Constraints on the Generalized Chaplygin Gas Model from Gamma-Ray Bursts
aa r X i v : . [ a s t r o - ph . C O ] A ug Constraints on the Generalized Chaplygin Gas Model fromGamma-Ray Bursts
R. C. Freitas a, , S. V. B. Gon¸calves a, and H. E. S. Velten a,b, , a Grupo de Gravita¸c˜ao e Cosmologia, Departamento de F´ısica,Universidade Federal do Esp´ırito Santo, 29075-910, Vit´oria, Esp´ırito Santo, Brazilb Fakult¨at f¨ur Physik, Universit¨at Bielefeld, Bielefeld 33615, Germany. Abstract
We study the generalized Chaplygin gas model (GCGM) using Gamma-ray bursts ascosmological probes. In order to avoid the so-called circularity problem we use cosmology-independent data set and Bayesian statistics to impose constraints on the model parameters.We observe that a negative value for the parameter α is favoured in a flat Universe and theestimated value of the parameter H is lower than that found in literature.PACS number: 98.80.Es, 98.70.Rz One of the most important problems of Modern Cosmology is the determination of the mattercontent of the Universe. The rotation curve of spiral galaxies [1], the dynamics of galaxy clusters[2] and structure formation [3], indicate that there is about ten times more pressureless matterin the Universe than can be afforded by the baryonic matter. The nature of this dark mattercomponent remains unknown. Moreover, the Type Ia supernovae (SNe Ia) data indicates thatthe Universe is accelerating [4]. Models considering matter content dominated by an exotic fluidwhose pressure is negative [5], modified gravity theories such as f ( R ) [6] and the evolution ofan inhomogeneous Universe model described in terms of spatially averaged scalar variables withmatter and backreaction source terms [7] are some of the proposals to explain this current phaseof the Universe. At the same time, the position of the first acoustic peak in the spectrum of CMBanisotropies, as obtained by WMAP, favours a spatially flat Universe[8]. Combining all thesedata and if we consider the matter content of the Universe dominated by a fluid with negativepressure we have a scenario with a proportion of Ω m ∼ .
27 and Ω de ∼ .
73, with respect to thecritical density, for the fractions of the pressureless matter and dark energy, respectively. Thisscenario is usually called as the concordance cosmological model.The question is to know what is the nature of the dark matter and dark energy components.For dark matter many candidates have been suggested such as axions, a particle until nowundetected which would be a relic of a phase where the grand unified theory was valid [9],the lightest supersymmetric particle (LSP) like neutralinos [10] and the Kaluza-Klein particles[11] that are stable viable Weakly Interacting Massive Particles (WIMPs) and arise in twoframeworks: In Universal Extra Dimensions [12] and in some warped geometries like Randall-Sundrum [13]. For the dark energy, in the hydrodynamical representations of matter, the mostnatural candidate is a cosmological constant, but there is a discrepancy of some 120 orders ofmagnitude between its theoretical and observed values [14]. For this reason, other candidateshave been suggested like quintessence models that involve canonical kinetic terms of the self-interacting scalar field with the sound speed c s = 1 [15] and k-essence models that employrather exotic scalar fields with non-canonical (non-linear) kinetic terms which typically lead toa negative pressure [16]. More recently, a string-inspired fluid has been evoked: The Chaplygingas [17], that appears as a promising candidate for the dark sector of the Universe. e-mail: rc [email protected] e-mail: [email protected] e-mail: [email protected] p c = − Aρ c , (1)where p c represents the pressure, ρ c the fluid density and A is a parameter connected with thesound speed. This equation of state is suggested by a brane configuration in the context ofstring theories [18]. However, a more general equation of state has been suggested [19]: p c = − Aρ αc , (2)where again p c and ρ c stand for the generalized Chaplygin gas component and α is a newparameter, which takes the value 1 for the traditional Chaplygin gas but values larger than 1,or even negative may be considered. This is the so-called generalized Chaplygin gas.Much observational data that has been used for comparison with the theoretical cosmologicalmodels like the generalized Chaplygin gas model (GCGM). The spectra of anisotropy of cosmicmicrowave background radiation [20], baryonic acoustic oscillations [21], the integrated Sachs-Wolfe effect [22], the matter power spectrum [23], gravitational lenses [24], X-ray data [25]and ages estimates of high- z objects [26] have been used in this sense. Also, constraints fromcombined data sources have been obtained in [27]. Another tool used to make this comparisonis the Hubble diagram, the plot of redshift z versus luminosity distance d L = p L / π F , where L is the luminosity (the energy per time produced by the source in its rest frame) and F is themeasured flux, i.e., the energy per time per area measured by a detector. Normally, the SNeIa data are considered good standard candles and are used to construct the Hubble diagram,because their luminosity are well known [4, 28]. In particular, constraints on the GeneralizedChaplygin gas have been studied in [29]. These assumptions rest on a foundation of photometricand spectroscopic similarities between high- and low-redshift SNe Ia. But this discussion is notyet finished [30]. The other problem comes from the fact that there still does not exist SNe Iadata with z > .
