Physical constraints on interacting dark energy models
PPhysical constraints on interacting dark energy models
J. E. Gonzalez,
1, 2, ∗ H. H. B. Silva, † R. Silva, ‡ and J. S. Alcaniz
2, 4, § Departamento de F´ısica, Universidade Federal de Sergipe, 49100-000, Aracaju - SE, Brasil Observat´orio Nacional, 20921-400, Rio de Janeiro, RJ, Brasil Universidade Federal de Campina Grande, 58900-000, Cajazeiras, PB, Brasil Universidade Federal do Rio Grande do Norte, 59072-970, Departamento de F´ısica, Natal, RN, Brasil (Dated: September 5, 2018)Physical limits on the equation-of-state (EoS) parameter of a dark energy component non-minimally coupled with the dark matter field are examined in light of the second law of ther-modynamics and the positiveness of entropy. Such constraints are combined with observationaldata sets of type Ia supernovae, baryon acoustic oscillations and the angular acoustic scale of thecosmic microwave background to impose restrictions on the behaviour of the dark matter/dark en-ergy interaction. Considering two EoS parameterisations of the type w = w + w a ζ ( z ), we derivea general expression for the evolution of the dark energy density and show that the combination ofthermodynamic limits and observational data provide tight bounds on the w − w a parameter space. I. INTRODUCTION
The physical mechanism behind the late-time cosmicacceleration is currently one of the major open problemsin the field of cosmology. This phenomenon has beenevidenced from analysis and interpretation of differentobservational data sets [1–9] and, in the context of thegeneral relativity theory, can be explained either if oneadmits the existence of an exotic field, the so-called darkenergy, or if the matter content of the universe is subjectto dissipative processes [10, 11] (see [12–14] for a review).The lack of knowledge on the nature of the dark sectorhas motivated several approaches to unveil the physicalproperties of both dark matter and dark energy. In prin-ciple, a thermodynamic analysis should be relevant toconstrain the behavior of these dark components or evento restrict the range of acceptable values of their parame-ters. Many approaches of this kind have been formulatedin the literature (see, e.g., [15–22] and references therein).For instance, the thermodynamics of a dark energy com-ponent described by a varying equation-of-state parame-ter (EoS) ω = ω ( a ) with null chemical potential ( µ = 0)was discussed in [23] whereas a general treatment for darkenergy thermodynamics considering a non-zero chemicalpotential ( µ (cid:54) = 0) was presented in [24], generalising theresults of Refs. [15–17, 23]. On the other hand, moti-vated by a possible solution of the so-called coincidenceproblem [25], interacting models of dark matter and darkenergy constitute an alternative description of the darksector which have been largely investigated (see, e.g., [26]and references therein). This class of models are based onthe premise that there is currently no known symmetryin Nature preventing a non-minimal coupling in the darksector and, therefore, such possibility as well as its cos-mological consequences must be explored. In models of ∗ [email protected] † [email protected] ‡ raimundosilva@fisica.ufrn.br § [email protected] this kind the non-gravitational interactions between thefluids also contribute to their density evolution, therebyviolating the usual assumption of adiabaticity (for a re-cent observational analysis of a large class of interactingmodels, see [27]).In this paper, we extend the thermodynamic analy-ses of [23, 24] to a more general framework which as-sumes a phenomenological energy exchange between thedark energy and the cold dark matter components. Us-ing the approach of [28, 29] to obtain the interactionterm, we derive the evolution of dark energy density fortwo equation-of-state (EoS) parameterisations of the type w = w + w a ζ ( z ) [30–32] and impose physical constraintson its parameters from both the second law of thermo-dynamics and the positiveness of entropy. We also per-form a joint statistical analysis using current observa-tional data from distance measurements to type Ia super-novae (SNe Ia) from the JLA compilation [33], measure-ments of θ ( z ) obtained from the baryon acoustic oscilla-tions (BAO) signal using the angular two-point correla-tion function (2PACF) [34–37] and the angular acousticscale of the cosmic microwave background (CMB) pro-vided by the Planck Collaboration 2015 [38]. In ouranalysis, we also use the latest measurement of the lo-cal expansion rate H , as reported in [39]. We show thatthe usual constraints on the w − w a parametric space aresignificantly enhanced when the thermodynamic boundsare incorporated in the observational analysis. Through-out this paper a subscript 0 stands for present-day quan-tities and a dot denotes time derivative. We assume aflat background and work with units where the speed oflight c = 1. II. INTERACTING MODELS
First let us consider that the energy-momentum tensorof the cosmic fluid T µν consists of two perfect fluid parts,i.e., T µν = T µν + T µν , (1) a r X i v : . [ a s t r o - ph . C O ] S e p with T µνi = ( p i + ρ i ) u µ u ν + p i g µν , where ρ i is the en-ergy density and p i is the equilibrium pressure of thespecies i = 1 ,
2. By considering the Friedmann-Lemaitre-Robertson-Walker space-time and a coupling betweenthese components, the condition ∇ ν T µν = 0 leads to˙ ρ dm + 3 ˙ aa ρ dm = − ˙ ρ x − aa (1 + ω ) ρ x = Q , (2)where ρ dm and ρ x are the energy densities of cold darkmatter (DM) and dark energy (DE), respectively, while Q is the coupling function. For Q >
Q < ρ dm ∝ a − . However, if this component inter-acts with dark energy, such interaction necessarily causesa deviation from standard evolution, which may be char-acterised by the (cid:15) parameter, i.e. [28, 29] ρ dm = ρ dm, a − (cid:15) , (3)which is equivalent to a coupling term of the type Q = (cid:15)Hρ dm . (4)where H = ˙ a/a is the Hubble parameter. In [29], itwas shown that the (cid:15) parameter must be positive, whichmeans from Eq. (4) that Q > w ( a ). Replacing this intoEq. (2) one finds ρ x = ˜ ρ x, − (cid:15)ρ dm, (cid:82) exp (cid:104) (cid:82) ω ( a ) a da (cid:105) a − (cid:15) da exp (cid:104) (cid:82) ω ( a ) a da (cid:105) , (5)where ˜ ρ x, is an integration constant and ω = ω ( a ) ≡ p x /ρ x is the time-dependent EoS parameter of dark en-ergy fluid. In order to proceed further, we will as-sume the following form for the EoS parameter: ω ( a ) = ω + ω a ζ ( a ), with ζ ( a ) obeying two functional forms thathas been widely discussed in the literature [30–32] : ζ ( a ) = (1 − a ) (P1) − a a − a +1 (P2)Substituting the above parameterisations into Eq. (5) wefind, respectively, ρ (1) x ˜ ρ x, = 1 − A (cid:15) (cid:82) a ω + ω a )+ (cid:15) − exp [3 ω a (1 − a )] daa ω + ω a ) exp [3 ω a (1 − a )] , (6a) For a recent comparative study between these w ( a ) parameteri-sations, we refer the reader to [40]. ρ (2) x ˜ ρ x, = 1 − A (cid:15) (cid:82) a ω + (cid:15) − (cid:104) a (2 a − a +1) (cid:105) ω a / daa ω ) (cid:104) a (2 a − a +1) (cid:105) ω a / , (6b)where A ≡ ρ dm, / ˜ ρ x, is a constant.Now, considering that the baryonic and radiation com-ponents are separately conserved, the Friedmann equa-tion can be written as E j ( z ) = (cid:20) Ω r a + Ω b a + Ω dm a − (cid:15) + ˜Ω x f ( j ) ( a ) (cid:21) / , (7)where E j = H j /H , the density parameters follow theusual definition, and f ( j ) stands for the ρ ( f ) x / ˜ ρ x, ratiogiven by Eqs. (6a) and (6b). Note that the so-calleddynamical Λ models (see, e.g., [41]) are fully recoveredfor values of w = − w a = 0. III. THERMODYNAMIC ANALYSIS
In general, the thermodynamic description of the inter-action between two perfect fluids requires the knowledgeof three quantities: the energy-momentum tensor T µνi ,given by Eq. (1), and the particle flow vector N µi andthe entropy flux S µi defined, respectively, as N µi = n i u µ , (8) S µi = n i σ i u µ , (9)where n i ≡ N i /a is the particle number density and σ i ≡ S i /N i the specific entropy (per particle) for eachspecies [42, 43]. By considering that the decay into DMor DE affects only the particle mass (the particle numberis unaltered), the fluids are composed by variable-massparticles [44]. Therefore, the particle flow vector is con-served as follows ∇ µ N µi = ˙ n i + Θ n i = 0 , (10)where Θ ≡ ∇ µ u µi = 3 ˙ a/a is the fluid expansion rate. Thespecific entropy obeys the Gibbs equation, i.e., n i T i dσ i = dρ i − ρ i + p i n i dn i , (11)Now, assuming that ρ i = ρ i ( n i , T i ) and p i = p i ( n i , T i ), itcan be shown that the temperature evolution law is givenby [23, 42, 43]˙ T i T i = (cid:18) ∂p ,i ∂ρ i (cid:19) n i ˙ n i n i + (cid:18) ∂ Π i ∂ρ i (cid:19) n i ˙ n i n i . (12)The fact that DM is pressureless means that there is notemperature evolution law for this component. There-fore, only the DE temperature evolution law is relevantfor the thermodynamic analysis that follows. The mid-dle and right-hand sides of Eq. (2), on the other hand,can be rewritten as ˙ ρ x + 3( ρ x + p ) ˙ aa = − ˙ aa , where wehave split the dark energy pressure into two components: p = ω ρ x and Π given byΠ ≡ w a ζ ( a ) ρ x + (cid:15) ρ dm , (13)which mimics a fluid with bulk viscosity (see Refs. [23,24] for a discussion). Therefore, the entropy source ofthe DE fluid is [43] ∇ µ S µx = − ΠΘ T x . (14)Considering that the DE temperature is always posi-tive and growing in the course of the universe expansion(see, e.g., [15, 23, 24]), the second law of thermodynamicsimplies that 3 ω a ζ ( a ) ρ x (cid:15)ρ dm ≤ − . (15)Along with Eqs. (3), (6a) and (6b), the above inequalityprovides our first thermodynamic constraint on the DEquantities. For parameterisations (P1) and (P2), theyare written, respectively, as ω a ≤ − A (cid:15) a ω + ω a )+ (cid:15) exp [3 ω a (1 − a )] (1 − a ) − (cid:8) − A (cid:15) (cid:82) a ω + ω a )+ (cid:15) − exp [3 ω a (1 − a )] da (cid:9) ,ω a ≤ − A (cid:15) a ω + (cid:15) (cid:104) a a − a +1 (cid:105) ω a / (cid:16) − a a − a +1 (cid:17) − (cid:26) − A (cid:15) (cid:82) a ω + (cid:15) − (cid:104) a a − a +1 (cid:105) ω a / da (cid:27) , which clearly are not defined at a = 1, where ω = ω . Onthe other hand, using the well- known Euler relation withnull chemical potential: T x S x = ( ρ x + p x ) V x (where V x ∝ a is the comoving volume) the positiveness of entropy requires that [1 + ω ( a )] ρ x ≥ . (16)which provides our second set of thermodynamic con-straints. For parameterisations (P1) and (P2), it is writ-ten as [1 + ω + ω a (1 − a )] ρ (1) x ≥ . (17) (cid:20) ω + ω a − a a − a + 1 (cid:21) ρ (2) x ≥ , (18)respectively. When the dark energy density satisfies theweak energy condition, i.e., ρ x ≥
0, for all values of the As stated by the statistical microscopic concept of entropy: S = k B ln W > scale factor a in the interval of study, the second set ofthermodynamic constraints is exactly equal to the oneobtained for non-interacting models [23, 24]:[1 + ω ( a )] ≥ . (19)For the case in which the dark matter and dark energycomponents are not coupled ( (cid:15) → IV. OBSERVATIONAL DATA
In order to test the class of models discussed in theprevious section, we perform a Bayesian statistical ana-lysis using different cosmological observables taking intoaccount the above sets of thermodynamic constraints.The primary data set used in this analysis is the typeIa supernovae (SNe Ia) compilation named Joint Light-curve Analysis (JLA), which comprises 740 observationaldata obtained by SDSS-II and SNLS collaborations [33].The distance modulus is standardised using the modelˆ µ = m ∗ B − ( M B − α × X + β × C ) (20)where m ∗ B is the observed peak magnitude in the restframe B band, C is the color at the maximum brightness, X is the time stretching of the light-curve and α and β are nuisance parameters. The absolute magnitude M B isdependent on the host galaxy properties and the effectsof this dependence are corrected by the step function: M B = (cid:26) M B if M stellar < M (cid:12) ,M B + ∆ M if M stellar ≥ M (cid:12) (21)being M stellar the stellar mass of the SN host galaxy and∆ M another nuisance calibration parameter [33]. Thedistance modulus is related to the cosmological modelvia the luminous distance by µ model ( z ) = 5 log (cid:18) d L ( z )1Mpc (cid:19) + 25 , (22)where d L ( z ) is the luminosity distance. Note that boththe cosmological and SNe calibration parameters are fit-ted simultaneously.In our analysis we also use recent BAO data obtainedfrom a 2-point angular correlation function analysis ofthe SDSS luminous red galaxies and quasars (hereafter θ BAO ) [34–37]. The θ BAO data are obtained by mea-suring the angular separation between pairs for a definedcomoving acoustic scale, considering thin redshift shellsof order δ z = 0 . − .
