Reconstruction of the interaction term between dark matter and dark energy using SNe Ia 2 1 0 Freddy Cueva Solano and Ulises Nucamendi 2 n Instituto de F´ısica y Matema´ticas a Universidad Michoacana de San Nicola´s de Hidalgo J Edificio C-3, Ciudad Universitaria, CP. 58040 7 Morelia, Michoaca´n, M´exico 2 E-mail: [email protected],[email protected] ] O C Abstract. We apply a parametric reconstruction method to a homogeneous, . isotropic and spatially flat Friedmann-Robertson-Walker (FRW) cosmological model h p filled of a fluid of dark energy (DE) with constant equation of state (EOS) parameter - interactingwithdarkmatter(DM).Thereconstructionmethodisbasedonexpansions o r of the general interaction term and the relevant cosmological variables in terms of t s Chebyshev polynomials which form a complete set orthonormal functions. This a interactiontermdescribes anexchangeofenergyflowbetweenthe DE andDMwithin [ darksector. Toshowhowthemethodworkswedothereconstructionoftheinteraction 2 function expanding it in terms of only the first six Chebyshev polynomials and obtain v 3 the best estimation for the coefficients of the expansion assuming three models: (a) a 0 DE equationofthe stateparameterw=−1(aninteractingcosmologicalΛ),(b) aDE 3 equation of the state parameter w = constant with a dark matter density parameter 1 fixed,(c)aDEequationofthestateparameterw =constantwithafreeconstantdark . 9 matter density parameterto be estimated, andusing the Union2 SNe Ia dataset from 0 1 “The Supernova Cosmology Project” (SCP) composed by 557 type Ia supernovae. In 1 both cases, the preliminary reconstruction shows that in the best scenario there exist v: the possibility of a crossing of the noninteracting line Q=0 in the recent past within i the1σand2σerrorsfrompositivevaluesatearlytimestonegativevaluesatlatetimes. X This means that, inthis reconstruction,there is anenergytransfer fromDE to DMat r a early times and an energy transfer from DM to DE at late times. We conclude that this fact is an indication of the possible existence of a crossing behavior in a general interaction coupling between dark components. PACS numbers: 95.36.+x, 98.80.-k,98.80.Es Submitted to: Journal of Cosmology and Astroparticle physics Dark energy interacting with dark matter 2 1. Introduction In the last years the accelerated expansion of the universe has now been confirmed by several independent observations including those of high redshift (z ≤ 1) type Ia Supernovae (SNeIa) data at cosmological distances [1]-[2]. This has been verified by precise measurements of the power spectrum of the cosmic microwave background (CMB)anisotropies[3]-[4], thegalaxypowerspectrumdetectionandthebaryonacoustic peak in the large-scale correlation function of luminous red galaxies in the experiment Sloan Digital Sky Survey (SDSS) [5]-[6]. To explain these observations, it has been postulated the existence of a new and enigmatic component of the universe so-called dark energy (DE) [7]-[30] from which the cosmological constant is the simplest model [22], [31]-[37]. Recent observations [2], [4], [6], [38]-[39] show that if it is assumed a dark energy(DE)equationofstate(EOS) withconstant parameterw = P /ρ , thenthere DE DE remains little room for departure of DE from the cosmological constant. In addition these observations indicate that our universe is flat and it consists of approximately 70% of Dark Energy (DE) in the form of a cosmological constant, 25% of Dark Matter and 5% of baryonic matter. However the cosmological constant model has two serious problems: the first of them is the cosmological constant problem [20], [22], [31]-[37] which consists in why the observed value of the Cosmological Constant ρobs ∼ (10−12 Gev)4 is so-small compared Λ with the theoretical value ρPl ∼ (1018 Gev)4 predicted from local quantum field theory Λ if we are confident in its application to the Planck scale?. The second problem is the so named The Cosmic Coincidence problem [12]-[30] consisting in why, in the present, the energy density of DE is comparable with the density of dark matter (DM) while the first one is subdominant during almost all the past evolution of the universe?. In the last decade, in order to solve the The Cosmic Coincidence problem, several researchers haveconsideredapossiblephenomenological interactionbetweentheDEand DM components [40]-[87]. As far as we