Modeling dark energy through an Ising fluid with network interactions
aa r X i v : . [ h e p - t h ] A p r Modeling dark energy through an Ising fluid with network interactions
Orlando Luongo
Physics Department, University of Naples ”Federico II”, I-80126, V. Cinthia, Naples, Italy,INFN, Section of Naples, I-80126, Naples, Italy andInstitute of Nuclear Sciences, UNAM, AP 70543, Mexico, DF 04510, Mexico.
Damiano Tommasini
Institute of Physics and MTA-DE Particle Physics Research Group,University of Debrecen, H-4010, Debrecen, P.O. Box 105, Hungary.
We show that the dark energy effects can be modeled by using an
Ising perfect fluid with net-work interactions, whose low redshift equation of state, i.e. ω , becomes ω = − PACS numbers: 98.80.-k, 98.80.Jk, 98.80.Es
I. INTRODUCTION
Recent cosmological observations pointed out that theuniverse is undergoing an accelerated expansion [1, 2].Unfortunately, the physical mechanism which drives theobserved cosmic speed up is so far unclear. Moreover,standard pressureless baryonic matter is inadequate byitself to characterize the universe acceleration, even byassuming the additional presence of cold dark matter [3–6]. As a standard landscape, it is possible to postulatethe existence of a further ingredient, dubbed dark en-ergy (DE) [7–11]. Dark energy behaves as a weakly in-teracting anti-gravitational fluid, described by a negativeequation of state (EoS). Even though the nature of sucha fluid has not been clarified, a wide number of differ-ent paradigms have followed each other, spanning fromslowly rolling scalar field, known as quintessence [12, 13],lattice of topological defeats [14], to barotropic fluids[15] or modifications of Einstein gravity [16–18] and soforth (for further details see [19] and references therein).One of the simplest way to explain the observed cosmicspeed up is provided by the so-called ΛCDM paradigm.In particular, the ΛCDM model leads to the introduc-tion of a vacuum energy cosmological constant Λ, andassumes a total matter content, Ω m , given by the sum ofbaryonic and dark matter densities [20, 21]. The corre-sponding EoS is constant as the universe expands, lead-ing moreover to a negative and constant pressure [22].Even though the model is in a fairly good agreementwith current observations, it suffers from two profoundshortcomings, i.e. the fine tuning and coincidence prob-lems [13, 19, 23, 24]. To alleviate these two issues, theDE density, ρ , and its corresponding pressure, P ( ρ ), arethought to evolve separately with significative departures from standard matter. Thus, possible extensions of theΛCDM model are frequently characterized by assuming atime-variable EoS, ω ( z ) ≡ P ( ρ ) ρ , evolving as − ≤ ω < z = 0.In this work, we investigate how to model DE by con-sidering an Ising network-interacting fluid on a lattice[12, 25–27]. We show that a fluid, whose interaction isprovided by a series of networks, may predict effects dueto a negative pressure, at z ≪ II. THE ISING FLUID WITH NETWORKINTERACTIONS
In this section, we describe the cosmological conse-quences of assuming an Ising fluid with network inter-actions, as a source of DE. Let us first notice that instandard lattice models, the particle description is for-mally represented by assuming a grid with an occupa-tional variable σ i . Its value is zero, if the site is empty,and one if occupied by particles. This simple picture isalso known in the literature as ”bit“ gas model, in anal-ogy to computational science [30, 31]. Its use is usuallyadopted in different fields of physics, especially in order tomodel extended sites by numerical computations throughMonte Carlo simulations, ranging from condensed matterto quantum computing and particle physics [32–40]. Un-der these hypotheses, the occupational variables σ i areactually analogous to spin variables in the well knownIsing model [41, 42]. However, the physical meaning of σ i is basically different from Ising spins. In particular, σ i is not an intrinsic physical property of the system,but only a way to discriminate the presence and absenceof particles. Moreover, once the particle interaction isspecified, our lattice fluid may lead to a negative EoS,in particular regions of the phase space [43, 44]. We willconsider this property, in order to describe the DE effectsin a homogeneous and isotropic universe. A. The entropy representation
In the Hamiltonian formalism, we can write down theHamiltonian of our model, by defining a chemical poten-tial µ . We have H = − X i,j J i,j σ i σ j − X i µσ i , (1)where µ > σ i is given by J i,j >