02. Differently from the usual mea-surements of the BAO signal obtained from the 2-pointcorrelation function (which assume a fiducial cosmologyin order to transform the measured angular positionsand redshifts into comoving distances), the 2PACF mea-surements of θ BAO are almost model-independent, which (cid:15) w o w a Ω dm H (P1) 0 . ± . − . ± . − . ± .
85 0 . ± .
012 71 . ± . . ± . − . ± . − . ± .
53 0 . ± .
012 71 . ± . H = 73 . ± . . s − . Mpc − . (cid:15) w o w a Ω dm H (P1) 0 . ± . − . ± . − . ± .
99 0 . ± .
021 64 . ± . . ± . − . ± . − . ± .
71 0 . ± .
021 64 . ± . makes them a robust quantity to test cosmological mod-els. The theoretical value of θ BAO for a given cosmologyis given by θ BAO ( z ) = r s (1 + z ) d A ( z ) , (23)where d A = d L / (1 + z ) and the sound horizon scale isobtained from the expression: r s ( z ) = 1 √ (cid:90) ∞ z drag (cid:18) b γ (1 + z (cid:48) ) (cid:19) − / dz (cid:48) H ( z (cid:48) ) , (24)with z drag being determined by the fitting formula in[45] and Ω γ corresponding to the present photon densityparameter. The data points used in the analysis are takenfrom [34–37].Finally, we use the information of the CMB data fromthe Planck Collaboration encoded in the position of thefirst peak of the temperature power spectrum, l . Thefirst peak at the CMB power spectrum can be calculatedusing the expression [46]: l = l A (cid:40) − . (cid:20) ρ r ( z ∗ )0 . ρ b ( z ∗ ) + ρ dm ( z ∗ )) (cid:21) . (cid:41) , (25)where l A is the acoustic scale given by: l A = π (1 + z ∗ ) d A ( z ∗ ) r dec . (26)In the above expressions, z ∗ is the decoupling redshiftfitting in [45] and r dec is the sound horizon scale at thedecoupling epoch. We use l = 220 . ± . V. ANALYSIS AND RESULTS
We perform a bayesian statistical analysis with theabove mentioned sets of data where our posterior distri-bution is written in terms of the likelihood distribution, L (Θ | d ) and the prior distribution, π (Θ), as: P (Θ | D ) ∝ L ( D | Θ) π (Θ) , (27) where Θ is the set of parameters and D the data con-sidered. Our Markov Chain Monte Carlo (MCMC) sim-ulations are made using the emcee Python module [47]assuming a Gaussian likelihood distribution, L ( D | Θ) ∝ exp( − χ T / , (28)where the total chi-square function is the sum of the con-tribution of each cosmological observable, χ T = χ SNe + χ BAO + χ CMB . For the SNe Ia data we consider χ SNe = ( ˆ µ − µ model ) T C − SN ( α, β )( ˆ µ − µ model ) T . (29)We also take into account statistical and system-atic errors encoded in the SNe covariance matrix C SN ( α, β ) [33].In our statistical analysis, we use the most recentestimate of the Hubble constant H = 73 . ± . . s − . Mpc − [39] as a Gaussian prior and flat priorsfor the other parameters. In particular, taking into ac-count the constraint imposed on the (cid:15) parameter by[29], only positive epsilon values are allowed. We fixthe baryon content at the Planck Collaboration valueΩ b h = 0 . r = 4 . × − h − and the photon density param-eter is Ω γ = 2 . × − h − for a CMB temperature T CMB = 2 .
