Observational constraints on the free parameters of an interacting Bose-Einstein gas as a dark-energy model
Hiram E. Lucatero--Villaseñor, German Izquierdo, Jaime Besprosvany
OObservational constraints on the free parameters ofan interacting Bose-Einstein gas as a dark-energymodel
Hiram E. Lucatero-Villase˜nor and Germ´an Izquierdo ‡ Facultad de Ciencias, Universidad Aut´onoma del Estado de M´exico, Toluca5000, Instituto literario 100, Estado de M´exico, M´exico
Jaime Besprosvany
Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, ApartadoPostal 20-364, Ciudad de M´exico 01000, M´exico
Abstract.
Dark energy is modelled by a Bose-Einstein gas of particles with anattractive interaction. It is coupled to cold dark matter, within a flat universe, forthe late-expansion description, producing variations in particle-number densities.The model’s parameters, and physical association, are: Ω G , Ω m , the dark-energy rest-mass energy density and the dark-matter term scaling as a massterm, respectively; Ω i , the self-interaction intensity; x , the energy exchange rate.Energy conservation relates such parameters. The Hubble equation omits Ω G ,but also contains h , the present-day expansion rate of the flat Friedman-Lemˆaitre-Robertson-Walker metric, and Ω b , the baryon energy density, used as a prior.This results in the four effective chosen parameters Ω b , h , Ω m , Ω i , fit with theHubble expansion rate H ( z ), and data from its value today, near distance, andsupernovas. We derive wide 1 σ and 2 σ likelihood regions compatible with definitepositive total CDM and IBEG mass terms. Additionally, the best-fit value ofparameter x relieves the coincidence problem, and a second potential coincidenceproblem related to the choice of Ω G .
1. Introduction
The observed accelerated expansion of the Universe suggests that the source withthe greatest contribution to the total energy density of the homogeneous Friedman-Lemaˆıtre-Robertson-Walker (FLRW) metric is an unknown form of energy withnegative pressure called dark energy (DE) [1]. The second source with the greatestcontribution is pressureless cold dark matter (CDM), followed by ordinary (baryonic)matter. Both dark sources presumably only interact with ordinary matter andradiation through gravitation, but it is reasonable to assume that there is aninteraction between the dark sources, as it avoids or relieves the coincidence problem[2], associated to DE as a cosmological constant/scalar field. In addition, galaxy-cluster dynamics data [3] and the integrated Sachs-Wolfe effect [4] support the idea ofan interacting dark energy (IDE). Several IDE models exist in the literature, as those ‡ E-mail address: [email protected] a r X i v : . [ a s t r o - ph . C O ] J a n motivated by particle physics, thermodynamics, and phenomenological assumptions[5–7].The non-relativistic Bose-Einstein gas (IBEG) model, applied to the late universe,is an example of an IDE with a detailed microscopic description. Its attractiveinteraction component between the particles generates a negative pressure. In anapplication to the early universe [8], it leads to an accelerated expansion under certainconditions. Including a general energy-exchange mechanism, the IBEG explains thepresent accelerated expansion of the Universe, and avoids (or greatly relieves) thecoincidence problem. The background dynamics of the IBEG model is similar to thatof the ΛCDM model for certain free-parameter choices [2].Observational data can constrain the free parameters of the DE models by meansof Bayesian methods [9]. Several IDE models have been tested in the literature,e.g., [10–12]. Some observational data are model independent such as the localHubble constant measurements [13, 14], the data from the history of the Hubbleconstant H ( z ) vs z obtained by the cosmic chronometer approach [15–18] or the typeIa supernova [19,20]. Other data sets are model dependent, as the anisotropy of cosmicmicrowave background measurements [1, 21], the baryonic acoustic oscillation (BAO)data [22], the gas mass fraction [23], and the evolution of the growth function [24].In this paper we find bounds on the free parameters of the IBEG model by meansof observational data that is model independent. Additionally, we compare theobservational bounds with other constraints of theoretical origin. This kind of analysisis of vital importance to the IBEG model given the physical nature and detailedinformation of the free parameters.The paper is organized as follows. In section 2, we review the IBEG model of lateaccelerated expansion, defining the corresponding Hubble constant H ( z ). In section3, we obtain the observational constraints on the model’s free parameters by means ofBayesian methods. The observational data sets we use are the local Hubble constantmeasurements, and the Hubble-parameter history data, including type Ia supernovaluminosity-distance data. In section 4, we compare the observational constraints withtheoretical bounds. Finally, in section 5, we summarize the work. From now on,we assume units for which c = (cid:126) = k B = 1. As usual, a zero subindex refers to avariable’s present value; likewise, we normalize the present day FLRW metric scalefactor by setting a = 1.