know, the first models of dark energy coupled with dark matter were proposed by Wetterich [9], [11] in the framework of a scalar field with an exponential potential (named The Cosmon) coupled with the matter. Some years later, with the discovery of the recent accelerated expansion of the universe [1]-[2] and in order to solve the coincidence problem, several authors put forward the idea of a coupled scalar field with dark matter named Coupled Quintessence [40]-[46], [56]-[58], [66], [71]-[72], [82]. On the other hand, the theory of dynamical systems have been applied to different models of coupled dark energy in order to clarify the cosmological evolution ofthesolutions ofevery model withemphasis inthestudy ofthecritical points [88]-[94]. Some recent studies have claimed that, for reasonable and suitably chosen interaction terms, the coincidence problem can be significantly ameliorated in the sense that the rate of densities r ≡ ρ /ρ either tends to a constant or varies more slowly DM DE than the scale factor, a(t), in late times [66], [76]. However, the existence or not of some class of interaction between dark components is to be discerned observationally. To this Dark energy interacting with dark matter 3 respect, constraints on the strength of such interaction have been put using different observations [96]-[134]. Recently, it has been suggested that an interacting term Q(z) dependent of the redshift crosses the noninteracting line Q(z) = 0 [133]-[134]. In [133], this conclusion have been obtained using observational data samples in the range z ∈ [0,1.8] in order to fit a scenario in which the whole redshift range is divided into a determined numbers of bins and the interaction function is set to be a constant in each bin. They found an oscillatory behavior of the interaction function Q(z) changing its sign several times during the evolution of theuniverse. Onthe other hand, in[134] isreported acrossing of the noninteracting line Q(z) = 0 under the assumption that the interacting term Q(z) is a linearly dependent interacting function of the scale factor with two free parameters to be estimated. They found a crossing from negative values at the past (energy transfers from dark matter to dark energy) to positive values at the present (energy transfers from dark energy to dark matter) at z ≃ 0.2−0.3. While it is not totally clear if an interaction term can solved the The Cosmic Coincidence problem or if such crossing really exists, we can yet put constraints on the size of such assumed general interaction and on the probability of existence of such crossing using recent cosmological data. We will do this postulating the existence of an general nongravitational interaction between the two dark components. We introduce phenomenologically this general interaction term Q into the equations of motion of DE and DM, which describes an energy exchange between these components [40]-[87]. In order to reconstruct the interaction term Q as a function of the redshift we expand it in terms of Chebyshev Polynomials which constitute a complete orthonormal basis on the finite interval [-1,1] and have the nice property to be the minimax approximating polynomial (this technique has been applied to the reconstruction of the DE potential in [137]-[138]). At the end, we do the reconstruction using the observations of “The Supernova Cosmology Project” (SCP) composed by 557 type Ia supernovae [2]. Due to that in this paper our principal goals are: (i) the development of the formalism of reconstruction of the interaction and (ii) the recent reconstruction of the evolution of that interaction, we do not include another data sets like CMB anisotropies, the galaxy power spectrum or the baryon acoustic peak (BAO) measured in the experiment SDSS. Clearly the use of these data sets implies special considerations such as the application of the cosmological perturbation theory in the reconstruction method which is beyond the scope of this paper. We will do the total reconstruction in the evolution of the interaction in our future work. In our reconstruction process we assume two interacting models: (a) a DE equation of the state parameter w = −1 (an interacting cosmological Λ) and (b) a DE equation of the state parameter w = constant (as far as know the only reference proposing a reconstruction process of coupled dark energy using