aa r X i v : . [ a s t r o - ph . C O ] J un Prepared for submission to JCAP
Interacting warm dark matter
Norman Cruz, a Guillermo Palma, a David Zambrano a and ArturoAvelino b a Departamento de F´ısica, Facultad de Ciencia,Universidad de Santiago de Chile,Casilla 307, Santiago, Chile. b Departamento de F´ısica, DCI, Campus Le´on,Universidad de Guanajuato,CP. 37150, Le´on, Guanajuato, M´exico.E-mail: [email protected]
Abstract.
We explore a cosmological model composed by a dark matter fluid interactingwith a dark energy fluid. The interaction term has the non-linear λρ α m ρ β e form, where ρ m and ρ e are the energy densities of the dark matter and dark energy, respectively. The parameters α and β are in principle not constrained to take any particular values, and were estimatedfrom observations. We perform an analytical study of the evolution equations, finding thefixed points and their stability properties in order to characterize suitable physical regionsin the phase space of the dark matter and dark energy densities. The constants ( λ, α, β ) aswell as w m and w e of the EoS of dark matter and dark energy respectively, were estimatedusing the cosmological observations of the type Ia supernovae and the Hubble expansion rate H ( z ) data sets. We find that the best estimated values for the free parameters of the modelcorrespond to a warm dark matter interacting with a phantom dark energy component, witha well goodness-of-fit to data. However, using the Bayesian Information Criterion (BIC)we find that this model is overcame by a warm dark matter – phantom dark energy modelwithout interaction, as well as by the ΛCDM model. We find also a large dispersion on thebest estimated values of the ( λ, α, β ) parameters, so even if we are not able to set strongconstraints on their values, given the goodness-of-fit to data of the model, we find that alarge variety of theirs values are well compatible with the observational data used. Keywords:
Interacting model, phantom dark energy, warm dark matter ontents r w m = 0 52.4 Case w m > w e and w m The existence of a dark component with an exotic equation of state, i. e., with a ratio w = p/ρ negative and close to −
1, which drives an accelerated expansion, is consistent withdata coming from type Ia Supernovae (SNe Ia) [1], large scale structure formation (LSS) [2],cosmic microwave background radiation (CMB) [3], baryon acoustic oscillations (BAO) [4]and weak lensing [5].Cosmic observations show that densities of dark energy (DE) and dark matter (DM)are of the same order today, despite their different decreasing rates. To solve the so–calledcoincidence problem [6] an evolving dark energy field with a non-gravitational interactionwith matter [7] is proposed (decay of dark energy to dark matter). In this case both darkfluids interact via an additional coupling term in the fluid equations. In the current researchdifferent forms of coupling have been considered. Most of them study the coupling betweencold DM and DE. In general, the interactions investigated are particular cases the form λ m Hρ m + λ e Hρ e , where H is the Hubble parameter, and ρ m , ρ e are the dark matter anddark energy densities respectively [8].Nevertheless, non linear interactions of the form λ ρ m ρ e ρ m + ρ e were considered in [9]. Amore plausible interaction is inspired by the situation of two types of fluids interacting,where the interaction is proportional to the product of the powers of the energy density ofboth components. In this case, the interaction rate goes to zero as one or both densitiesbecome zero, and increases when each of the densities grows. We then consider in thispaper a general interaction of the form λρ αm ρ βe , where the parameter λ has dimensions of[ energy - density α + β − × time ] − . – 1 –his type of interaction was investigated in the framework of an holographic dark en-ergy [10] and in coupled quintessence [11, 12], where the evolution of the energy densitiesof the dark interacting components was investigated for different values of the parameters α and β . A cyclic scenario for the present situation ρ e ∼ ρ m was found as a possible solutionto the coincidence problem. The particular case α = β = 1 was studied in [13]. In this case,if energy is transferred from dark energy to dark matter ( λ > w e < − r = ρ m /ρ e require a phantom dark energy [14].Nevertheless, these stationary solutions do not guarantee a solution of the coincidenceproblem. For example, in [15] a DE modelled by a phantom field was studied in the frameworkof interaction terms proportional to a density. The results of this investigation indicatesthat in all cases the late-time solutions correspond to a complete DE domination and asa consequence the