8. To know the properties and behavior of dark energy for high values of z wewill have to wait for new data of the SNe Ia or to find other distance indicators. In this sense, toextend the comparison between observational data and theoretical models at very high redshiftwe propose to use Gamma-ray bursts (GRBs) due to the fact that they occur in the range ofhigh z beyond the SNe data found today [31].The GRBs are jets that release ∼ − ergs or more for a few seconds and becomes,in this brief period of time, the most bright object in the Universe. They were discovered inthe sixties by the Vela satellites in the “Outer Space Treaty” that monitored nuclear explosionsin space [32]. Launched in 1991 The Burst and Transient Source Experiment on the ComptonGamma-Ray Observatory (BATSE on the Compton GRO) [33] observations concluded that theangular distribution of the GRBs on the sky is isotropic within statistical limits. This studyruled out the idea that the GRBs are galactic objects, but it is consistent with the bursts beingextra-galactic sources at cosmological distances. More recently, the SWIFT mission (launchedin 2004) has provided the most accurate GRB data, available in the Swift BAT Catalog.The search for a self-consistent method to use the GRBs in cosmological problems is intenseand promising. But the possibility of using GRBs as standard candles is not a simple question.GRBs are known to have several light curves and spectral properties from which the luminosityof the burst can be calculated once calibrated, and these can make GRBs into standard candles.Just as with SNe Ia, the idea is to measure the luminosity indicators, deduce the source lumi-nosity, measure the observed flux and then use the inverse-square law to derive the luminositydistance. The difficulty arises when these indicators are a priori established through some cos-mological model like the concordance one. This means that the parameters of the calibratedrelations of luminosity/energy are still coupled to the cosmological parameters derived from a2iven cosmological model. This is the so called circularity problem. This problem appears inseveral works that have made use of these GRBs luminosity indicators as standard candles atvery high redshift [34]. It is possible to treat the circularity problem with a statistical approach[35]. On the other hand, many papers have dealt with the use of so called Amati relation, orthe Ghirlanda relation for this purpose [36]. However, as argued recently in [37], these pro-cedure involve many unjustified assumptions which if not true could invalidate the results. Inparticular, many evolutionary effects can affect the final outcome. However, recently Liang etal. [38, 39, 40] made a study considering SNe Ia as first-order standard candles for calibratingGRBs, the second-order standard candles. The sample in reference [38] was calibrated from the192 supernovae obtained in [41]. The updated sample used in [39, 40] has been obtained andcalibrated cosmology-independently from the Union2 (557 data points) compilation [42] releasedby the Supernova Cosmology Project Collaboration. In these articles the authors found rele-vant constraints on the Cardassian and Chaplygin gas model by adding to the GRB data theSNe Ia (Union2), the Shift parameter of the Cosmic Microwave Background radiation from theseven-year Wilkinson Microwave Anisotropy Probe and the baryonic acoustic oscillation fromthe spectroscopic Sloan Digital Sky Survey Data Release galaxy sample. The sample obtainedin [39] will be used in our analysis. These authors obtain the distance moduli µ of GRB inthe redshift range of SNe Ia and extend this result to very high redshift GRB ( z > .
