, the age of universeneeds to satisfy the lower bound t > − Gyr . Thiscondition is fulfilled by our model with t = 19 . Gyr ,as it can be seen in Fig. 5.The energy density perturbation of the model growsin the first stage of the universe showing that initial in-stabilities in the primordial universe could leads to theformation of structure in the later universe. At late timeswe have found a Liapunov function which indicates thatthe perturbation decreases asymptotically reaching δ = 0in the limit t → ∞ . Acknowledgments
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