coincidence problem remains unsolved. A suitable coupling, with theform ρ e ρ m ρ e + ρ m , was chosen in [16] for a phantom field evolving under an exponential potential.An accelerated late-time phase with a stationary ratio of the energies densities of the twocomponents was found.The requirement of phantom DE to obtain stationary solutions can be seen as a kindof theoretical need for this type of matter in the framework of an interacting dark sector.From the observational point of view, phantom DE is supported by new SN Ia data [17] [18],analysis of the cosmic microwave background (CMB) and large-scale structure. This modelis also preferred by WMAP data combined with either SNe Ia or baryon acoustic oscillations(BAO). Nevertheless, from the theoretical point, fluids that violate the null energy condition ρ + p ≥ w e , its the late time behavior is characterized byan increasing energy density which becomes infinite in a finite time (Big Rip). The interactionof a phantom fluid with dark matter can prevent this type of behavior, leading to finite energydensities during the cosmic evolution and avoiding of future singularities.The aim of this paper is to study the time evolution of the dark sector densities whenthe interaction mentioned above is considered. This evolution is driven by a highly non-linearcoupled differential equations when the parameters α and β are left free. We solve numericallythese equations and also study analytically the stability of the fixed-points. Since it has beenargued that the second law of thermodynamics and Le Chˆatelier’s principle implies λ > r when the dark matter is assumed not to be dust.In section 3 we study the two coupled differential equations corresponding to continuityequations of both interacting fluids. We study analytically the fixed-points including theirstability. In section 4, we obtain numerical solutions using a numerical method of adaptivestep-size algorithm called Bulirsch-Stoer method. We explore the behavior of the fixed pointsvarying the parameters α , β and the EoS of both dark sector components. In section 5 weuse result from cosmological observation to find the best values for the free parameters,( α, β, ¯ λ, w m ), of the corresponding theoretical model. Finally, in section 6 the main resultsare summarized and different physical scenarios consistent with cosmological observationsare discussed. In the following we assume a flat FRW universe filled basically with the fluids of the darksector. We consider a warm dark matter of density ρ m and a dark energy component describedby the density ρ e . For simplicity we also assume that both fluids obey a barotropic EoS, sowe have p m = w m ρ m for the warm dark matter and p e = w e ρ e for the dark energy. In whatfollows we restrict our model to the late time of cosmic evolution, which implies that theothers components of the universe, like radiation and baryons are negligible. In this case thesourced Friedmann equation is given by3 H = ρ m + ρ e , (2.1)where 8 πG = 1 has been adopted. We will assume that the dark matter component isinteracting with the dark energy component, so their continuity equations take the form˙ ρ m + 3(1 + w m ) Hρ m = + Q (2.2)˙ ρ e + 3(1 + w e ) Hρ e = − Q , (2.3)where H = ˙ a/a is the Hubble parameter, and a ( t ) is the scale-factor. Here, an overdotindicates a time derivative. Q represents the interaction term, despite that we do not use aspecific functional dependence at this stage, we will only assume that Q do not change itssign during the cosmic evolution. Let us discuss briefly the behavior of the dark components in terms of an effective EoS drivenby the interacting term. Rewriting the continuity equation (2.2) in the usual form˙ ρ m + 3(1 + w m eff ) Hρ m = 0 , (2.4)where w m eff represents the effective EoS for the interacting dark matter, which is given by w m eff = w m − Q Hρ m . (2.5)– 3 –ote that the behavior of w m eff can be quite different if w m is non zero. With theusual assumption of cold dark matter ( w m = 0) and Q >
0, which implies that energy istransferred from dark energy to dark matter, some kind of exotic dark matter with a negativeEoS is driven, assuming, of course, that we are in an expanding universe
H >