4) in acompletely cosmological model-independent way. This approach has been also studied in [43].Some analysis have been made with the GCGM and the GRBs as distant markers [44]. In thereference [45] the authors build a specific distribution of GRB to probe the flat GCGM and theXCDM model. While the GCGM has an equation of state given by expression (2) the XCDMmodel is considered in terms of a constant equation of state ω = p/ρ <
0. The main conclusionof this article is that the use of GRBs as a dark energy probe is more limited when compared toSNe Ia. We anticipate that we shall arrive at a similar conclusion. Moreover the XCDM modelis better constrained than the GCGM. On the other hand, in [46] the GCGM and the ΛCDMmodel are compared by using the GRB and SNe Ia data to build the Hubble diagram. Theseauthors show through the statistical analysis that the Chaplygin gas model (they use α = 1)have the best fit when compared with the data. Also they verify that the transition redshiftbetween the decelerated and the accelerated state of the Universe occurs at z ∼ . − . z ∼ . − et al. [39], calibrated cosmology-independently from the Union2 compilation of SNe Ia, toconstraint the cosmological parameters of the GCGM. We want to show how GRB data couldconstraint different Chaplygin cosmologies.This paper is organized as follows. In next section, we described a brief review of GCGM.In section 3 the luminosity distance d L is obtained for the GCGM and compared with theobservational data. Finally, in section 4 we present our discussion and conclusions. We consider here an homogeneous and isotropic Universe described by the Friedmann’s equation (cid:18) ˙ aa (cid:19) + ka = 8 πG ρ m + ρ c ) , (3)where the density ρ has the subscripts m for the matter pressureless fluid and c for the generalizedChaplygin gas with equations of state p m = 0 and p c = − A/ρ αc , respectively. Dot meansderivative with respect to the cosmic time t . Flat, closed and open spatial sections correspondto k = 0 , , − ρ m + 3 ˙ aa ρ m = 0 → ρ m = ρ m a , (4)˙ ρ c + 3 ˙ aa (cid:18) ρ c − Aρ αc (cid:19) = 0 → ρ c = ρ c (cid:18) ¯ A + 1 − ¯ Aa α ) (cid:19) / (1+ α ) , (5)where ρ m = ρ m ( a ), ρ c = ρ c ( a ) = ( A + B ) / (1+ α ) with a ( t = 0) = a = 1 being the scalefactor today. The new definition of the constant A is given by ¯ A = A/ρ αc and it is connectedto the sound velocity today in the gas by the expression v s = p ∂p c /∂ρ c (cid:12)(cid:12)(cid:12) t = √ α ¯ A .Initially, the GCGM behaves like a dust fluid, with ρ ∝ a − , while at late times the GCGMbehaves as a cosmological constant term, ρ ∝ A / (1+ α ) . Hence, the GCGM interpolates a matterdominated phase (where the formation of structure occurs) and a de Sitter phase. At the sametime, the pressure is negative while the sound velocity is positive, avoiding instability problemsat small scales [47].In order to proceed with data comparison we need to calculate the luminosity distance inthe GCGM. Using the expression for the propagation of light and the Friedmann’s equation (3),we can express the luminosity distance as d L = a a r = (1 + z ) S [ f ( z )] , (6)where r is the co-moving coordinate of the source and S ( x ) = x for ( k = 0) ,S ( x ) = sin x for ( k = 1) ,S ( x ) = sinh x for ( k = − . (7)The function f ( z ) is given by f ( z ) = 1 H Z z dz ′ { Ω m ( z ′ + 1) + Ω c [ ¯ A + (1 − ¯ A )( z ′ + 1) α ) ] / (1+ α ) − Ω k ( z ′ + 1) } / , (8)with the definitions Ω m = 8 πG ρ m H , Ω c = 8 πG ρ c H , Ω k = − kH , (9)and Ω m + Ω c + Ω k = 1. The final equations have been also expressed in terms of the redshift z = − a .In our numerical calculations we relax the restriction that the pressureless matter componentis entirely given by baryons. We consider the nucleosynthesis results for the baryonic componentof the Universe and assume the total pressureless matter density as Ω m = Ω b + Ω dm , whereΩ b h = 0 . H = 100 hKms − M pc − . Then, in our notation Ω dm means the extra darkmatter contribution. The observational data set used in this article is composed by 42 GRBs from [39, 40]. As told inthe introduction, this sample has been obtained and calibrated cosmology-independently fromthe Union2 compilation. This fact is of crucial importance to admit GRBs as cosmological4robes since the circularity problem described above is avoided. At the same time, this dataset allow us to analyse the free parameters of the GCGM for a redshift range larger than theavailable data from SNIa reaching up to z ≈
6. It is important to emphasize that with thissample the authors of [39, 40] have obtained stark constraints on the Cardassian and Chaplygingas model by combining the GRB data with other cosmological probes.If we want to have a reliable sample of GRBs to make our analysis, the Hubble diagram forthe GRBs should be calibrated from the SN at z ≤ .