0. The Hamiltonian of Eq.(1) is formally analogous to the Ising Hamiltonian in themean field approximation, with the substitution [45] σ i = S i + 12 , (2)where S i = ± Z and to evaluate the thermodynamicalvariables. The information of network interactions iscontained in J i,j . Hence, by assuming that the volume and temperature are functions of the redshift, we choosean unitary lattice spacing and a hard sphere interaction,which consists in introducing an excluded volume by pro-hibiting the multiple occupancy of particles on a givenlattice site [46]. In order to obtain the EoS of our model,we can take into account a finite region of volume V at agiven temperature T , where N particles are confined tomove only on d -dimensional discrete lattice points, insidethe region under interest [47]. The volume is expressedby V = h · L d − with h the height and L d − the area ofthe side wall. Thus, the entropy of the system becomes S = k B ln Ω( N , V ) , (3)where Ω( N , V ) represents the total permutations ofputting N particles in a volume V . In the absence ofexcluded volume interactions, the entropy of the systemis factorized S = k B ln Ω( N , V ) = k B ln( ω N i ) = k B N ln( ω i ) , (4)where Ω = ω N i , and ω i represents the single wall proba-bility associated to the i − th particle in a unitary volume.In the simplest case of one single particle, the entropy re-duces to k B ln ω . In classical statistical mechanics, thepartition function is in general Z = exp (cid:16) − k B T H (cid:17) andreduces to a simple form when the Hamiltonian is nota functional. Note that, the contributions to the parti-tion function come from two sources: one is associatedto the kinetic energy of the system, while the other oneis represented by the total potential energy. B. The equilibrium configuration
In the equilibrium configuration, the contribution com-ing from the kinetic energy is decoupled from the con-figurational statistics [48]. Hence, the equilibrium pres-sure can be determined by considering only the configu-rational properties of the system. In the case of a latticeconfiguration, the free energy is computed exactly, be-cause the hard sphere interaction is formally equivalentto introducing an excluded volume [49]. Hence, by pro-hibiting multiple particle occupancy on a given latticesite [42], we getΩ( N , V ) = V ! N !( V − N )! . (5)By assuming that the DE density is ρ ≡ NV ρ Λ , makinguse of the Stirling approximation, we infer S ≈ − k B V [ ρ ln ρ + (1 − ρ ) ln(1 − ρ )] , (6)where we plugged into Eq. (4), the definition of Ω( N , V ),i.e. Eq. (5). Moreover, the scaling ruler ρ Λ , is associatedto the minimal size of lattice sites. For the sake of clear-ness, ρ Λ determines whether the approximation of latticefluid holds. Thus, the term ρ Λ defines a border constant of the equilibrium configuration, whose physical mean-ing has nothing to do with the number density of latticesites. For our purposes, it is convenient to assume here-after ρ Λ = 1. We replace ρ Λ , in the incoming sections.Moreover, the pressure of our fluid, i.e. P = ωρ , is givenby P = −T ∂ S ∂ V = − k B T ln(1 − ρ ) , (7)where ω represents the EoS of our model. As shown in[50], the same results could be found in the context ofthe grand partition ensemble, without losing generality.The expressions of the pressure and density are in-terpreted in our picture as sources of DE. They enterthe energy momentum tensor, in addition to standardpressureless matter. Thus, for a spatially flat homoge-neous and isotropic universe, i.e. ds = dt − a ( t ) ( dr + r sin θdφ + r dθ ), the Friedmann equations are H = 8 πG ρ t , (8)˙ H + H = − πG P + ρ t ) , where both the total pressure and density, respectively P and ρ t , are composed by pressureless matter, i.e. ρ m ,and by DE density , i.e. ρ . III. COSMOLOGICAL CONSEQUENCES OFNETWORK INTERACTING ISING FLUID
In this section, we are interested in finding the EoS ofDE, modeling it as a network interacting fluid throughthe use of standard thermodynamic recipe. Hence, per-fect fluids are modeled by an energy momentum tensorof the form T αβ = ( ρ + P ) u α u β − P g αβ , (9)from that it naturally follows the conservation laws forenergy and particle number densities. In particular, byassuming the redshift definition in terms of the cosmictime and H ( z ), i.e. dzdt = − (1 + z ) H ( z ) , (10)we get, from Eqs. (9) and (10), the continuity equationfor ρ in terms of z dρdz = 3 P + ρ z . (11) Notice that the total pressure P is the pressure of DE, becausestandard matter is supposed to be pressureless. From Eq. (11), one gets the functional form of ρρ ∝ exp (cid:20) Z ω ( z )1 + z dz (cid:21) , (12)and, in addition, it is possible to show that T ∝ exp (cid:20) Z ω ( z )1 + z dz (cid:21) , (13)which represents the DE temperature. It is remarkableto notice that possible temperature measurements at dif-ferent epochs of the universe could discriminate whetherDE dominates over other species or not. Current ob-servations seem to indicate negligible departures frompresent temperature [51]. This does not permit us tofix constraints on the DE temperature today, and onits evolution. According to Eqs. (12) and (13), in thesmall redshift approximation, one recovers the equiparti-tion principle between ρ and T , having T ≈ αρ , with α a constant to be determined. Plugging the temperaturedefinition in terms of ρ into the Eq. (7), we get P = αρ log[1 − ρ ] . (14)Once the degrees of freedom are fixed, the correspond-ing cosmological model depends on the today DE value.From Eq. (11), we obtain the continuity equation for DE(1 + z )3 dρdz − ρ | {z } = αρ log h − ρρ Λ i| {z } , (15) Dynamics of DE Source of DE where we restored the definition of ρ Λ . In Eq. (15), wesplit the dynamics of DE in terms of the redshift z (leftside), from the DE source (right side), as determined byour model. We evaluate the Hubble rate, by expandingaround ρ Λ and we get H ( z ) = H s Ω m (1 + z ) + h αρ Λ − Υ(1 + z ) i − , (16)with Υ an integration constant, given byΥ = αρ Λ − − Ω m . (17)It is worth noticing that the error due to the approxi-mations made in Eqs. (6) and (16) is negligibly small.The numerical solution, corresponding to our logarithmiccorrection of the pressure, well approximates the exactsolution, with an error ≤ z . Thus, the EoSof DE can be rewritten as ω ( z ) = − αρ Λ αρ Λ − Υ(1+ z ) , (18)whose value today is ω = − αρ Λ (1 − Ω m ) , (19)which becomes ω = − αρ Λ → − − Ω m . From theformer relation for ω , it is evident that our model re-duces to ΛCDM as a limiting case, showing that the roleplayed by the cosmological constant is actually mimicked through the term ∝ αρ Λ . To guarantee that the universeis accelerating today, we should constrain the ratio αρ Λ .The degeneracy between α and ρ Λ is alleviated by thefact that one needs to evaluate the ratio αρ Λ , instead of α and ρ Λ separately. Using the Friedmann equations,i.e. Eqs. (8), by keeping in mind the definition of theacceleration parameter: q = − z ) H dHdz , (20)we find for our model q = 3Ω m (1 + z ) − ρ α ω ( z ) m (1 + z ) + 2 ρ Λ (1+ z ) α ω ( z ) − . (21)The cosmological acceleration starts at the transition red-shift z acc , i.e. z acc ≈ ρ Λ + 3(1 − Ω m ) α − Ω m ) α h − (1 − Ω m ) αρ Λ i , (22)which approximatively reads z acc ≃ . α/ρ Λ ≈ . m = 0 .
3. Equation (22)corresponds to the case q = 0, and leads to the epochin which DE dominates over standard pressureless mat-ter [52]. The acceleration parameter becomes negative,according to current observations, when z < z acc . More-over, the transition redshift, given by Eq. (22), is in anexcellent agreement with the value predicted by ΛCDM,i.e. z acc, Λ ≃ .
75. The variation of q with respect to thecosmological redshift z , measures the rate of change ofthe acceleration, i.e. j ( z ). For our model, it is easy toget j ( z ) = − ρ Λ β + β Z ( z ) + β Z ( z ) + β Z ( z ) ( ρ Λ + Z ( z )) ( ρ Λ + Ω m Z ( z )) , (23)where for simplicity we defined the following constants β = − ρ − α (1 − Ω m ) + 9 α (1 − Ω m ) ρ Λ ,β = (7 − m ) ρ Λ − α (1 − Ω m ) ,β = − − m , (24) β = − Ω m ρ Λ . and the position Z ( z ) = α (1 − Ω m ) h ( z + 1) − i . (25)In particular, j ( z ) is also known in literature as the jerkparameter and it is given by the definition j = ¨ HH − q − z = 0. It is expected that j > z Μ H z L FIG. 1. Representation of supernova magnitude µ ( z ) VS red-shift z . Data and the best fit, both related to the correspond-ing uncertainties (red lines). The union 2.1 dataset has beenplotted. guarantee that for z > z acc the acceleration parameter ispositive [54, 55]. The jerk parameter today reads j = ρ − αρ Λ (1 − Ω m ) + 9 α (1 − Ω m ) ρ , (26)and the expected acceleration parameter today reads q = 12 h m − − Ω m ) i − . (27)By the definitions of q and j , it is possible to find outviable priors for the free parameters, α , ρ Λ and Ω m . Inparticular, we summarize the priors, that we impose inour computational analysis, in Tab. I, taking into ac-count the results obtained in Eqs. (22), (26) and (27). IV. COSMOLOGICAL CONSTRAINTS
In this section, we constrain the free parameters of ourmodel, through a numerical analysis based on currentcosmological data. We rely on three statistical sets ofparameters. In doing so, we define three different max-imum order of parameters, assuming a hierarchy amongthe three sets. The sets are summarized as follows A = (cid:26) H , Ω m , αρ Λ (cid:27) , B = (cid:26) Ω m , αρ Λ (cid:27) , (28) C = (cid:26) αρ Λ (cid:27) . The hierarchy of Eqs. (28) predicts a broadening ofthe sampled distributions, adding the cosmological co-efficients to constrain. In the Gaussian regime, the error W m H - W m ΑΡ L ΑΡ L H FIG. 2. 1-dimensional marginalized contour plots for H , αρ Λ and Ω m , using A . We considered the union 2.1 compilation. Here,we report 1 σ , 2 σ and 3 σ of confidence levels. propagation becomes higher as hierarchy, between pa-rameters, increases. To alleviate the error propagationand possible systematics, we consider the numerical pri-ors reported in Tab. I. A. Priors on the cosmological parameters and theinitial condition on H Viable cosmological priors are need in order to allevi-ate the so called degeneracy between cosmological dis-tances. In fact, the luminosity distance by itself is notenough to separately constrain all the cosmological den-sities. It follows that the EoS cannot be constrained witharbitrary accuracy with SNeIa data only. By fixing vi- able priors, one needs to reduce the total phase space.This permits us to complement distance measurementsby different constraints. In other words, once the priorshave been determined, it is possible to infer cosmologi-cal bounds from one class of measurements. Moreover,the simple choice of using different distances does notguarantee a priori that the physical region for constrain-ing the free parameters is actually reduced. For thosereasons, we rely on three combined tests, performed byusing SNeIa, BAO measurements and the CMB surveys.Our choice determines a combination of cosmological andgeometrical procedures, which allows to circumscribe thephase space with higher precision. In addition, spatialgeometry is also set to be geometrically flat, while theinitial condition on H is determined as follows: 1) first, - W m ΑΡ L W m ΑΡ L FIG. 3. 