725 K.The results of the analysis are presented in the TableI, and in Figures 1 and 2 for the parameterisation P1 andP2, respectively. The figures show 1 σ and 2 σ confidencecontours of the cosmological parameters and their poste-rior distribution marginalised over all other parameters.We also perform a statistical analysis using flat priors forthe entire set of parameters, whose results are presentedin Table II. We note that the bounds on (cid:15) are significantlyreduced when the Gaussian prior is used, in agreementwith the anticorrelation between H and (cid:15) exhibited inFigs. 1 and 2.In order to combine the observational and the thermo-dynamic constraints, we also perform a statistical anal-ysis with fixed (cid:15) values. We analyse the cases with . . . . w o . . . . . w a . . . . Ω d m .
015 0 .
030 0 .
045 0 . † . . . . H . . . . w o . . . . . w a .
20 0 .
22 0 .
24 0 . Ω dm . . . . H FIG. 1. The results of our statistical analysis. Confidence contours (68.3% and 95.4%) and the posterior distribution for thecosmological parameters assuming P1 and considering the Gaussian prior H = 73 . ± .
62 km . s − . Mpc − . . . . . w o . . . . . w a . . . . Ω d m .
015 0 .
030 0 .
045 0 . † . . . . H . . . . w o . . . . . w a .
20 0 .
22 0 .
24 0 . Ω dm . . . . H FIG. 2. The same as in Figure (1) for P2. (cid:15) = 0 .
002 and (cid:15) = 0 .
018 for the parameterisation P1 andwith (cid:15) = 0 .
002 and (cid:15) = 0 .
020 for the parameterisationP2, which correspond to the 1 σ limits on (cid:15) provided byour statistical analysis (see Table I). In Figures 3 and 4,we present the new 1 σ and 2 σ confidence contours in the w − w a plane and the thermodynamic constraints (16)- (18) for the mentioned (cid:15) values. The redshift intervalused in the thermodynamic constraints is z ∈ (0 . , . z interval inside the plane region in Figures 3 and 4, therefore the second constraint set similar bounds tothe ones derived in [23].We also find that the first thermodynamic constraint issensitive to the values of the (cid:15) parameter. Figures 3 and4 show that in order to satisfy both thermodynamic con-ditions inside the 2 σ confidence level, the value of (cid:15) mustbe very small. Indeed, it should be smaller than the 1 σ upper limit allowed by the complete statistical analysis(see Table I). In the analysed cases with fixed (cid:15) values, wefind an intersection region between both thermodynamicconstraints and the 2 σ observational confidence contourdelimited approximately by the triangles with vertices w w a † =0 . w w a † =0 . FIG. 3.
Left panel:
Confidence contours (68.3% and the 95.4%) on the w − w a plane for parameterisation P1. The hatchedand shaded regions correspond to the first and second sets of thermodynamic constraints, respectively. The confidence contoursand the thermodynamic constraints assume (cid:15) = 0 . Right panel:
The same as in the previous panel for (cid:15) = 0 . π (Θ) to perform the statistical analysis. w w a † =0 . w w a † =0 . FIG. 4. The same as in Figure (3) for P2 with (cid:15) = 0 .
002 and (cid:15) = 0 . (for P1 and P2, respectively):( w o , w a ) = (cid:15) = 0 . − . , − . , ( − . , − . , ( − . , − . (cid:15) = 0 . − . , − . , ( − . , − . , ( − . , − . w o , w a ) = (cid:15) = 0 . − . , − . , ( − . , − . , ( − . , − . (cid:15) = 0 . − . , − . , ( − . , − . , ( − . , − . VI. CONCLUSIONS
Relaxing the usual assumption of a minimal couplingbetween the components of the dark sector introducessignificant changes in the predicted evolution of the uni-verse. In this paper we have firstly discussed thermo-dynamic constraints on a class of interacting models as-suming two parameterisations of the dark energy EoS(Eq. 6). The constraints on w come from the secondlaw of thermodynamics and positiveness of entropy andare combined with current observational data through aBayesian analysis. We have shown that this combinationof physical and observational constraints on w imposevery tight limits on the w − w a parametric space, asshown in Figs. 3 and 4. The thermodynamic analysisperformed in this work generalises several cases previ-ously discussed in the literature. ACKNOWLEDGEMENTS
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