0. For warmdark matter we would have, depending on the type of interaction considered and the strengthof the coupling constant appearing in Q , a possible change of the sign on the effective EoSduring the cosmic evolution. For the dark energy, the effective EoS is given by w Λ eff = w e + Q Hρ e . (2.6)For Q > w e = − w e < − r In order to address the coincidence problem in terms of the dynamics of the parameter r ≡ ρ m /ρ e , we will make in what follows a similar analysis along the line describes in ref. [14].The dynamics of the parameter r is given by˙ r = r (cid:18) ˙ ρ m ρ m − ˙ ρ e ρ e (cid:19) , (2.7)where the dot indicates derivative with respect to the cosmic time. Changing the timederivatives by a derivative with respect to ln a , which will be denoted by a prime, i.e.˙ ρ = ρ ′ H , eqs (2.2) and (2.3) go into ρ ′ m ρ m = − (1 + w m ) + Q Hρ m , ρ ′ e ρ e = − (1 + w e ) − Q Hρ e . (2.8)In terms of total density ρ = ρ e + ρ m , we obtain ρ ′ = − (cid:20) w m r + w e r + 1 (cid:21) ρ. (2.9)For the coincidence parameter r the evolution equation reads r ′ = r (cid:20) ( w e − w m ) + Q H (1 + r ) rρ (cid:21) . (2.10)Before specifying any particular type of interaction, we will discuss some general prop-erties of these equations. The critical point of Eq. (2.9) is given by r c = − (1 + w e ) w m + 1 . (2.11)If w m > r must be positive, it follows that w e < − r ′ = 0 in Eq.(2.10) and the valueof r c one finds ρ c = − ( w m − w e ) w m + 1 Q H (1 + w e ) . (2.12)Depending on the value of w m ≥ .3 Case w m = 0In this case, Eq. (2.12) simplifies to ρ c = w e w e + 1 Q H . (2.13)This case was analyzed in [14], concluding that a positive stationary energy density ρ c requires Q > w e < −
1. This means that independent on the interaction type, theexistence of critical points requires a positive exchange from dark energy to dark matter(DM). w m > (warm dark matter) For this case the expression for ρ c becomes ρ c = w e − w m w m + 1 Q H (1 + w e ) . (2.14)The condition w e < − Q . This result also holds for − < w m < t andsubstituting for ˙ ρ m and ˙ ρ e gives the auxiliary equation2 ˙ H = − (1 + w m ) ρ m − (1 + w e ) ρ e . (2.15)The acceleration is given by the relation ¨ a = a ( ˙ H + H ). From (2.1) and (2.15), weobtain ¨ a = − a ρ m (1 + 3 w m ) + (1 + 3 w e ) ρ e ) . (2.16)The condition ¨ a > ρ e > − w m w e ρ m . (2.17)Since we shall consider w e < − w m / | w e | − ρ m , ρ e ). Introducing the interaction term Q = λρ αm ρ βe in Eqs. (2.2) and (2.3) yields˙ ρ m = − w m ) Hρ m + λρ αm ρ βe (3.1)˙ ρ e = − w e ) Hρ e − λρ αm ρ βe . (3.2)The time evolution of the dark matter and dark energy densities is given by the highlynon-linear coupled differential equations (3.1) and (3.2). We rewrite them as follows:˙ ρ i = f i ( ρ m , ρ e ), (3.3)– 5 –here i = m, e . The functions f m and f e are defined by the following expressions f m ( ρ m , ρ e ) = −√ w m ) ρ m ( ρ m + ρ e ) / + λρ αm ρ βe (3.4) f e ( ρ m , ρ e ) = −√ w e ) ρ e ( ρ m + ρ e ) / − λρ αm ρ βe . (3.5)From numerical results we expect that the above equations have some non-trivial fixed-points (¯ ρ m , ¯ ρ e ), which we want to study analytically including their stability properties. Inspite of the non-linearities and according to ref. [23], it is still possible to analyze the stabilityof fixed points by using the linearized piece of the original differential equations˙ ρ i = X j a ij ρ j + R i ( ρ m , ρ e ) (3.6)where R i ( ρ m , ρ e ) includes all the non linearities of the original system of equations (3.4) and(3.5), provided the inequality | R i ( ρ m , ρ e ) | ≤ N X i ρ i ! + γ (3.7)is fulfilled in a neighbour region around the fixed points, for some positive constants N and γ . To achieve this goal we expand both the dark matter and dark energy densities aroundtheir fixed point values ( ¯ ρ m , ¯ ρ e ) as follows: ρ αm = (¯ ρ m + µρ m ) α = ¯ ρ αm + αµρ m + O (cid:0) µ ρ m / ¯ ρ m (cid:1) ρ βe = ( ¯ ρ e + µρ e ) β = ¯ ρ βe + βµρ e + O (cid:0) µ ρ e / ¯ ρ e (cid:1) ,it also holds( ρ m + ρ e ) / = (cid:16) ¯ ρ αm + ¯ ρ βe (cid:17) / + 12 (cid:16) ¯ ρ αm + ¯ ρ βe (cid:17) / µ ( ρ m + ρ e ) + O (cid:0) µ (cid:1) . Inserting the above perturbative expressions into the differential equation system oneobtains up to first order in µ : ˙ ρ i = X j a ij ρ j (3.8)or explicitly˙ ρ m = −√ ω m ) " ¯ ρ m ( ρ m + ρ e )2 ( ¯ ρ m + ¯ ρ e ) / + ρ m ( ¯ ρ m + ¯ ρ e ) / + λ h βρ e ¯ ρ αm + αρ m ¯ ρ βe i (3.9)˙ ρ e = −√ ω e ) " ¯ ρ e ( ρ m + ρ e )2 ( ¯ ρ m + ¯ ρ e ) / + ρ e (¯ ρ m + ¯ ρ e ) / − λ h βρ e ¯ ρ αm + αρ m ¯ ρ βe i . (3.10)The tree level values are implicitly defined by the relations √ w m ) ¯ ρ m (¯ ρ m + ¯ ρ e ) / = λ ¯ ρ αm ¯ ρ βe (3.11) √ w e ) ¯ ρ e (¯ ρ m + ¯ ρ e ) / = − λ ¯ ρ αm ¯ ρ βe , (3.12)– 6 –r equivalently, ¯ ρ e = − ω m ω e (3.13)¯ ρ m = " ( − β λ √ ω m ) β − (1 + ω e ) / − β ( ω e − ω m ) / (3 / − α − β ) − . (3.14)Now we are prepared to analyze the different numerical results obtained from the directnumerical solution of the system of eqs. (3.3). The numerical solutions were obtained byusing a very accurate numerical method of adaptive step-size algorithm called Bulirsch–Stoermethod, which will be explained in the next section.We will use the numerical values ω m = 0 and ω e = − .