2. Interacting Bose-Einstein gas as dark energy
The IBEG model consists of a Bose-Einstein gas of spin-zero non-relativistic particlesinteracting through an attractive contact local potential between particle pairs. Thequasiparticle energy is composed of a kinetic and a potential-energy terms, and thisfeature is maintained for the total energy, where the kinetic contribution is independentof the particle number (see [2, 8] for a detailed model description). If the IBEG is setat a temperature below the critical one T c , particles start becoming condensate, with n c accounting the condensate number density, and n (cid:15) , n = n c + n (cid:15) , the non-condensateand total number densities, respectively. The gas energy density then reads [8] ρ g = mn + ε / (128 g ) − / m − n / (cid:15) + 12 v n , (1)where m is the IBEG particle mass, providing the energy density at rest, ε =3 ζ ( ) / (2 ζ ( )) (cid:39) . g = 1 for spin-zero particles and is related to the kineticenergy term of the gas, and v is a negatively-defined potential term that accountsfor the interaction between the particles. The gas pressure can be obtained from itsdescription in the thermodynamical limit and, using p = − ( ∂E∂V ) N,S , taking the form p = 23 An / (cid:15) + 12 v n , (2)where A = ε / (128 g ) − / m − .Although the IBEG gas by itself can be used in a early acceleration cosmologicalscenario [8], it is necessary to assume that it is coupled to CDM and that the numberdensity is not constant in order to obtain a late acceleration solution in a FLRWframe [2]. In this case, the model is defined in an expanding flat FLRW metric as ds = − dt + a ( t ) (cid:2) dr + r d Ω θ (cid:3) , (3)where a ( t ) is the scale factor and t , r and Ω θ are the time, the radius and solid-angle comoving coordinates of the metric, respectively. The energy-momentum sourceis composed of pressureless baryonic matter with energy density ρ b , pressureless colddark matter (CDM) with energy density ρ dm (Ref. [2] uses ρ m for ρ dm ,) and the IBEGwith energy density given by eq. (1). The latter two sources are coupled in such away that the IBEG non-condensate number density is changed as a Markoff’s processfollowing the phenomenological law n (cid:15) = n (cid:15)i a − + n (cid:15) a x − , (4)where n (cid:15) and x ≤ n (cid:15)i = 0 isassumed (as the influence of the first component diminishes as a increases).Note that the IBEG particles created by the coupling are non-condensated and,additionally, no further condensation occurs at that epoch as T c scales as T (see [2] fordetails). Consequently, it is reasonable to set the IBEG condensate particle numberdensity to null, and, then, n = n (cid:15) . The energy-density conservation equation for thethree sources read˙ ρ b + 3 Hρ b = 0 , (5)˙ ρ dm + 3 Hρ dm = − Q, (6)˙ ρ g + 3 H ( ρ g + p g ) = Q, (7)where the dot means derivation with respect to time, Q is the coupling term. Thissystem is completed with the Hubble equation H = (8 πG/ ρ b + ρ dm + ρ g ).Using (1,2) and from the derivation of (4), the coupling term Q can be obtained Q = 3 Hx (cid:18) ρ G a x − + 53 ρ c a x − + 2 ρ i a x − (cid:19) , (8)where we have defined the parameters ρ G = mn (cid:15) , ρ c = An / (cid:15) , ρ i = v n (cid:15) / . (9)(Ref. [2] uses c for n (cid:15) ).Consequently, the three-source energy densities evolve as ρ b = ρ b a − , (10) ρ dm = ρ m a − − ρ G a x − + 5 xρ c − x a x − + 2 xρ i − x a x − , (11) ρ g = ρ G a x − + ρ c a x − + ρ i a x − , (12)where ρ b is the baryonic matter present-day energy density, and ρ m is a present-dayCDM term that evolves as a mass term (i.e., the CDM energy-density component thatscales as a − ).The x = 1 case should be treated apart. Then, the non-condensate particlenumber evolves proportionally to the volume and scales as a in a FLRW scenario.Consequently, the IBEG energy density is constant, as the Λ term in the ΛCDMmodel. Similarly, the