1-dimensional marginalized contour plots for H , αρ Λ and Ω m , using set B . We considered the union 2.1 compilation.Here, we report 1 σ , 2 σ and 3 σ of confidence levels. - - - - z Ω H z L zH H z L FIG. 4. We show the behavior of the EoS of our model, in terms of the redshift z (left figure). Moreover, we plot the Hubblerate (right figure), comparing it with the ΛCDM model. Respectively red line for our model and dashed line for ΛCDM. Weused the indicative values Ω m = 0 . αρ Λ = 1 . - - zq H z L FIG. 5. Graphic of q ( z ) for our model (color line) and ΛCDM (dashed). We notice small differences in the redshift transitionsfrom the acceleration to deceleration phases and good agreement with the numerical value at small redshift with respect topresent cosmographic bounds.Flat priors0 . < h < . . < Ω b h < . . < Ω dm h < . . ρ Λ < α < ρ Λ Additional constraintsΩ k = 0Ω m < . we consider H free to vary, 2) second, we constrained itby WMAP 7-years results, 3) finally, we impose its value,by assuming the Hubble space telescope (HST) measure.Additional cosmological and geometrical priors, adoptedthroughout our numerical analysis, are summarized inTab. I, as already underlined. B. Cosmological datasets
In our numerical analysis, we consider three differentdatasets. In particular, we take into account the union2.1 compilation, the BAO measure and the CMB surveys,with the constraint on H given by the measurement ofthe WMAP 7-years and HST respectively. In the union2.1 compilation of the supernova cosmology project [56],the covariance matrix has been evaluated with and with-out systematics. The compilation includes previous sur-veys, i.e. union 2 [57] and union 1 [58]. The supernova measurements may be represented in the plane modulus-redshift, i.e. µ − z . They consist of 580 measurements of µ and z , spanning in the redshift range 0 . < z < . z is assumed to be negligibly small, whileto each supernova is associated a corresponding error on µ . This turns out to be important since supernovas arestandard indicators and represent the primary distanceindicators. The relevance of supernova measurements isrelated to the fact that their rest frame wavelength regionspans from 4000 to 6800 ˚ A for all the transient eventswhich characterize supernovas. By assuming that thesame rest frame wavelengths are measured at all z , onecan compare the supernova brightness as independent aspossible from a supernova model. Thence, to fix cosmo-logical constraints, we make use of a Bayesian method inwhich the best fits of parameters are inferred by maxi-mizing the following likelihood function L ∝ exp( − χ / , (29)where χ is the ( pseudo ) chi-squared function. To ob-tain the corresponding posterior distributions, we con-sider uniform priors in the interval ranges of Tab. I. It iseasy to show that the luminosity distance is d L ( z ) = (1 + z ) Z z dξ E ( ξ ) , (30)where E ≡ HH , and by defining the distance modulus µ for each supernova µ = 25 + 5 log d L M pc , (31)together with the corresponding 1- σ i error, we canrewrite the χ parameter of Eq. (29) as χ SN = X i ( µ theor i − µ obs i ) σ i . (32) TABLE II. Best fits of the free parameters of our model, tested by SNeIa. The quoted errors show the 68.3%, 95.4% and99.7% confidence level uncertainties.Parameter Set A Set B Set B Set C Set C χ min = 0 . χ min = 0 . χ min = 0 . χ min = 0 . χ min = 0 . H . +1 . . . − . − . − . H ≡ . H ≡ . H ≡ . H ≡ . m . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . Ω m ≡ .
274 Ω m ≡ . αρ Λ . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . Notes. H is given in Km/s/Mpc. TABLE III. Best fits of the free parameters with BAO. Note that with BAO it is not possible to estimate H . The quotederrors show the 68.3%, 95.4% and 99.7% confidence level uncertainties.Parameter Set A Set C Ω m . +0 . . . − . − . − . Ω m ≡ . αρ Λ . +0 . . . − . − . − . . +0 . . . − . − . − . A simple test with SNeIa minimizes the term χ SN , whichcorresponds to maximize Eq. (29).The large scale galaxy clustering observations providethe signatures of the BAO [59]. The theoretical back-ground employs that the universe consisted of a hotplasma of photons, electrons, protons, baryons and otherlight nuclei, at a certain epoch of its evolution. As a con-sequence, the Thompson scattering between photons andelectrons leads to oscillations in the hot plasma. As theuniverse becomes neutral, it is possible to consider theinitial perturbation patterns which are imprinted on thematter distribution. Hence, by observing the spectrumof galaxy correlations today, it is possible to focus onobservations of BAO. The corresponding measurement isrepresented by a sound horizon length, whose physicalmeaning deals with the distance traveled by an acous-tic wave by the time of plasma recombination. The BAOpeak is considered a standard cosmological ruler, becausesuch a peak is independent of the choice of a particularcosmological model. We use the peak measurement ofluminous red galaxies, denoted by A BAO . It reads A BAO =(Ω b + Ω dm ) h E ( z BAO ) i ×× (cid:20) z BAO Z z BAO E ( ξ ) dξ (cid:21) , (33)with Ω b and Ω dm respectively the baryonic and dark mat-ter densities and z BAO = 0 .