1, which are of physical interestas we will discuss it in the next section. For these particular values, the above equations havea fixed point given by ¯ ρ e = 10¯ ρ m and¯ ρ m = (cid:18) λ β √ (cid:19) (3 / − α − β ) − (3.15)These fixed points are displayed in figures 3 and 4, and their loci agree remarkably wellwith the corresponding ones of the numerical results.In particular, for α = 0 . β = 1 . λ = 1, eq. (3.15) leads to the relation ¯ ρ m ≈ . α = 0 . β = 1 . λ = 1, eq (3.15) gives ¯ ρ m ≈ .
16, which again perfectly agree with thenumerical result shown in figure 4.Now we comeback to the linearized system of eqs. (3.9) and (3.10). We rewrite itexplicitly as the homogeneous system of differential equations˙ ρ m = − p (3) (1 + ω m ) " ¯ ρ m ρ m + ¯ ρ e ) / + ( ¯ ρ m + ¯ ρ e ) / + λα ¯ ρ βe ! ρ m + − p (3) (1 + ω m ) " ¯ ρ m ρ m + ¯ ρ e ) / + λβ ¯ ρ αm ! ρ e (3.16)˙ ρ e = − p (3) (1 + ω e ) " ¯ ρ e ρ m + ¯ ρ e ) / + ( ¯ ρ m + ¯ ρ e ) / − λβ ¯ ρ αm ! ρ e + − p (3) (1 + ω e ) " ¯ ρ e ρ m + ¯ ρ e ) / − λα ¯ ρ βe ! ρ m . (3.17)The stability of the fixed points of the above equations depend on the eigenvalues givenby characteristic equation associated to the system: k − ( a + a ) k + ( a a − a a ) = 0. (3.18)As it is well known, depending on the roots of eq. (3.18), the trajectories around thefixed-point (˜ ρ m = 0 , ˜ ρ e = 0) will be stable or unstable, (see for instance [23]).For example, if all roots of the characteristic eq. (3.18) have negative real parts, thenthe trivial solution ( ρ m ρ e ) T = (0 0) T of the linearized system and also of the non-linearsystem (3.3) is asymptotically stable. On the other side, if at least one of the roots of eq.(3.18) has a positive real part then both systems have an unstable fixed point at (0 0) T .– 7 –or the interacting model described by eq. (3.3), there are five physical parameters, w m , w e , λ , α and β , which should be chosen according to both, physical stability propertieson one side, and compatibility with observational data on the other side. The solutions of eq.(3.18) were numerically evaluated for different regions of the parameter space, and found theinteresting physical region defined by the inequalities: 0 ≤ ω m ≤ / − − ω m < ω e < − λ >
1, 0 . < α < .
222 and 0 . < α < .
02 for ω m = 0 and β > .
8. From it, wewill considered the subregion 0 ≤ ω m ≤ / ω e = − . λ = 1, α = 0 . β = 1. Inparticular, in these regions the condition of equation (3.7) holds, which guaranties that thelinear analysis of stability also apply to the non-linear differential equation considered. In this section we present numerical results obtained by using the Bulirsch–Stoer method tosolve the non-linear coupled system of eqs. (3.3). The Bulirsch–Stoer method uses an adap-tive step-size control parameter, which ensures extremely high accuracy with comparativelylittle extra computational effort. In the past, this method has proven to be very accuratefor solving non-linear differential equations [24]. In addition, we have computed the fixedpoint trajectories for different values of the five physical parameters w m , w e , λ , α and β .The corresponding trajectories within the stability region discussed in the previous sectionwill be shown in the figures below.The evolution of matter and energy densities ρ m and ρ e depends critically on the valueof the parameter w e . In particular, for w e > −
1, the system exhibits a smooth evolutionof the densities towards the fixed point. On the other side, if w e < − α and β leads to spiral orbits as it is shown in figures 3 and 4. w e and w m We will now consider the particular values α = β = 1, which has been claimed to give thebest fit to observations [10] for an interaction term of the form λρ αm ρ βe . As explained above,we have used the Bulirsch–Stoer method to solve numerically 3.3.For λ = 1, 0 < w m < / − . < w e < − . w m = 0 and w e within the interval [ − . , − . ρ m = 0 .