IBEG pressure is constant with p g (cid:54) = − ρ g , so the coupling termis not null but reads Q = − H (cid:18) ρ G + 53 ρ c + 2 ρ i (cid:19) , (13)while the CDM in our model evolves as ρ m = ρ m a − − ρ G − ρ c − ρ i . (14)In such aspects, we safely conclude that the IBEG-model dynamics differs from thatof the ΛCDM model in the x = 1 case.We note that the present-day CDM total energy density is ρ dm = ρ m − ρ G + 5 x − x ρ c + 2 x − x ρ i . (15)The free parameters of the IBEG model are then ρ G , ρ c , ρ i , ρ m , ρ b and x .The Hubble equation for H can be rewritten in terms of the redshift z = (1 /a ) − H H = (Ω b +Ω m )(1+ z ) + 2Ω c − x (1+ z ) − x − + Ω i − x (1+ z ) − x − , (16)where H is the present-day Hubble constant, and we have defined Ω a =(8 πGρ a ) / (3 H ) with a = b, m, G, c, i . The above Hubble constant does not dependon the free parameter Ω G . A relation between the free parameters can be obtainedfrom Eq. (16) evaluated at z = 0Ω c = 2 − x (cid:18) − Ω b − Ω m − Ω i − x (cid:19) , (17)and, from (16) and (17), it is possible to rewrite the equation for H as H ( z ) = h (cid:40) (1 + z ) x − + (Ω b + Ω m ) (cid:2) (1 + z ) − (1 + z ) x − (cid:3) +Ω i z ) − x − − (1+ z ) x − − x (cid:41) , (18)where H ( z ) is given in units of km s − Mpc − and h = H / (100 km s − Mpc − )replaces Ω c as a free parameter of the model. Regarding the x = 1 case again,we point at similarities of the IBEG and ΛCDM models in H ( z ); given the abovedefinition, for x = 1 the Hubble factor is formally identical for both models. UsingHubble-factor related data in the x = 1 case would make the parameters Ω c and Ω i indistinguishable for the fit, with the combination − c / − Ω i playing an identicalrole as the Ω Λ term.The validity of the model’s free parameters was discussed in [2], and the analysisused the total CDM energy density ρ dm and the total IBEG energy density ρ g ( a = 1),fixed with ΛCDM model best-fit values (0 .
24% and 0 .
72% of the total energy densityof the Universe, respectively). Additionally, the Hubble expansion rate was set at75km s − Mpc − , based on supernova data. These values were set in order to comparethe IBEG and ΛCDM models. On the other hand, in this work, no such parametersare fixed beforehand and the model’s validity is discussed as well.The observational data directly related to the Hubble constant with z < z dec =1024 (as the data of the local value of the Hubble constant, the history of Hubbleconstant data and the type IA supernova data) can be used to constrain the freeparameters h , Ω i and x directly. Parameters Ω b and Ω m have both the samedependence on z , so they cannot be constrained separately by Hubble-constant relatedobservations, unless some priors are used. On the other hand, no information on Ω G can be obtained directly from the Bayesian analysis of this kind of data, as it is absentfrom the Hubble expansion rate expression (18).
3. Constraints on the IBEG-model free parameters by the Hubbleconstant
In order to constrain the model’s free parameters, we use the Bayesian approachand Monte Carlo Markov Chains (MCMC) integrated within the code SimpleMC [25]developed by A. Slosar and J. Vazquez for [26].We define χ sud = (cid:88) i (cid:18) y ( x i | θ sud ) − D i σ i (cid:19) (19)for a given set of uncorrelated data (sud) consisting of N points of the type ( x i , D i ) , i ∈ [1 , N ], and corresponding to a theoretical function of the type y ( x, θ sud ) ( θ sud beinga list of free parameters of a given model,) where σ i is the error of the data point i .The choice of free-parameter values that minimize χ sud has the largest probability tobe true, given the sud [9], and we refer to them as the best-fit parameter values θ bfsud .We also derive the parameter-space confidence regions: the 1 σ likelihood is theparameter-space region around the best-fit value for which any choice of θ sud has a68 .