parameterizations of the coupling function is [139]). The organization of this paper is a follow. In the second section we introduce the general equations of motion of the DE model interacting with DM. In the third Dark energy interacting with dark matter 4 section, we write the cosmological equations for both interacting dark components. In the forth section we develop the reconstruction scheme of the interaction term in terms of a expansion of Chebyshev polynomials. In the fifth section, we briefly describe the application of the type Ia Supernova data cosmological test and the priors used on the freeparametersofthereconstructiontogether withabriefdiscussion oftheresultsofour reconstruction and the best estimated values of the parameters fitting the observations. Finally, in the last section we discuss our main results and present our conclusions. 2. General equations of motion for dark energy interacting with dark matter. We assume an universe formed by four components: the baryonic matter fluid (b), the radiation fluid (r), the dark matter fluid (DM) and the dark energy fluid (DE). Moreover all these constituents are interacting gravitationally and additionally only the dark components interact nongravitationally through an energy exchange between them mediated by the interaction term defined below. The gravitational equations of motion are the Einstein field equations G = 8πG Tb +Tr +TDM +TDE , (1) µν µν µν µν µν whereas that the equation(cid:2)s of motion for each fluid(cid:3)are ∇νTb = 0, (2) µν ∇νTr = 0, (3) µν ∇νTDM = −F , (4) µν µ ∇νTDE = F , (5) µν µ where the respective energy-momentum tensor for the fluid i is defined as (i = b,r,DM,DE), Ti = ρ u u +(g +u u )P (6) µν i µ ν µν µ ν i here u is the velocity of the fluids (assumed to be the same for each one) where as ρ µ i and P are respectively the density and pressure of the fluid i measured by an observer i with velocity uµ. F is the cuadrivector of interaction between dark components and µ its form is not known a priori because in general we do not have fundamental theory, in case of existing, to predict its structure. We project the equations (2)-(5) in a part parallel to the velocity uµ, uµ∇νTb = 0, (7) µν uµ∇νTr = 0, (8) µν uµ∇νTDM = −uµF , (9) µν µ Dark energy interacting with dark matter 5 uµ∇νTDE = uµF , (10) µν µ and in other part orthogonal to the velocity using the projector h = g +u u acting βµ βµ β µ on the hypersurface orthogonal to the velocity uµ, hµβ∇νTb = 0, (11) µν hµβ∇νTr = 0, (12) µν hµβ∇νTDM = −hµβF , (13) µν µ hµβ∇νTDE = hµβF , (14) µν µ using (6) in (7)-(10) we obtain the mass energy conservation equations for each fluid, uµ∇ ρ +(ρ +P )∇ uµ = 0, (15) µ b b b µ uµ∇ ρ +(ρ +P )∇ uµ = 0, (16) µ r r r µ uµ∇ ρ +(ρ +P )∇ uµ = uµF , (17) µ DM DM DM µ µ uµ∇ ρ +(ρ +P )∇ uµ = −uµF , (18) µ DE DE DE µ µ at the other hand it introducing (6) in (11)-(14) it permits to have the Euler equations for every fluid, hµβ∇ P +(ρ +P )uµ∇ uβ = 0, (19) µ b b b µ hµβ∇ P +(ρ +P )uµ∇ uβ = 0, (20) µ r r r µ hµβ∇ P +(ρ +P )uµ∇ uβ = −hµβF , (21) µ DM DM DM µ µ hµβ∇ P +(ρ +P )uµ∇ uβ = hµβF , (22) µ DE DE DE µ µ Finally we closed the system of equations assuming the following state equations for the respectively baryonic, dark matter, radiation components, P = 0 (23) b P = 0 (24) DM 1 P = ρ (25) r r 3 while for the dark energy we assume a state equation with constant parameter w, P = wρ (26) DE DE Dark energy interacting with dark matter 6 3. Cosmological Equations of motion for dark energy interacting with dark matter. We assumed that the background metric is described by the flat Friedmann-Robertson- Walker (FRW) metric written in comoving coordinates as supported by the anisotropies of the cosmic microwave background (CMB) radiation measured by the WMAP experiment [3] ds2 = −dt2 +a2(t) dr2 +r2dΩ2 , (27) where a(t) is the scale factor and(cid:0)t is the cosm(cid:1)ic time. In these coordinates we choose for the normalized velocity, uµ = (1,0,0,0) (28) and therefore we have, a˙ ∇ uµ = 3 ≡ 3H (29) µ a uµ∇ uβ = 0 (30) µ where H is the Hubble parameter and the point means derivative respect to the cosmic time. In congruence with the symmetries of spatial isotropy and homogeneity of the FRW spacetime, the densities and pressures of the fluids are depending only of the cosmic time, ρ (t), P (t), and at the same time the parallel and orthogonal components