4. For the fixed value w e = − . w m within the interval [0 , / ρ e = 3 . w m = 0and w e = −
1, which correspond to a crossover value (see figure 1).
From the previous subsection, closed trajectories are obtained only for the region defined by w e < −
1. Depending on the values of α and β , converging or diverging spirals are obtained.In fact, α < β > α > β < ,0 0,1 0,2 0,3 0,4 0,50,51,01,52,02,53,03,54,04,55,0 e m w e = -1.00 w e = -1.02 w e = -1.04 w e = -1.06 w e = -1.08 w e = -1.10 e m Figure 1 . (Color on-line) For the parameters α = β = λ = 1, w m = 0 and w e = [ − . , − , ρ m when the range of w e is swept. The dots representthe fixed points given by eq. (3.13) and (3.14). Stability Region w m w e λ α β [0, 1/3] [-2- w m , -1] > > Table 1 . Stability region obtain from eq. (3.18).
In a spatially flat FRW Universe, the Hubble constraint and the conservation equations forthe matter and dark energy fluids are given by H = 8 πG ρ m + ρ e ) , (5.1)˙ ρ m + 3 H ( ρ m + p m ) = λρ α m ρ β e , (5.2)˙ ρ e + 3 H ( ρ e + p e ) = − λρ α m ρ β e , (5.3)where λ is a constant to quantify the strength of the interaction between the matter withthe dark energy. These equations can be written in terms of the scale factor a as– 9 – ,0 0,5 1,0 1,5 2,03456789 e m w m = 0 w m = 0.066 w m = 0.133 w m = 0.200 w m = 0.266 w m = 0.333 e m Figure 2 . (Color on-line) For the parameters α = β = λ = 1, w e = − . w m = [0 , − / ρ e when the range of w m is swept. The dots representthe fixed points given by eq. (3.13) and (3.14). dρ m da + 3 a ρ m (1 + w m ) = λ ρ α m ρ β e aH , (5.4a) dρ e da + 3 a ρ e (1 + w e ) = − λ ρ α m ρ β e aH . (5.4b)where λ > λ related with λ as λ = ¯ λ H ( ρ ) α − ( ρ ) β , (5.5)where ρ ≡ H / (8 πG ) is the critical density evaluated today and H is the Hubble con-stant. We define also the dimensionless parameter density ˆΩ i ≡ ρ i /ρ with i = m, e .Using these definitions, the Friedmann constraint equation (5.1) can be expressed as H = H p ˆΩ m + ˆΩ e , and the conservation eqs. (5.4) become d ˆΩ m da + 3 a ˆΩ m (1 + w m ) = ¯ λ ˆΩ α m ˆΩ β e a p ˆΩ m + ˆΩ e , (5.6a) d ˆΩ e da + 3 a ˆΩ e (1 + w e ) = − ¯ λ ˆΩ α m ˆΩ β e a p ˆΩ m + ˆΩ e . (5.6b)– 10 – e m Figure 3 . (Color on-line) For initial values ρ m = 0 . ρ e = 3 . α = β = λ = 1, w m = 0 and w e = − .
1, a closed orbit is shown –blue line– around its fixed point –blue dot–.Starting for the same initial values but changing the value of the power beta , spiral trajectories areshown for the evolution of the densities –red and green lines–, for β = 0 . β = 1 . Using the relationship between the scale factor and the redshift z given by a = 1 / (1 + z )we express eqs. (5.6) in terms of the redshift as d ˆΩ m dz = 11 + z " w m ) ˆΩ m − ¯ λ ˆΩ α m ˆΩ β e p ˆΩ m + ˆΩ e , (5.7a) d ˆΩ e dz = 11 + z " w e ) ˆΩ e + ¯ λ ˆΩ α e ˆΩ β e p ˆΩ m + ˆΩ e . (5.7b)We solve numerically this ordinary differential equation system (ODEs) for the functionsˆΩ m ( z ) and ˆΩ e ( z ), with the initial conditions ˆΩ m ( z = 0) ≡ Ω m0 = 0 . e ( z = 0) ≡ Ω e0 = 0 . E ≡ H/H becomes E ( z, w m , w e , ¯ λ, α, β ) = q ˆΩ m ( z ) + ˆΩ e ( z ) , (5.8)where ˆΩ m ( z ) and ˆΩ e ( z ) are given by the solution of the ODEs (5.7). We constrain the values of the free parameters ( w m , w e , ¯ λ, α, β ) using cosmological obser-vations that measure the expansion history of the Universe, which will be explained in thefollowing sections. We compute their best estimated values through a minimizing process of a– 11 – ,00 0,25 0,50 0,75 1,000123 e m = 0.9 = 0.8 = 0.7 = 0.6 = 0.3 = 0.1 Figure 4 . (Color on-line) Phase diagrams for ρ m and ρ e (starting from the same initial conditions)are given by the color lines for different values of α , and for λ = β = 1, w m = 0 and w e = − .