3% probability for the sud be measured; and the 2 σ likelihood parameter-spaceregion has a 95 .
4% probability. These regions are used to constrain the model’s freeparameters.In order to evaluate the fit quality of the best-fit parameter values, it is possible todefine the degrees of freedom (dof) of a given sud as the number of data points N minusthe number of free parameters and, then, to compute χ sud,red = χ sud ( θ bfsud ) / ( dof ).The theoretical function with the best-fit parameter values y ( x, θ bfsud ) poorly fits thegiven sud when χ sud,red >>
1, while χ sud,red of order of unity represents a good fit [9].We also can compare the fit quality of y ( x, θ bfsud ) with other theories by comparing thecorresponding χ sud,red .In this section we consider three sud related to H ( z ). Given that Ω b cannotbe determined with this kind of data, and that it is not directly related to the IBEGmodel, we will set it as a Gaussian prior with mean value Ω b h = 0 . . x is assumed 0 . ≤ x ≤ m is relatedto the CDM mass-like term of the energy density, so it is defined positive and in therange 0 ≤ Ω m ≤
50. Ω i is related to v and is defined in the range − ≤ Ω i ≤ m and Ω i are not total energy-density terms and they shouldnot be constrained beforehand to be smaller than unity, as the Ω a parameters of theΛCDM model. Finally, h is taken in the range 0 . ≤ h ≤
1, as it is related to the ofthe Hubble-constant local value H .We proceed to obtain the constraints on the free parameters x , Ω m , Ω i and h . A local measurement of the Hubble constant H was reported from the HubbleSpace Telescope (HST) data with a 2.4% precision in [13]. The obtained value is H = 73 . ± . − Mpc − . We use it as a Gaussian prior to obtain a firstparameter constraint. The result is given in table 2 and the 1 σ and 2 σ contours canbe appreciated in figure 1. Ω b h h Ω m x − − −
100 0 Ω i Ω b h h Ω m x Figure 1.
Likelihood of the free parameters of the model for the Hubble SpaceTelescope Gaussian prior. The darker shaded region corresponds to 1 σ , whilethe lighter shaded region corresponds to 2 σ . The plots at the right of thelikelihoods represent the free-parameter probability distribution, with appropriatescale. Similarly for Figures 2-4. An independent measurement of the local Hubble constant was recently obtainedby the gravitational wave (GW) detectors LIGO and Virgo, jointly with the 1M2H, theDark Energy Camera GW-EM, DES, DLT40, Las Cumbres Observatory, VINROUGE,and MASTER Collaborations [14]. They measured it both in the GW spectrum andin the electromagnetic spectrum, from the merger of a binary neutron-star system.The value obtained is H = 70 +12 − km s − Mpc − , which is compatible with the othermeasurement considered above to 1 σ . We proceed as previously, setting a Gaussianprior to h with mean value 0 .
70 and standard deviation 0 .
08. The IBEG free-parameterconstraints are presented in table 2 under the tag GW. Given that both valuesconsidered in this section are compatible to 1 σ , the IBEG free-parameter contoursobserved are almost identical, with the exception of h .There are other local Hubble-constant estimates including cosmic microwavebackground (CMB) measurements from Planck [1] (67 . ± . − Mpc − ), andBarionic Acoustic Oscillation (BAO) measurements from SDSS30, strong lensingmeasurements from H0LiCOW31, and high-angular-multipole CMB measurementsfrom SPT32. All of them are model dependent via the linear-perturbation evolutionand the background dynamics. We decided not to consider them as we restrictourselves to measurements related to the background dynamics of the model. We next consider 37 observational data on the Hubble-constant history, H ( z ) vs z ,represented in table 1 with their respective uncertainties and sources in the literature.The data from [15–18] are obtained by the cosmic-chronometer approach and are modelindependent. The remaining data are related to BAO observations that can be modeldependent. In this case and given the large uncertainties in the measurements, weassume that the deviation of the acoustic peak and distance scale of the IBEG modelwill not be very different from that of the ΛCDM model, and thus, the correspondingdata can be used as well.The constraints on the free parameters from the H ( z ) data can be found in table2 and the likelihood is represented in figure 2. The parameter Ω i is poorly constrictedby this sud, as can be appreciated in 2.For this sud, the dof= 32 and, consequently, χ H ( z ) ,red = 0 .