35. In addition, the observed A BAO is estimated to be A BAO,obs = 0 . (cid:18) . . (cid:19) − . = 0 . , (34)with an error σ A = 0 . χ BAO = (cid:18) A − A obs σ A (cid:19) . (35) The third test is represented by CMB observations. Thiskind of measurements has recently reached much interest,as a nearly isotropic background was discovered in 1965.Measures of the WMAP satellite found new data, ableto alleviate the cosmological degeneracy between mod-els. The underlying philosophy relies on CMB radiation,which can be directly detected by keeping in mind thatit is influenced by two cosmological epoches, i.e. the lastscattering era and present time [60, 61]. For the CMBtest, we define the so-called CMB shift parameter R CMB .Its standard definition reads R CMB = (Ω b + Ω dm ) Z z rec dξ E ( ξ ) , (36)with z rec ≈ R CMB ≡ l l ′ , where l is the posi-tion of the first peak on the CMB TT power spectrum ofthe model under consideration. Moreover, l ′ is the firstpeak in a flat homogeneous and isotropic universe withΩ b = 1 − Ω dm . Hereafter, l is written as l = D A ( z rec ) s ( z rec ) − , (37)where D A ( z rec ) is called co-moving angular distance, i.e. D A ( z rec ) = Z z rec (1 + ξ ) dξ , (38)with s ( z rec ) representing the sound horizon at recombi-nation s ( z rec ) = 1 H Z ∞ z rec v ( ξ ) E ( ξ ) − dξ . (39)We defined v ( ξ ) as the sound speed of the photon-to-baryon fluid. Its expression reads v = (cid:16) ρ b ρ γ (cid:17) − . . TABLE IV. Best fits of the free parameters with CMB. Note that with CMB it is not possible to estimate H . The quotederrors show the 68.3%, 95.4% and 99.7% confidence level uncertainties.Parameter Set A Set C Ω m . +0 . . . − . − . − . Ω m ≡ . αρ Λ . +0 . . . − . − . − . . +0 . . . − . − . − . TABLE V. Best fits of the parameters for the three considered models with SNeIa, BAO and CMB. The quoted errors showthe 68.3%, 95.4% and 99.7% confidence level uncertainties.Parameter Set A Set B Set B Set C Set C χ min = 0 . χ min = 0 . χ min = 0 . χ min = 0 . χ min = 0 . H . +1 . . . − . − . − . H ≡ . H ≡ . H ≡ . H ≡ . m . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . Ω m ≡ .
274 Ω m ≡ . αρ Λ . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . . +0 . . . − . − . − . Notes. H is given in Km/s/Mpc. The importance of using CMB leads to its complemen-tary with respect to SNeIa and BAO measurements. Thiscomes from the fact that SNeIa and BAO are confinedin low redshift regimes, constrained to z <
2, while forCMB, the recombination redshift is higher of three ordersof magnitude [63]. According to the CMB measurement,we minimize the following χ CMB = (cid:18) R − R obs σ R (cid:19) . (40)Since the three different sets of observations are not cor-related between them, the total χ reads χ = X i =1 χ i = χ SN + χ BAO + χ CMB . (41)From one hand, the SNeIa test needs to fix H , as ini-tial condition, in addition to the free parameters of ourmodel, i.e. α and ρ Λ . It is prominent to assume gaus-sian priors on the Hubble constant, in order to fulfillthe initial condition on H . In doing so, we considerthe WMAP 7-years measurements, which suggests H =70 . ± . km/s/M pc and H = 74 . ± . km/s/M pc [64],as measured by the HST. Following these constraints, andthose of Tab. I, we perform three separate tests withSNeIa, BAO and finally CMB, by combining them to-gether. On the other hand, the cosmological tests BAOand CMB do not directly depend on H . It follows thatthe unique bound on H should be imposed when weperform the supernova test only. V. MODEL COMPARISON THROUGH AICAND BIC SELECTION CRITERIA
As already stressed in Sec. III, the degeneracy prob-lem, between cosmological models, plagues standard techniques of numerical analyses. The task of discrim-inating which models better fit current data is actually athorny issue of statistics. The main disadvantage lies onassuming, in a numerical computation, a particular formof H ( z ) for the cosmological model. In other words, eachcosmological test assumes a priori that the cosmologi-cal model under exam is statistically favored. This isa consequence of the χ analysis, which is able to con-strain the free parameters of a given model, although itdoes not provide any information about the validity ofthe model itself. To alleviate this problem, one coulduse model independent procedures of statistical analysis.Among various possibilities, there exist in the literaturestatistical methods, able to understand which model isreally favorite, than others, once the same data surveyis used. It is easy to show that comparing different chisquares among alternative models, with the same num-bers of free parameters, it is possible to check whichmodel is statistically favored than others. In lieu of lim-iting our attention to simply fit our model, as in Sec. IV,we compare our approach with two relevant paradigms,i.e. a variable quintessence model: the Chevallier, Po-larsky, Linder (CPL) parametrization [75, 76]) and thestandard ΛCDM model. Naively, we can expect that acombination of lowest chi squares and fewest numbers ofparameters provides the model which better reproducescosmological data. We summarize below the ΛCDM andCPL Hubble rates respectively, i.e. E = h Ω m (1 + z ) + 1 − Ω m i , (42) E = h Ω m (1 + z ) + (1 − Ω m ) f( z ) i , with f( z ) = (1 + z ) ω + ω a ) exp (cid:16) − ω a z z (cid:17) . In partic-ular, ω = − ω = ω + ω a (cid:16) z z (cid:17) , respectively for0 TABLE VI. BIC and AIC analysis, performed by assuming model A with SNeIa data.Model Num. of Par. (k) Parameters χ min ∆ BIC ∆ AIC
ΛCDM 1 Ω m m , αρ Λ m , ω , ω a ΛCDM and CPL models. In what follows, we describethe statistical methods that we are going to use, in orderto make a comparison between our model and the abovecited ΛCDM and CPL paradigms.