4. This allows to obtain the followingluminosity/energy relations: The τ lag − L relation, the V − L relation, the L − E p relation, the E γ − E p relation, the τ RT − L relation, the E iso − E p relation, and the E iso − E p − t b relation.In general these relations can be written as log( y ) = a + b log( x ) (two-variable relations) andlog( y ) = a + b log( x )+ b log( x ) (multi-variable relation). In this relations y is the luminosity inunits of erg s − or energy in units of erg and x is the GRB parameter measured in the rest frame;in the latter expression x and x are E p (1 + z ) / (300 keV) and t b / (1 + z ) / (1 day) respectively,and b and b are the slopes of x and x respectively. The calibration’s process is achieved usingtwo methods: linear interpolation (the bisector of the two ordinary least-squares) and the cubicinterpolation (the multiple variable regression analysis). The variables a and b i are determinatedwith 1 − σ uncertainties. With the linear interpolation, the error of the interpolated distancemodulus can be calculated by σ µ = ([( z i +1 − z ) / ( z i +1 − z i )] ǫ µ,i + [( z − z i ) / ( z i +1 − z i )] ǫ eµ,i +1 ) / ,where µ is the interpolated distance modulus of a source at redshift z , ǫ µ,i and ǫ eµ,i +1 are errorsof the SNe, µ i and µ i +1 are the distance moduli of the SNe at nearby redshifts z i and z i +1 ,respectively. In the case of the cubic interpolation method the error can be estimated by theexpression σ µ = ( A ǫ µ,i + A ǫ µ,i +1 + A ǫ µ,i +2 + A ǫ µ,i +3 ) / , where ǫ µ,i + j are errors of the SNeand µ i + j are the distance moduli of the SNe at nearby redshifts z i + j (index j run from 0 to 3)with: A = [( z i +1 − z )( z i +2 − z )( z i +3 − z )][( z i +1 − z i )( z i +2 − z i )( z i +3 − z i )] ; A = [( z i − z )( z i +2 − z )( z i +3 − z )][( z i − z i +1 )( z i +2 − z i +1 )( z i +3 − z i +1 )] ; A = [( z i − z )( z i +1 − z )( z i +3 − z )][( z i − z i +2 )( z i +1 − z i +2 )( z i +3 − z i +2 )] ; A = [( z i − z )( z i +1 − z )( z i +2 − z )][( z i − z i +3 )( z i +1 − z i +3 )( z i +2 − z i +3 )] . (10)The results obtained by the cubic interpolation method are almost similar to the results obtainedby the linear interpolation method. It is important to emphasize again that the calibrationresults are completely independent of cosmological models used (for further discussion, see [38]).In order to compare the GCGM with the observational data, the first step is to compute thetheoretical luminosity distance µ , µ th = 5 log (cid:18) d L Mpc (cid:19) + 25 , (11)with the relations for the GCGM described above. Here, as in [39, 40], by using only linearinterpolating we have the 27 GRBs at z ≤ . z > . τ lag − L, V − L, L − E p , E γ − E p , τ RT − L ) calibratedwith the sample at z ≤ . L ) or energy ( E γ ) of each burst at high redshift ( z > . µ = ( P i µ i /σ µ i ) / ( P i σ − µ i ), with its uncertainty5 i = ( P i σ − µ i ) − / , where the summations run from 1 to 5 over the five relations describedabove.Considering a set of free parameters { p } the agreement between theory and observation ismeasured by minimizing the quantity, χ ( p ) = X i =1 (cid:2) µ thi ( p ) − µ obsi ( p ) (cid:3) σ i , (12)where µ th and µ obs are the theoretical value and the observed value of the luminosity distancefor our model, respectively, and σ means the error for each data point. We use Bayesian analysisto obtain the parameters estimations through the probability distribution function (PDF) P = B e − χ p )2 , (13)where B is a normalization constant. A full Bayesian analysis is made by considering all freeparameters of the