1. Theywere obtain by solving numerically eqs. (3.4) and (3.5). For this figure and all the following ones, theblack dashed line represent the fixed-point trajectory given in eqs. (3.13) and (3.14). The full blackline is the zero-acceleration line given by the expression (2.15) with ¨ a = 0, the region above this linecorrespond to an accelerating universe while the region below this line correspond to a decelerationuniverse. χ function defined below, and calculate the marginalized confidence intervals and covariancematrix of the five parameters. We use the type Ia supernovae (SNe Ia) of the “Union2.1” data set (2012) from the SupernovaCosmology Project (SCP) composed of 580 SNe Ia [25]. The luminosity distance d L in aspatially flat FRW Universe is defined as d L ( z ) = c (1 + z ) H Z z dz ′ E ( z ′ ) (5.9)where E ( z ) corresponds to the expression (5.8), and “ c ” to the speed of light in units ofkm/sec. The theoretical distance moduli µ t for the k-th supernova at a distance z k is givenby µ t ( z ) = 5 log (cid:20) d L ( z )Mpc (cid:21) + 25 (5.10)So, the χ function for the SNe is defined as χ ( w m , w e , ¯ λ, α, β ) ≡ n X k =1 (cid:18) µ t ( z k , w m , w e , ¯ λ, α, β ) − µ k σ k (cid:19) (5.11)– 12 – ,0 0,1 0,2 0,3 0,4 0,50,00,51,01,52,02,53,03,5 e m Figure 5 . (Color on-line) Phase diagrams for the parameters α = 0 . λ = β = 1, w m = 0 . w e = − .
1. Spiral evolution of the densities towards the fixed points is shown.
Best estimates,
SNe + H ( z ) w m w e ¯ λ α β χ χ . o . f . . +10 . − . − . +1 . − . . +775 . − . . ± .
06 0 . ± . Table 2 . The best estimated values for the parameters ( w m , w e , ¯ λ, α, β ), computed using the jointSNe + H ( z ) data sets together. The sixth and seventh columns show the minimum of the χ functionand its corresponding χ function by degrees of freedom, χ . o . f . , defined as χ . o . f . ≡ χ / ( n − p )where n is the number of data ( n = 592) and p the number of free parameters estimated ( p = 5). Theerrors correspond to 68.3% of confidence level. The covariance matrix is given in expression (5.14) andthe figure 7 shows the marginal confidence intervals for pairs of the ( w m , w e , ¯ λ, α, β ) parameters. Thenuisance Hubble constant H parameter was marginalized assuming a flat prior probability function.From the computed value of χ . o . f . = 0 .
97, we find that the model has a good fit to data. We alsofind a very large dispersion on the values of the power parameters (¯ λ, α, β ), therefore, we are not ableto set stronger or useful constraints on these two parameters, both positive and negative values for( α, β ) in a a large range are almost equally likely. For ¯ λ we limit ourselves to values of ¯ λ >
0. For w m we find a non vanishing value as best estimate, suggesting a warm nature for the dark matter fluid.And for w e the best estimated value lies in the phantom regime, however, given the statistical errorin its estimation we cannot be conclusive about the phantom nature of the dark energy component.We find interesting that we obtain the stronger constraint (i.e., less dispersion) in its value, comparedto the other parameters. where µ k is the observed distance moduli of the k-th supernova, with a standard deviationof σ k in its measurement, and n = 580. – 13 – z z Figure 6 . (Color on-line) Evolution for ˆΩ m ( z ) (blue) and ˆΩ e ( z ) (red), with the initial conditionsˆΩ m ( z = 0) ≡ Ω m0 = 0 .
274 and ˆΩ e ( z = 0) ≡ Ω e0 = 0 .
726 (present time z = 0). The parameters are α = 0 . β = 1, w e = − . λ = 1. Full lines are for w m = 0 . w m = 0 . w m = 0 .