A. Selection criteria
In this subsection, we propose two statistical methodswhich follow the guidelines given in Sec. V. They arethe so-called AIC [65–68] and BIC [69] selection crite-ria. The first one has reached a widely accepted con-sensus, becoming a common diagnostic tool [70–73]. Ithas been largely used for regression models [74], sinceits first applications [65–68]. The basic demands of AICand BIC consist in postulating two statistical distribu-tions, i.e. f ( x ) and g ( x | θ ). The first distribution f ( x )is imposed to be the exact reconstruction of a particu-lar model, while the second distribution, i.e. g ( x | θ ), isthought to approximate f ( x ), by bounding N parame-ters, that are included within the vector θ . Once f ( x )and g ( x | θ ) are defined by numerical procedures, the setof parameters θ is estimated, minimizing the departuresbetween f ( x ) and g ( x, θ min ).Obviously, the function f ( x ) is not known a priori.Hence, both the AIC and BIC criteria are meaning-less if evaluated for a single model. In other words,once the minima of AIC and BIC are determined, i.e. AIC min and
BIC min respectively, we should investigatethe differences ∆
AIC ≡ AIC − AIC min and ∆
BIC ≡ BIC − BIC min . A general form for the
AIC function isAIC = − L max + 2 k , (43)and by following [69], we define the BIC function asBIC = − L max + k ln N , (44)where for both the methods AIC and BIC, L max is themaximum likelihood function, corresponding to the min-imum of χ . Moreover, k is the number of parametersto be estimated, and N is the number of data pointsused to perform the cosmological fits. To Gaussiandistributed errors corresponds χ min = − L max andwe can simplify ∆AIC = ∆ χ min + 2∆ k and ∆BIC =∆ χ min + ∆ k ln N .In Tab. III, we report the numerical results of using theAIC and BIC criteria. In particular, we find for E and E respectively: Ω m = 0 . +0 . − . , H = 69 . +1 . − . , χ SN,min = 0 . m = 0 . +0 . − . , H = 70 . +1 . − . , ω = − . +0 . − . , ω a = 1 . +1 . − . and χ SN,min = 0 . VI. FINAL REMARKS
In this work, we investigated the possibility to modelDE through an Ising fluid on a lattice with network in-teractions. A negative pressure, associated to DE andcompatible with present observations, emerged by con-sidering the barotropic EoS of our model in the equilib-rium configuration. We inferred theoretical constraints ina fairly good agreement with current observations. Wedemonstrated that, at low redshift, it is possible to obtaina viable Hubble rate which well mimics the DE effects, re-ducing to ΛCDM at the zero order expansion, in terms ofthe DE density. The corresponding acceleration parame-ter and its variation, namely the jerk parameter, providedinteresting results which properly fitted with modern ob-servations. In particular, the acceleration starts at a red-shift which is excellently close to the one predicted byΛCDM. In addition, the present value of j ( z ) confirmedthat the acceleration parameter has changed its sign, asthe universe expands. From such considerations, it fol-lowed that one of the main advantages of our model re-lied on interpreting the DE nature as an emergent Isingfluid with network interactions. In addition, we fixedconstraints on the free parameters of our model, by em-ploying three cosmological datasets, i.e. the SNeIa, BAOand CMB surveys. In particular, we first used SNeIa,BAO and CMB separately and then we combined SNeIawith BAO and CMB, constraining the free parameters ofour model in tighter intervals, though viable geometri-cal and cosmological priors. In doing so, we chose for thecomputational analysis three cosmological sets of