model. However, we will study some particular Chaplygin configurationsbefore a detailed analysis with 5 free parameters. With this strategy we hope to gain someintuition about the GRB data from the partial outcomes. Below, we will describe differentChaplygin-based cosmologies investigated in the present work. We show our results in Table 1-2and in Figures 1-7.Our first step is to study the Chaplygin gas ( α = 1). We remenber that this equation ofstate, as cited above, has also raised interest in particle physics thanks to its connection withstring theory and its supersymmetric extension [18]. We shall consider the prior information:0 ≤ ¯ A ≤
1, 0 ≤ Ω dm ≤ .
957 and 0 ≤ H ≤ dm = 0 . +0 . − . , ¯ A = 0 . +0 . − . and H = 51 . +9 . − . at 1 σ level. However,the dispersion in the GRB data is quite high. We compute these same estimatives using theSupernovae Constitution sample [48] in order to compare the dispersion of these two samples.For the SN we found Ω dm = 0 . +0 . − . , ¯ A = 0 . +0 . − . and H = 59 . +2 . − . at 1 σ level. Someconstraints on the Generalized Chaplygin gas have been placed using the Constitution data set[49]. This allows a comparison between some of our results and the ones from Supernovae. Ingeneral, GRBs recover the results from SNe but with a high dispersion.In our next analysis we relax the prior information about the Hubble parameter and leaveit free to vary. We show the two-dimensional PDFs in the Fig. (5). In Fig. (6) the solid linesare the corresponding one-dimensional probabilities.The above choice for the priors in the parameter α is conservative. With this choice we wantto avoid a super luminal propagation in the sound speed formula. However, as argued in [50]the formula v s = α ¯ A represents the group sound velocity. Actually, in order to violate causalitythe wavefront velocity should exceed 1 [51]. Considering this possibility we assume now α ≥ k to vary between [-0.6,0.6].For this case, we show the results in Fig. 7. 6ase α ¯ A Ω dm H Ω k CGM ( α = 1) → Fig. 2 1 0 . +0 . − . . +0 . − . . +9 . − . h = 0 . → Fig. 4 << . +0 . − . . +0 . − . . ≤ α ≤ → Fig. 6 − . +4 . − . . +0 . − . . +0 . − . . +9 . − . α ≥ → Fig. 6 − . +4 . − . . +0 . − . . +0 . − . . +9 . − . k = 0 (0 ≤ α < → Fig. 7 1 . +5 . − . . +0 . − . . +0 . − . . +10 . − . − . +0 . − . GCGM Ω k = 0 ( α ≥ → Fig. 7 1 . +5 . − . . +0 . . . +0 . − . . +8 . − . − . +0 . − . Table 1: For the different Chaplygin-based cosmologies in the first column we show the final 1Destimation for the free parameters. The errors are computed at 1 σ level.Figure Ω dm × H ¯ A × H ¯ A × Ω dm ¯ A × α Ω dm × α H × α
30 40 50 60 70 H W d m A - Marginalized
30 40 50 60 70 80 H A (cid:143)(cid:143) W dm Marginalized W dm A (cid:143)(cid:143) H Marginalized
Figure 1: Two-dimensional probability distribution function (PDF) for the free parameters inthe CGM. The curves represent 99.73%, 95.45% and 68.27% contours of maximum likelihood.The darker the region, the smaller the probability. W dm P D F H P D F A (cid:143)(cid:143) P D F Figure 2: One-dimensional PDFs for the three free parameters of the CGM.7 Α A (cid:143)(cid:143) W dm Marginalized W dm A (cid:143)(cid:143) Α Marginalized -20 -15 -10 -5 0 Α W d m A - Marginalized