075 and dash-doted lines are for w m = 0 . e ( z ) (red) with respect to w m remains almost constant compared to ˆΩ m ( z ) (blue)along the interval − < z < For the Hubble parameter H ( z ) measured at different redshifts, we use the 12 data listed intable 2 of Busca et al. (2012) [26], where 11 data come from references [27]–[29]. We assumed H = 70 km s − Mpc − for the data of Blake et al. (2011) [27] as Busca et al. suggest. The χ function is defined as χ ( w m , w e , ¯ λ, α, β ) = X i (cid:18) H ( z i , w m , w e , ¯ λ, α, β ) − H obs i σ Hi (cid:19) (5.12)where H obs i and H ( z i , w m , w e , ¯ λ, α, β ) = H · E ( z i , w m , w e , ¯ λ, α, β ) are the observed and theo-retical values of H ( z ) respectively. E ( z, w m , w e , λ, α, β ) is given by the expression (5.8) and σ Hi is the standard deviation of each H obs i datum.We construct the total χ function that combine the SNe and H ( z ) data sets together,as χ = χ + χ , (5.13)where χ and χ are given by expressions (5.11) and (5.12) respectively.We numerically minimize it to compute the best estimate values for the five ( w m , w e , ¯ λ, α, β )parameters together, and measures the goodness-of-fit of the model to the data. For that,we use a combination of some built-in functions of the c (cid:13) Mathematica software as well as– 14 – .00 0.05 0.10 0.15 0.200100200300400500600 w m Λ Confidence Intervals - - - w m Α Confidence Intervals - - - w m Β Confidence Intervals - - - - - - w m w e Confidence Intervals - - - - Λ Α Confidence Intervals 0 20 40 60 80 - Λ Β Confidence Intervals - -
50 0 50 100 - - - - Λ w e Confidence Intervals -
50 0 50 - - Α Β Confidence Intervals - -
100 0 100 200 - - - - - - Α w e Confidence Intervals - - -
50 0 50 100 150 - - - - Β w e Confidence Intervals
Figure 7 . (Color on-line). Marginal confidence intervals (CI) from the five parameters space( w m , w e , ¯ λ, α, β ), computed together using the joint SNe + H ( z ) data sets. In each panel, the contourplots were computed marginalizing over the other two remaining parameters. The CI correspond to68 . .
4% and 99 .
7% of confidence level. The best estimated values for ( w m , w e , ¯ λ, α, β ) are shownin table 2 and the covariance matrix C is given in (5.14). For the dark matter barotropic index w m ,we find that non vanishing values are compatible with the observations of SNe + H ( z ). Moreover,marginalizing over the other four parameters we find a value of w m = 0 . +0 . − . suggesting a pref-erence for a warm dark matter instead of a cold one from the present model and data used. On theother hand, we notice a large dispersion in the contour plots, in particular for the power parameters(¯ λ, α, β ), indicating that a large range of positive and negative values are allowed for both ( α, β ) withalmost the same statistical confidence level; we are not able to set more useful constraints on thesethree parameters. the Levenberg-Marquardt Method described in the Numerical Recipes book [30], to minimizethe χ function (5.13).We use also the definition of “ χ function by degrees of freedom ”, χ . o . f . , defined as– 15 – . o . f . ≡ χ / ( n − p ) where n is the number of total combined data used and p the numberof free parameters estimated. For our case ( n = 592 , p = 4).The numerical results are summarized in table 2 and figure 7. The computed covariancematrix that we found corresponds to C = . − . − .
29 11625 . − − . . . − .
28 41731 . − .
29 2980 .
78 601931 − . × . − . − . × − . × − . . × − . × . × (5.14) In order to shed some light on the coincidence problem we have explored a cosmological modelcomposed by a dark matter fluid interacting with a dark energy fluid. Motivated by veryrecent investigations we have considered a warm dark matter. Since non-linear interactionsrepresent a more physical plausible scenario for interacting fluid we studied an interactionwhich is given by the term λρ α m ρ β e . We have found a general result which indicates the positivecritical points of the coincidence parameter r = ρ m /ρ e exist if w e < −
1, independently ofthe interaction chosen and the particular EoS used to describe the dark matter. We haveconsidered from the beginning that the energy is transferred from dark energy to dark matter( λ > ρ m , ρ e ), the behaviour of the fixed points in terms of the parameters λ, α, β, w m and w e .Closed orbits were found for w e < − . α = β = 1. If α or β are different from theunity, these closed orbits transform into spiral trajectories, evolving towards the origin for α < β = 1, and away for the fixed point for α = 1 and β >
1. This analysis allowed toconstrain the parameters in order to have physically reasonable scenarios, that is acceleratedexpansion in the late time phase of the cosmic evolution and far future evolution with finiteDM and DE densities, which corresponds to spiral trajectories propagating from the fixedpoints.The parameters ( w m , w e , λ, α, β ) were estimated using the cosmological observations ofthe Union 2.1 type Ia supernovae and the Hubble expansion rate H ( z ) data sets, that measurethe late time expansion history of the Universe. A summary of these results are shown ontable 2 and figure 7.For the barotropic index w m of the EoS of the dark matter, we found non vanishing andpositive values for w m that are well compatible with the SNe + H ( z ) observations. Marginal-izing over the other four parameters, we found that w m = 0 . +10 . − . which indicates thata warm dark matter is well compatible with the observations used here. This is also inagreement with other models and observations indicating a warm nature of the dark matter– 16 –uid. However, we notice a dispersion on the value of w m larger than the allowed by otherobservations.For the barotropic index w e of the dark energy component, we find that the best es-timated value of − . +1 . − . lies in the phantom regime, however, given the magnitude ofthe statistical error it is not possible to claim that the phantom nature of the dark energycomponent is favoured by the observations. We can only claim that the phantom regime iswell allowed for the considered values of w e .For the interacting coefficient λ , we defined a dimensionless ¯ λ for convenience. Using thecosmological observations it is found that the possible values for the interacting coefficientare in the vast range of 0 < ¯ λ <
800 with 68 .