observ-ables, ordered in a hierarchial way, evaluating the corre-sponding errors up to 3 σ of confidence level. The resultsconfirmed that our model could be viewed as a viable al-ternative to ΛCDM, in order to describe the DE effects atpresent time. To this end, we showed that, through theuse of the AIC and BIC selection criteria, our model pro-vided small departures than ΛCDM, behaving smoother1than the so called CPL parametrization. Future effortscould be devoted to apply the network interacting Ising model to other stages of the universe evolution, wonder-ing whether the model could be modeled for differentepochs of the universe evolution. [1] Riess A. G., et al. , 1998, Astron. J., , 1009.[2] Perlmutter S. et al. , 1999 Astrophys. J., , 565.[3] de Bernardis P. et al. , 2000, Nature, , 955.[4] Percival W. J. et al. , 2002, Mon. Not. Roy. Astron. Soc., , 1068.[5] Croft, R. A. C., Weinberg, D. H., Katz, N., Hernquist,L., 1998, Astrophys. J., , 44.[6] Mc Donald P. et al. , 2006, Astrophys. J. Suppl., ,80; S. Capozziello, L. Consiglio, M. De Laurentis, G. DeRosa, C. Di Donato, 2011, The missing matter problem:from the dark matter search to alternative hypotheses ,ArXiv[astro-ph]:1110.5026.[7] Sahni V., Starobinsky A., 2010, Int. J. Mod. Phys. D, ,2105, (2006); S. Capozziello, J. Matsumoto, S. Nojiri, S.D. Odintsov, Phys. Lett. B, 693, 198-208; S. Capozziello,M. De Laurentis, S. Nojiri, S. D. Odintsov, 2009, Phys.Rev. D, , 124007; S. Capozziello, P. Martin-Moruno,C. Rubano, 2008, Phys. Lett. B, , 12-15.[8] Clarkson, C., Ellis, G., Larena, J., Umeh, O., 2011, Rept.Prog. Phys., , 112901.[9] Kolb, E. W., Matarrese, S., Notari, A., Riotto, A., 2005,Phys. Rev. D, , 023524.[10] Rasanen, S., 2006, JCAP, , 003; Bamba, K.,Capozziello, S., Nojiri, S., Odintsov, S. D., 2012, As-troph. and Sp. Sci., , 155-228.[11] Luongo, O, Quevedo, H., 2011, ArXiv[gr-qc]:1104.4758;Luongo, O., Quevedo, H., 2012, Astr. and sp. sci., 338,2, 345-349.[12] Luongo, O., Iannone, G., Autieri, C., 2010, Europh.Lett., , 39001.[13] Ratra, B., Peebles, P. J. E., 1988, Phys. Rev. D, , 3406.[14] Vilenkin, A., 1994, Phys. Rev. Lett., , 1016.[15] Linder, E. V., Scherrer, R. J., 2009, Phys. Rev. D, ,023008.[16] Capozziello, S., Carloni, S., Troisi, A., 2003, Recent Res.Dev. Astron. Astrophys., , 625; Capozziello, S., De Lau-rentis, M., 2011, Phys. Rept., , 167-321.[17] Carroll, S. M., Duvvuri, V., Trodden, M., Turner M. S.,2004, Phys. Rev. D, , 043528.[18] Carroll, S. M., De Felice, A., Duvvuri, V., Easson, D.A., Trodden, M., Turner M. S., 2005, Phys. Rev. D, ,063513.[19] Copeland, E. J., Sami, M., Tsujikawa, S., 2006, Int. J.Mod. Phys. D, , 1753.[20] Carroll, S. M., Press, W. H., Turner, E. L., 1992, Ann.Rev. Astron. Astrophys., , 499.[21] Tegmark, M., et al. , [SDSS Collaboration], 2004, Phys.Rev. D, , 103501.[22] Padmanabhan, T., 2009, Adv. Sci. Letts., , 174.[23] Caldwell, R. R., Dave, R., Steinhardt, P. J., 1998, Phys.Rev. Lett., , 1582.[24] Padmanabhan, T., 2012, The Physical Principle thatdetermines the Value of the Cosmological Constant ,ArXiv[hep-th]:1210.4174.[25] Prigogine, I. et al., 1988, Proc. Nat. Acad. [26] Prigogine, I., Geheniau, J., Gunzig, E., Nardone, P.,(1989), Gen. Rel. Grav., , 767.[27] Wainwright, J .,Ellis, G. F. R. 1997, D ynamical Systemsin Cosmology, Cambridge University Press, Cambridge.[28] Sanchez, I. C., Lamcombe, R. H., 1978, Macromolecules, , 1145.[29] Lee, T. D., Yang, C. N. 1952, Phys. Rev. 87, 410.[30] de Oliveira, P. M. C., Sa’ Martins, J. S., Stauffer, D.,Moss de Oliveira, S., 2004, Phys. Rev. E, , 051910.[31] Kauffman, S. A., 1993, Origins of Order: Self-Organization and Selection in Evolution , Oxford Univer-sity Press, Oxford.