69 which can beinterpreted as the model slightly ’over-fitting’ the data. It is possible to use the H ( z ) data with the ΛCDM model implemented on the SimpleMC code, and then, χ sud,red = 1 .
33 ( χ sud,red > H ( z ) data, as both have values close to 1.A comparison between the fit of both models can also be made by means of theAkaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC)[34, 35]. The first parameter is defined AIC = χ ( θ bfsud ) + 2 d, (20)where d is the model’s number of free parameters. For this sud in the ΛCDM case,the parameter is easily computed, AIC Λ = 51 .
84, while for the IBEG model itis
AIC
IBEG = 32 .
08. The observational data suggest that the ”preferred model”corresponds to that with the smaller AIC parameter. The difference ∆
AIC = AIC Λ − AIC
IBEG >
10 is interpreted as strong evidence against the ΛCDM model. z H ( z ) σ H ( z ) ref. z H ( z ) σ H ( z ) ref.0.07 69.0 19.16 [28] 0.570 100.3 3.7 [29]0.09 69.0 12 [30] 0.593 104 13 [15]0.12 68.6 26.2 [28] 0.6 87.9 6.1 [31]0.17 83 8 [30] 0.680 92 8 [15]0.179 75 4 [15] 0.730 97.3 7 [31]0.199 75 5 [15] 0.781 105 12 [15]0.2 72.9 29.6 [28] 0.875 125 17 [15]0.27 77 14 [30] 0.88 90 40 [16]0.28 88.8 36.6 [28] 0.9 117 23 [30]0.32 79.2 5.6 [29] 1.037 154 20 [15]0.352 83 14 [15] 1.3 168 17 [30]0.3802 83 13.5 [17] 1.363 160 33.6 [18]0.4 95 17 [30] 1.43 177 18 [30]0.4004 77 10.2 [17] 1.53 140 14 [30]0.4247 87.1 11.2 [17] 1.75 202 40 [30]0.44 82.6 7.8 [31] 1.965 186.5 50.4 [18]0.44497 92.8 12.9 [17] 2.34 222 7 [32]0.4783 80.9 9 [17] 2.36 226 8 [33]0.480 97 62 [16] Table 1.
Measurements of H vs z . The Hubble constant and the variance areexpressed in km s − Mpc − units. Another criterion is defined by means of the parameter
BIC = χ ( θ bfsud ) + d ln( N ) , (21)where N = 37 for this sud. Then, BIC Λ = 58 .
28 and
BIC
IBEG = 40 .
13, and thedifference is ∆
BIC = BIC Λ − BIC
IBEG >
10, which indicates strong evidence againstthe ΛCDM in comparison to the IBEG model.
The type Ia supernova distance modulus ( µ ) data is the more constrained model-independent data considered in this work. The distance modulus is directly relatedto H ( z ) as µ = 5 log (cid:18) d L M pc (cid:19) + 25 ,d L = (1 + z ) (cid:90) z dz (cid:48) H ( z (cid:48) ) . To get likelihoods of the IBEG parameters from this sud we use the joint lightcurves (JLA) from [36] and a different definition of the χ sud , given that the data isused in 30 correlated bins. In this case, χ JLA = (cid:88) ij ( D i − y ( x i | θ JLA )) Q ij ( D j − y ( x j | θ JLA )) , (22)where Q ij is the ij term of the reported correlation matrix.The results of the analysis are found in table 2, and the likelihood regions infigure 3. The 1 σ region of Ω i is more restricted from the JLA data than it is in Ω b h h Ω m x − − − − − Ω i Ω b h h Ω m x Figure 2.
Likelihood of the free parameters of the model obtained from the H ( z )data. The darker shaded region correspond to 1 σ , while the lighter shaded regioncorrespond to 2 σ . the rest of the data sets. On the other hand, the 2 σ region of the same parameter isstill similar to the other data sets. The dof in this case is 25 and the corresponding χ JLA,red = 1 .
30, which can be interpreted as a good fit of the data. The ΛCDMmodel reads χ JLA,red = 1 .