6% of confidence level. This gives us at leastan indication of favouring the data the interaction between the dark fluids in this model.For the power parameters ( α, β ), we found a large dispersion in their values that areconsistent with the SNe + H ( z ) observations with the same confidence level, from positiveto negative values of both parameters. So, in the present work we are not able to set strongconstraints on α and β but at least we can assert that a large variety of positive and negativevalues of the powers ( α, β ) are allowed according to the data.From the computed value of χ . o . f . = 0 .
97 ( χ = 573 . H ( z ) data.On the other hand, we use the Bayesian Information Criterion (BIC) [31] to determinewhich model formed from different cases of the values of ( w m , w e , ¯ λ, α, β ) is favoured by theobservations. The value of BIC, for Gaussian errors of the data used, is defined asBIC = χ + ν ln N (6.1)where ν and N are the number of free parameters of the model and the number of data usedrespectively. The model favoured by the observations compared to the others corresponds tothat with the smallest value of BIC, in addition to the criterion that the value of χ shouldbe about or smaller to the number of data used (in our case, N = 592) for that model.Computing the magnitude of the χ function to measure the goodness-of-fit of data whenit is evaluated at some values of interest for ( w m , w e , ¯ λ, α, β ), as well as the correspondingvalue of the BIC, we findi χ ( w m = 0 . , w e = − . , ¯ λ = 0 . , α = 0 . , β = 0 .
73) = 573 . . χ (0 , − , , ,
0) = 581 .
05; BIC = 593 .
82: This case corresponds to the ΛCDM model;there is not interaction between the dark components.iii χ (0 . , − . , , ,
0) = 580 .
52; BIC = 593 .
28: This case corresponds to a warmdark matter interacting with a phantom dark energy; there is not interaction betweenthe dark components.iv χ (0 . , − , , ,
0) = 581 . .
25: A warm dark matter and cosmologicalconstant model, without interaction.v χ (0 . , − . , . , ,
0) = 580 .
69; BIC = 599 .
84: A warm dark matter interact-ing with a phantom dark energy, where the values corresponds to the best estimatedin the present work. In this case the interacting term is just the constant ¯ λ = 0 . Q =constant. – 17 –i χ (0 . , − , . , ,
1) = 577 . . Q = λρ m ρ e . Thisparticular case corresponds to that studied by Lip [13].Using the BIC as a model selection criterion, we find that the model from the above listwith the smallest value of BIC, and therefore most favored by the observations, correspondsto that composed of a warm dark matter and a phantom dark energy, without interaction.However, the difference with ΛCDM in the BIC value is too small so that we can just concludethat the warm dark matter – phantom darkk energy model described above is as good asΛCDM model to fit the SNe+ H ( z ) data. The interacting model (i) is not as good as theothers models, despite it has a good fit to data ( χ d . o . f . = 0 . Q = λρ m ρ e . This conclusion is in conflict with that of Lip (2011) [13].In summary, from the dynamical system approach, the non linear interaction chosen inthis work leads to plausible scenarios that can alleviate the coincidence problem. The stablefixed points represent universes which end in a dark sector with non zero and finite energydensities in both fluids, despite the phantom behaviour of the dark energy fluid.On the other hand, using the SNe + H ( z ) observations, the best estimated values for thefree parameters of the model correspond to a warm dark matter interacting with a phantomdark energy component, with a well goodness-of-fit to data measured through the obtainedmagnitude of χ d . o . f . . However, using the BIC model criterion we find that this model isovercame by a warm dark matter – phantom dark energy model without interaction, as wellas by the ΛCDM model. Acknowledgments
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