i i of the cuadrivector of interaction with respect to the velocity are respectively, uµF = Q(a) (31) µ hµβF = 0 (32) µ where Q(a) is known as the interaction function depending on the scale factor. The introduction of the state equations (23)-(26), the metric (27) and the expressions (28)- (32) in the equations of mass energy conservation for the fluids (15)-(18) produces, ρ˙ +3Hρ = 0, (33) b b ρ˙ +4Hρ = 0, (34) r r ρ˙ +3Hρ = Q, (35) DM DM ρ˙ +3(1+w)Hρ = −Q, (36) DE DE At the other hand, the Euler equations (19)-(22) are satisfied identically and do not produce any new equation. From the Einstein equation (1) we complete the equations of motion with the first Friedmann equation, 8πG H2(a) = (ρ +ρ +ρ +ρ ). (37) b r DM DE 3 Its convenient to define the following dimensionless density parameters Ω⋆, for i = i b,r,DM,DE, as the energy densities normalized by the critical density at the actual epoch, ρ Ω⋆ ≡ i , (38) i ρ0 crit Dark energy interacting with dark matter 7 and the corresponding dimensionless density parameters at the present, ρ0 Ω0 ≡ i , (39) i ρ0 crit where ρ0 ≡ 3H2/8πG is the critical density today and H is the Hubble constant. crit 0 0 Solving (33) and (34) in terms of the redshift z, defined as a = 1/(1+z), we obtain the known solutions for the baryonic matter and radiation density parameters respectively: Ω⋆(z) = Ω0(1+z)3, (40) b b Ω⋆(z) = Ω0(1+z)4, (41) r r Theenergyconservation equations(35)and(36)forbothdarkcomponents arerewritten in terms of the redshift as: dρ 3 Q(z) DM − ρ = − , (42) DM dz 1+z (1+z)·H(z) dρ 3(1+w) Q(z) DE − ρ = , (43) DE dz 1+z (1+z)·H(z) Phenomenologically, we choose to describe the interaction between the two dark fluids as an exchange of energy at a rate proportional to the Hubble parameter: Q(z) ≡ ρ0 ·(1+z)3 ·H(z)·I (z), (44) crit Q The term ρ0 · (1 + z)3 has been introduced by convenience in order to mimic a rate crit proportional to the behavior of a matter density without interaction. Let be note that the dimensionless interaction function I (z) depends of the redshift and it will be the Q function to be reconstructed. With the help of (44) we rewrite the equations for the dark fluids (42)-(43) as, dΩ⋆ 3 DM − Ω⋆ = −(1+z)2 ·I (z), (45) dz 1+z DM Q dΩ⋆ 3(1+w) DE − Ω⋆ = (1+z)2 ·I (z), (46) dz 1+z DE Q 4. General Reconstruction of the interaction using Chebyshev polynomials. Wedotheparametrizationofthedimensionless couplingI (z)intermsoftheChebyshev Q polynomials, which form a complete set of orthonormal functions on the interval [−1,1]. They also have the property to be the minimax approximating polynomial, which means thathasthesmallest maximumdeviationfromthetruefunctionatanygivenorder[137]- [138]). Without loss of generality, we can then expand the coupling I (z) in the redshift Q representation as: N I (z) ≡ λ ·T (z), (47) Q n n n=0 X Dark energy interacting with dark matter 8 where T (z) denotes the Chebyshev polynomials of order n with n ∈ [0,N] and n N a positive integer. The coefficients of the polynomial expansion λ are real free n dimensionless parameters. Then the interaction function can be rewritten as N Q(z) = ρ0 ·(1+z)3 ·H(z)· λ ·T (z), (48) crit n n n=0 X We introduce (47) in (45)-(46) and integrate both equations obtaining the solutions, N z Ω⋆ (z) = (1+z)3 Ω0 − max λ ·K (x,0) , (49) DM DM 2 n n " # n=0 X N z Ω⋆ (z) = (1+z)3(1+w) Ω0 + max λ ·K (x,w) , (50) DE DE 2 n n " # n=0 X where we have defined the integrals x T (x˜) n K (x,w) ≡ dx˜ , (51) n (a+bx˜)(1+3w) Z−1 and the quantities, 2z x ≡ −1, (52) z max z max a ≡ 1 + , (53) 2 z max b ≡ , (54) 2 herez isthemaximum redshift atwhich observations areavailableso thatx ∈ [−1,1] max and |T (x)| ≤ 1, for all n ∈ [0,N]. n Finally, using the solutions (40)-(41) and (49)-(50) we rewrite the Friedmann equation (37) as H2(z) = H2 Ω0(1+z)3 +Ω0(1+z)4 +Ω⋆ (z)+Ω⋆ (z) , (55) 0 b r DM DE (cid:2) (cid:3) The Hubble parameter depends of the parameters (H , Ω0, Ω0, Ω0 , Ω0 , w) and the 0 b r DM DE dimensionless coefficients λ . However one of the parameters depends of the others due n to the Friedmann equation evaluated at the present, Ω0 = 1−Ω0 −Ω0 −Ω0 (56) DE b r DM At the end, for the reconstruction, we have the five parameters (H , Ω0, Ω0, Ω0 , w) 0 b r DM and the dimensionless coefficients λ . n To do a general reconstruction in (49)-(50) we must take N → ∞ and to obtain the solutions in a closed form. The