Figure 3: Two-dimensional PDFs for the GCGM fixing H = 72 km s − M pc − . The curvesrepresent 99.73%, 95.45% and 68.27% contours of maximum likelihood. The darker the region,the smaller the probability. -10 -8 -6 -4 -2 0 Α P D F A (cid:143)(cid:143) P D F W dm P D F Figure 4: One-dimensional PDFs for the three free parameters of the GCGM when H =72 km s − M pc − . In this study we have analyzed the Chaplygin gas model with a sample of 42 GRBs. Althoughthe use of GRBs as a cosmological tool is a promising way to probe cosmology at high redshiftswe have verified that the available data is still insufficient to impose precise constraints incosmological models. As observed in our analysis, the dispersion is still high when comparedwith others observational data sets. However we hope that with the future data from thefinal Swift BAT Catalog we will be able to put strong constraints on the dark energy/matterproperties.In our analysis, the unification scenario was not imposed from the beginning. This meansthat we allow an extra dark matter contribution (Ω dm ) in our calculations in order to probewhether the unification scenario is favoured. In our first analysis the free parameters ( ¯ A, Ω dm and H ) of the Chaplygin gas ( α = 1) were well constrained. Our results are in agreement withthe Supernova results [52]. The only difference is that we find a lower value for the Hubbleparameter, H = 51 . +9 . − . (1 σ ). However, it is possible to find in the literature similar resultsfor the parameter H [53].In our second analysis, in order to check the behaviour of the model when H = 72 km s − M pc − we leave α free, that is the so called Generalized Chaplygin Gas Model. From Figs. 3 and 4 theunification scenario is again favoured. However, the uncertainties are still high. The parameter α assumes a large negative value. There is no any peak in the parameter α distribution and8 Α A (cid:143)(cid:143) W dm and H Marginalized W dm A (cid:143)(cid:143) Α and H Marginalized W dm -12-10-8-6-4-202 Α A (cid:143)(cid:143) and H Marginalized
40 50 60 70 H W d m A (cid:143)(cid:143) and Α Marginalized
35 40 45 50 55 60 65 70 H A (cid:143)(cid:143) W dm and Α Marginalized
35 40 45 50 55 60 65 70 H Α A (cid:143)(cid:143) and W dm Marginalized
Figure 5: The same as Fig. (1) but considering four free parameters for the GCGM and theprior 0 ≤ α ≤ H P D F A - P D F W dm P D F -40 -30 -20 -10 0 Α P D F Figure 6: One-dimensional PDF for the GCGM free parameters when H is free to vary. Thesolid lines correspond to the prior choice 0 ≤ α ≤ α ≥
0. The final estimation for the parameter α does not depend on its prior information.the probability remains constant for negative values. For the background dynamics the region( α < −
1) represents a behavior different from the matter dominated phase when structures startto form. On the other hand, negative values for α imply an imaginary sound velocity, leadingto small scale instabilities at the perturbative level. Rigourously, the general situation is morecomplex: such instabilities for fluids with negative pressure may disappear if the hydrodynami-cal approach is replaced by a more fundamental description using, e.g., scalar fields. However,this is not true for the Chaplygin gas: even in a fundamental approach, using for example theBorn-Infeld action, the sound speed may be negative if α <
0. Perhaps the restriction α ≥ H free to vary, we confirm that the hypersurface9
20 40 60 80 100 H P D F A - P D F W dm P D F Α P D F -0.35-0.3-0.25-0.2-0.15-0.1 W k P D F -0.8 -0.6 -0.4 -0.2 0 W k P D F Figure 7: One-dimensional PDF for the GCGM free parameters when we allow a non-vanishingcurvature. The solid lines correspond to the prior choice 0 ≤ α < α ≥
0. The final estimation for the parameter α does not depend on its priorinformation. H = 72 km s − M pc − doesn’t represent the maximum probability in the 4-D parameters phasespace. Chaplygin gas models show values lower than H = 72 km s − M pc − [52]. We observealso that there is a significant difference in the final parameter estimation when we consider theprior α ≥