23, and we can conclude that both models have an almostidentical fit quality.As in subsection 3.2, we compare both models for this sud by means of theAIC and BIC parameters. The AIC parameter of both models is
AIC
IBEG = 42 . AIC Λ = 39 .
98, respectively. The difference between them reads ∆
AIC = AIC
IBEG − AIC Λ (cid:39) BIC
IBEG = 49 . BIC Λ = 45 . BIC = BIC
IBEG − BIC Λ (cid:39) Ω b h h Ω m x − − −
80 0 Ω i Ω b h h Ω m x Figure 3.
Likelihood of the free parameters of the model obtained from the JLAdata. The darker shaded region correspond to 1 σ , while the lighter shaded regioncorrespond to 2 σ . number) [34,35]. The tension between AIC and BIC criteria has been reported as wellfor the w CDM model and several interacting DE models [37]. + H ( z )+ JLA.
We define χ T = χ H + χ HST + χ JLA to obtain a total likelihood combining thethree sets of model-independent observations and run the MCMC implemented inSimpleMC to explore the free-parameter space. For the present-day Hubble parameter,we consider the Gaussian prior HST motivated, on one hand, by the fact that it isthe observation of present-day Hubble parameter with the smaller error, and, on theother hand, by the fact that the GW observation is 1 σ compatible with it. The totalconstraints and likelihoods of the free parameters are shown in table 2 and figure 4,respectively.1 Ω b h h Ω m x − − −
80 0 Ω i Ω b h h Ω m x Figure 4.
Likelihood of the free parameters of the model obtained from thecombined HST+ H ( z )+JLA data. The darker shaded region correspond to 1 σ ,while the lighter shaded region correspond to 2 σ .
4. Implications of the observational constraints on the IBEG model
Next, we compare the observational bounds with theoretical considerations, e.g., thatthe IBEG energy-density mass term be positive definite. Given the detailed physicaldescription of the free parameters, the observational constraints could be incompatiblewith physical bounds, and the IBEG model could be discarded. Other theoreticaldemands about the IBEG model are preferable but not compulsory, e.g., that theIBEG model solve or alleviate the coincidence problem.Most parameters that characterize the IBEG model are constrained and leadto the best-fit values from observations: Ω i = − .
60; Ω c = 4 .
79, where Ω c isrelated to the rest of the parameters by eq. (17). However, the model-independentobservations related to the Hubble constant, such as H , H ( z ) or JLA , cannot givedirect information about ρ G . Also, the parameters that define the microscopic nature2HST GW H ( z )Param. Value 1 σ Value 1 σ Value 1 σ Ω m b h × − × − × − h i − − x H ( z )+JLAParam. Value 1 σ Value 1 σ Ω m b h × − × − h i − − x Table 2.
Constraints on the IBEG free parameters from: the local Hubbleconstant obtained by HST and GW as Gaussian priors; history of the Hubbleconstant H ( z ); type Ia supernovae (JLA); and combined data (HST+ H ( z )+JLA). of the IBEG cannot be obtained with the above observables (the IBEG particlemass m , the non-condensate state particle number today, n (cid:15) , and finally, the self-interaction coupling v .) The CDM energy density today is also unknown. On theother hand, the energy-exchange term is constrained by observations to the best-fitvalue of x = 0 . ± .
01. As stated in [2], x > .
90 greatly relieves the coincidenceproblem in comparison to the ΛCDM model as r = ρ g /ρ m varies at a slow ratecompared with the rate of expansion H . In fact, x = 1 solves the coincidence problem.As x = 1 is compatible to 2 σ with the observations, we conclude that its allowed valuesconstrain the IBEG energy-exchange term in a way that they solve the coincidenceproblem.Given that Ω G is the IBEG mass term, it is positive definite. Theparallelism of the IBEG and ΛCDM models, and observations suggest that Ω dm =(3 ρ dm ) / (8 πGH ), with ρ dm defined in (15) is also positive, (or, at worst, null.)Then, using the relation (17), we put the bound over the free parametersΩ dm + Ω G = 5 x − x b + 2 − x m − x − x ) Ω i ≥ . (23)Although the best-fit values for such parameters, reported in table 2 from theHST+ H ( z )+JLA case, is incompatible with condition (23), there is a wide σ and2 σ region of compatible likelihoods.Figure 5 shows the bound over the parameters given by (23) for Ω b h = 0 . h = 0 .