details of the calculation of the integrals K (x,w) n in the right hand side of (49)-(50) are shown in detail in the Appendix A which shows the closed forms (A.9)-(A.10) for the integrals with odd and even integer n subindex, and valid for w 6= n/3 where n ≥ 0. Finally, we point out the formula we use for the reconstruction of other important cosmological property of the universe: Dark energy interacting with dark matter 9 • The deceleration parameter (1+z) dH(z) q(z) = −1+ · (57) H(z) dz 5. Reconstruction of the interaction up to order N = 5 using the type Ia Supernovae test. To simplify our analysis and to show how the method works, in this section we reconstruct the coupling function I (z) to different orders (N = 1,2,3,4,5), up to Q order N = 5, using the type Ia Supernovae test. The details of this reconstruction are describedintheAppendixB.WetestandconstrainthecouplingfunctionI (z)usingthe Q “Union2” SNe Ia data set from “The Supernova Cosmology Project” (SCP) composed by 557 type Ia supernovae [2]. As it is usual, we use the definition of luminosity distance d (see [1]) in a flat cosmology, L z dz′ d (z,X) = c(1+z) (58) L H(z′,X) Z0 where H(z,X) is the Hubble parameter, i.e., the expression (55), ”c” is the speed of light given in units of km/sec and X represents the parameters of the model, X ≡ (H ,Ω0,Ω0,Ω0 ,w,λ ,...,λ ) (59) 0 b r DM 1 N The theoretical distance moduli for the k-th supernova with redshift z is defined as k d (z ,X) µth(z ,X) ≡ m(z)−M = 5log L k +25 (60) k 10 Mpc (cid:20) (cid:21) where m and M are the apparent and absolute magnitudes of the SNe Ia respectively, andthesuperscript“th”standsfor“theoretical”. Weconstructthestatisticalχ2 function as n [µt(z ,X)−µ ]2 χ2(X) ≡ k k (61) σ2 k=1 k X where µ is the observational distance moduli for the k-th supernova, σ2 is the variance k k of the measurement and n is the amount of supernova in the data set. In this case n = 557, using the “Union2” SNe Ia data set [2]. With this χ2 function we construct the probability density function (pdf) as pdf(X) = A·e−χ2/2 (62) where A is a integration constant. 5.1. Priors on the the probability density function (pdf). In the models I, II and III shown in the Table 1 we marginalize the parameters Y = (H ,Ω0 ,Ω0,Ω0) in the pdf (62) choosing priors on them. In order to it, we 0 DM b r must compute the following integration, Dark energy interacting with dark matter 10 ∞ ∞ ∞ ∞ pdf(V) = pdf(X)pdf(Y)dH dΩ0 dΩ0dΩ0 (63) 0 DM b r Z0 Z0 Z0 Z0 where V = (w,λ ,...,λ ) represents the nonmarginalized parameters, pdf(X) is given 1 N by (62) and pdf(Y) is the prior probability distribution function for the parameters (H ,Ω0 ,Ω0,Ω0) which are chosen as Dirac delta priors around the specific values 0 DM b r Y˜ = (H˜ ,Ω˜0 ,Ω˜0,Ω˜0) measured by some other independent observations, 0 DM b r pdf(Y) = δ(H −H˜ )·δ(Ω0 −Ω˜0 )·δ(Ω0 −Ω˜0)·δ(Ω0 −Ω˜0) (64) 0 0 DM DM b b r r Introducing (64) in ((63) it produces, pdf(V) = A·e−χ˜2/2 (65) where we have defined a new function χ˜2 depending only on the parameters V = (w,λ ,...,λ ) as, 1 N 2 n µth(z ,V,Y˜)−µ k k χ˜2(V) ≡ (66) h σ2 i k=1 k X The specific values chosen for the Dirac delta priors are, • H˜ = 72 (km/s)Mpc−1 as suggested by the observations of the Hubble Space 0 Telescope (HST) [140]. • Ω˜0 = 0.233 DM • Ω˜0 = 0.0462 b • Ω˜0 = 4.62×10−5 r Once constructed the function χ˜2 (66), we numerically minimize it to compute the “best estimates” for the free parameters of the model: V = (w,λ ,...,λ ). The mini- 1 N mum value of the χ˜2 function gives the best estimated values of V and measures the goodness-of-fit of the model to data. For the Model IV, we leave too the parameter Ω0 free to vary and estimated it from DM the minimization of the χ˜2 function. In this case, the parameters to be marginalized are Y = (H ,Ω0,Ω0). Then, the marginalization will be as, 0 b r ∞ ∞ ∞ pdf(V) = pdf(X)pdf(Y)dH dΩ0dΩ0 (67) 0 b r Z0 Z0 Z0 where now V = (w,Ω0 ,λ ,...,λ ) represents the nonmarginalized parameters to be DM 1 N estimated, pdf(X) is given by (62) and pdf(Y) is the prior probability distribution function for the parameters (H ,Ω0,Ω0) which are chosen as Dirac delta priors around 0 b r the specific values Y˜ = (H˜ ,Ω˜0,Ω˜0) given above. 0 b r In the models II, III and IV the interaction function I (z) will be reconstructed up Q to order N = 5 in the expansion in terms of Chebyshev polynomials.