[32] Lehaut, G., Gulminelli, F., Lopez O., 2009, Phys. Rev.Lett., , 142503.[33] Wilding, N. B., Sollich, P., Buzzacchi, M. 2008, Phys.Rev. E, , 011501.[34] Lu, X. Y., Quan, S. W., Zhang, B. C., Hao, J. K., Zhu,F., Lin, L., Xu, W. C., Wang, E. D., et al. , 2008, Chin.Phys. Lett., , 3940.[35] de Souza, D. R., Tome, T., 2009, Physica A, 389, .[36] Simon, B. 1993, The Statistical Mechanics of LatticeGases , Vol. I, Princeton University Press.[37] Higuera, F. J., Jim´enez, J., 1989, Europhys. Lett., , 663.[38] Ruelle, D., 1968, Comm. math. phys., , 4.[39] Qian, Y. H., d’Humi´eres, D., Lallemand, P. 1992, J. Stat.Phys., , 3-4, 563-573.[40] Dieter, D., Wolf-Gladrow, A., 2000, Lattice-gas cellularautomata and lattice Boltzmann models: an introduction ,Springer.[41] Kennedy, T. 2010, J. Stat. Phys. , 409-426.[42] Binder, K. 1982, Phys. Rev. A, , 1699-1709.[43] Das, C.B. et al., 2005, Phys. Rep. 406, .[44] Campi, X., Krivine, H., 1997, Nucl. Phys. A, 620, .[45] Binder, K., Landau, D. P., 1984, Phys. Rev. B, , 1477.[46] Evertz, H. G., von der Linden, W., 2001, Phys. Rev.Lett., , 5164.[47] Das, C. B., Das Gupta, S., 2008, Nucl. Phys. A, ,149-157.[48] Gulminelli, F., 2004, Ann. Phys. Fr., 29, .[49] Huang, K., 2001, I ntroduction to statistical physics, Tay-lor and Francis, New York.[50] Hong, D.C., Mc Gouldrick, K., 1998, Physica A, ,3-4.[51] Lima, J. A. S., Alcaniz, J. S., 2004, Phys. Lett. B, ,191.[52] Farooq, O., Ratra, B., Hubble parameter measurementconstraints on the cosmological deceleration-accelerationtransition redshift , ArXiv: 1301.5243 [gr-qc], (2013).[53] Aviles, A., Gruber, C., Luongo, O., Quevedo, H., 2012,Phys. Rev. D, , 123516; A. Aviles, C. Gruber, O. Lu-ongo, H. Quevedo, Constraints from Cosmography in var-ious parameterizations , ArXiv[gr-qc]:1301.4044, (2013);M. Demianski, E. Piedipalumbo, C. Rubano, P. Scud-ellaro, Month. Notic. R. Astron. Soc. , 1396-1415,(2012); J. C. Carvalho, J. S. Alcaniz, Month. Notic. R.Astron. Soc., , 1873-1877, (2011); L. Xu, Y. Wang, Phys. Lett. B, , (2011), 114-120.[54] Cattoen, C., Visser, M., 2008, Phys. Rev. D, , 063501.[55] Luongo, O., 2011, Mod. Phys. Lett. A, , 20, 1459-1466.[56] Suzuki, N., Rubin, D., Lidman, C., Aldering, G., Aman-ullah, R., et al., 2012, Astrophys. J., , 85.[57] Amanullah, R., Lidman, C., Rubin, D., Aldering, G.,Astier, P., et al., 2010, Astrophys. J., , 712-738.[58] Kowalski, M., et al., 2008, Astrophys. J., , 749.[59] Percival, W. J., et al., 2010, Mon. Not. Roy. Astron. Soc., , 2148.[60] Salopek, D. S., 1992, Phys. Rev. Lett., , 3602.[61] Rubino-Martin, J. A., et al., 2003, MNRAS, , 1084.[62] Melchiorri, A., Griffiths, L. M., 2001, New Astron. Rev. , 321, (2001).[63] Eisenstein, D. J., et al, 2005, Astrophys. J. , 560.[64] Taylor, J. E., et al., 2012, Astrophys. J., , 127.[65] Akaike, H., 1981, J. Econometr., , 3-14.[66] Liddle, A. R., 2004, Mon. Not. R. Astron. Soc., , L49.[67] Schwarz, G., 1978, Ann. Stat., , 461-4. [68] Szydlowski, M., Kurek, A., Krawiec, A., 2006, Phys.Lett. B, , 171.[69] Kunz, M., Trotta, R., Parkinson, D., 2006, Phys. Rev.D, , 023503.[70] Akaike, H., 1997, IEEE Trans. Automat. Control, .[71] Biesiada, M., 2007, M., Jour. Cosm. Astrop. Phys.,JCAP, 02003.[72] Szydlowski, M., Godlowski, W., 2006, Phys. Lett. B, ,427.[73] Szydlowski, M., Kurek, A., 2006, AIP Conference Pro-ceedings, , 1031-1036.[74] Burnham, K. P., Anderson, D. R., 2002, Model Selectionand Multimodel Inference , New York, Springer.[75] Linder, E. V., 2003, Phys. Rev. Lett., , 09130.[76] Chevallier, M., Polarski, D., 2001, Int. J. Mod. Phys. D, , 213.[77] Komatsu, E., et al., 2011, ApJS, 192,18