70, and three different choices of x in the Ω m vs Ω i space, for the Ω dm +Ω G =0, where the space on the right-hand side of the line represents parameter choices withΩ dm + Ω G >
0. The lines correspond to: x = 0 .
85 and x = 1, respectively the upperand lower values considered for parameter x in the IBEG model [2]. Finally, x = 0 . H ( z )+JLA analysis. In the same plot, we show the1 σ and 2 σ Ω m vs Ω i likelihoods from the HST+ H ( z )+JLA analysis. We concludethat the IBEG model is consistent with condition 23.3 − − − − − − − − Ω i Ω m Ω dm + Ω G > x=0.85x=0.97x=1 Figure 5.
Bound Ω dm + Ω G = 0 given by (23) in the Ω m vs Ω i space forΩ b h = 0 . h = 0 .
70 and three different choices of x : x = 0 .
85 (green line), x = 0 .
97 (red line), and, x = 1 (black line). The space on the right-hand sideof the line represents parameter choices with Ω dm + Ω G >
0. The 1 σ and 2 σ likelihoods of the HST+ H ( z )+JLA case are also shown for comparison. Another, non-compulsory demand for the IBEG model is that ρ g ( a in ) = 0 forsome free-parameter choices and scale factor a in in the past. In [2], a in describes themoment when the CDM-IBEG energy interchange starts.If a in is near today’s value 1, a second coincidence problem arises, as theacceleration-producing substance exists only in the near past. Given that ρ G is notdetermined by the H ( z ) observations, it is not possible to limit a in from this work,but assuming the likelihood contours obtained by the data analysis, there is a widerange of Ω G choices that leads to an a in smaller enough than 1. Figure 6 showslog ( a in ) vs Ω G for x = 0 . h = 0 .
70, Ω m = 0 .
52 and different Ω i , in the regionwith Ω dm + Ω G >
0. For any chosen Ω i , there is a wide region of Ω G for which a in is smaller enough than 1, or even smaller than 10 − . Similar plots can be obtained fordifferent Ω m inside the likelihood region of figure 5. Despite the lack of informationon Ω G , we conclude that the coincidence problem associated to a in can be avoidedfor parameter choices in the allowed region of figure 5, for x = 0 .
5. Conclusions
The IBEG model consists of a flat FLRW metric containing baryonic, CDM, and IBEGsources, the latter for DE. The CDM, IBEG components exchange energy at a rate4
Figure 6.
The lines represent log ( a in ) vs Ω G for Ω b h = 0 . h = 0 . x = 0 .
97, Ω m = 0 .
52 and Ω i , in the region with Ω dm + Ω G > compatible with a Markoff’s process. The IBEG particles’ attractive self-interactionproduces negative presure responsible for an accelerated expansion of the metric [2],mimicking the ΛCDM-model dynamics. The IBEG model parameters have a directphysical meaning, and any additional bound on them produces information on itsconstituents. For certain parameter choices, the IBEG model solves or alleviates thecoincidence problem, and a second related problem, as consistent energy-exchangestarts can be found in the past. This effect is also consistent with the applicationof the model to the early universe [8], which avoids the energy-exchange componentthen.Observational data on the background dynamics from the Hubble parameter H ( z )constrain the IBEG-model’s free parameters. By setting limits on their range andcomparing with physical requirements, one may maintain or discard the model. Thedata used in this work are: the local measurements of the Hubble constant H [13,14],the history of H ( z ) [28]- [33], and the modulus distance of type Ia supernova in thejoint light curves (JLA) from [36]. A Gaussian prior is used on the baryonic energydensity based on universe measured values [27].The data strongly bound the parameter x , related to the IBEG energy-exchangerate, finding the best fit x = 0 . ± .
01, which relieves the coincidence problem inthe IBEG model. The 1 σ and 2 σ likelihoods for the parameters related to CDM massenergy density (Ω m ) and the IBEG self-interaction (Ω i ) are wide, not too restrictive,5and compatible with the theoretical and observational condition ρ dm + ρ G ≥
0. Thedata bounds are also compatible with parameter choices not affected by the secondcoincidence problem related to the time when the IBEG-CDM energy exchange starts,defined in terms of the scale factor a in . In this sense, we can conclude that the IBEGmodel remains consistent with the observational data used in this work and Bayesiananalysis, which restrict the IBEG-model free parameter space, relieving the traditionaland decay-start coincidence problems. The AIC and BIC parameters computed for thehistory of H ( z ) data tend to favor the IBEG model compared to the ΛCDM model. Onthe other hand, the JLA data present a tension between both criteria, AIC parametersfavoring the IBEG model while BIC one favoring the ΛCDM model. Such a tension ispresent in other DE models (such as the w CDM or some interacting DE models), andcan be explained in terms of the larger free-parameter number of the latter [37]. Ingeneral, we conclude that the IBEG model is favoured by the observational data, inthe same way as other interacting DE data recently studied in the literature [7,37–39].Finally, the IBEG particle mass-related parameter ρ G cannot be bounded bythe kind of data used in this work. Other observational data, related to the linearperturbation evolution of the model considered, can constrain it. Those model-relatedobservations include the CMB anisotropy measurements [1, 21], the BAO data [22],the gas-mass fraction [23], the evolution of the growth function [24]. These data arebeing analyzed and will be presented in a future work. Acknowledgments
The authors acknowledge financial support from DGAPA-UNAM, project IN112916.
References [1] Ade P A, Aghanim N, Arnaud M, Ashdown M, Aumont J, Baccigalupi C, Banday A, BarreiroR, Bartlett J, Bartolo N et al.
Astronomy & Astrophysics
A13[2] Besprosvany J and Izquierdo G 2015
Classical and Quantum Gravity Physics Letters B
Physical Review D International Journal of Modern Physics D Astrophysics and Space Science
Reports on Progress in Physics Classical and Quantum Gravity Lectures on Cosmology (Springer) pp 147-177[10] Dur´an I, Pav´on D and Zimdahl W 2010
Journal of Cosmology and Astroparticle Physics
Universe arXiv preprint arXiv:1705.09278 [13] Riess A G, Macri L M, Hoffmann S L, Scolnic D, Casertano S, Filippenko A V, Tucker B E,Reid M J, Jones D O, Silverman J M et al. The Astrophysical Journal et al. , The LIGO Scientific Collaboration et al.et al.
Nature et al.
Journal of Cosmology and Astroparticle Physics
Journal of Cosmologyand Astroparticle Physics [17] Moresco M, Pozzetti L, Cimatti A, Jimenez R, Maraston C, Verde L, Thomas D, Citro A, TojeiroR and Wilkinson D 2016 Journal of Cosmology and Astroparticle Physics
Monthly Notices of the Royal Astronomical Society: Letters
L16-L20[19] Riess A G, Filippenko A V, Challis P, Clocchiatti A, Diercks A, Garnavich P M, Gilliland R L,Hogan C J, Jha S, Kirshner R P et al.
The Astronomical Journal et al.
The Astrophysical Journal et al.
The Astrophysical Journal Supplement Series [23] Allen S, Rapetti D, Schmidt R, Ebeling H, Morris R and Fabian A 2008
Monthly Notices of theRoyal Astronomical Society et al.
Physical Review D et al. APS April Meeting Abstracts [27] Cooke R J, Pettini M, Jorgenson R A, Murphy M T and Steidel C C 2014
The AstrophysicalJournal
Research in Astronomy andAstrophysics et al. Monthly Notices of the Royal AstronomicalSociety
Physical Review D et al. Monthly Notices of the Royal Astronomical Society et al.
Astronomy & Astrophysics
A59[33] Font-Ribera A, Kirkby D, Miralda-Escud´e J, Ross N P, Slosar A, Rich J, Aubourg ´E, Bailey S,Bhardwaj V, Bautista J et al.
Journal of Cosmology and Astroparticle Physics
Mon. Not. R. Astron. Soc. , L49[35] A. R. Liddle, 2007
Mon. Not. R. Astron. Soc. , L74[36] Betoule M, Kessler R, Guy J, Mosher J, Hardin D, Biswas R, Astier P, El-Hage P, Konig M,Kuhlmann S et al.
Astronomy & Astrophysics
A22[37] Arevalo F, Cid A, Moya J, 2017
The European Physical Journal C Phys. Rev. D Mon. Not. R. Astron. Soc.463