0, instead of 0 ≤ α <
1. For instance, the unification scenario (Ω dm = 0) is favouredonly with the choice α ≥
0. Moreover, there is now a peak in the α distribution at α = − . +4 . − . but with a high dispersion. Again, negative values for α are favored, despite the two-dimensionalPDF ( α x H ) in Fig.5 indicates a high probability for α >
6. Such a contradiction seems to bean artifact of the marginalization process as can be seen also in the two-dimensional PDFs (Ω dm x H ) and ( ¯ A x H ) in Fig. 5. These plots confirm that the final 1D estimation can be verydifferent from the partial 2D ones. This diferrence is due to the integration of the probabilityfunction over the adopted prior values of the remmaning parameters.The analysis with five free parameters confirm some of the previous results. Negative cur-vature is prefered as well as Sn data data [52]. Also, the parameter α is now estimated with apositive value, in constrast with the previous results.The Chaplygin gas parameters have been estimated in many papers, considering differentanalysis and several observational data sets. Constraints critically depend on whether one treatsthe Chaplygin gas as true quartessence (replacing both dark matter and dark energy) or if oneallows it to coexist with a normal dark matter component. The former situation is widelyconsidered in the literature. As we leave the density parameter Ω dm free to vary in all thecases analysed here, it is not possible to directly compare our results with unified Chaplygincosmologies unless we assume the prior Ω dm = 0. This case has been studied using GRBs andother probes in reference [40]. For a comparison with this reference, figure 8 shows the two-dimmensional probability for the free parameter of the unified (Ω dm = 0) GCG model. The bestfit occurs at ( α = 0 . , ¯ A = 0 . σ ) with the joint analysis showed in[40].The analysis of section 3 can be compared with [52], where the influence of a free Ω dm parameter on the final estimations was taken into account. Our results have high confidence10 Α A (cid:143)(cid:143) H Marginalized
Figure 8: Constrainst on the free parameters of the unified (Ω dm = 0) GCG model.with the results obtained in [52].Finally, we remark that, perturbative analysis of Chaplygin models, for instance, revealsa large positive value ( α >> α [54] while kinematic tests show valuesnegatives or close to zero. At the background level, the crossing of different data sets (includingfor example SNe, CMB, BAO, H(z) data and galaxy cluster mass fraction) will provide a moreaccurate scenario for each Chaplygin-based cosmology studied in this work. We leave thisanalysis, including the perturbative study, for a future work. Acknowledgements
R.C.F., S.V.B.G. and H.E.S.V. thank DAAD (Germany) and CNPq, CAPES and FAPES(Brazil) for partial financial support. S.V.B.G. thanks the Laboratoire d’Annecy-le-Vieux dePhysique Theorique (France) and H.E.S.V. thanks the Fakult¨at f¨ur Physik, Universit¨at Bielefeld(Germany) for kind hospitality during part of the elaboration of this work. We thank WinfriedZimdahl and Christian Byrnes for the comments and suggestions.
References [1] M. Persic, P. Salucci and F. Stel, Mon. Not. Roy. Astron. Soc. , 27 (1996); A. Borriello and P. Salucci,Mon. Not. Roy. Astron. Soc. , 285 (2001).[2] C. S. Frenk, A. E. Evrard, S. D .M. White and F. J. Summers, Astrophys.J. , 460 (1996).[3]
The Best Theory of Cosmic Structure Formation is Cold + Hot Dark Matter (CHDM) , J. Primack, Pro-ceedings of the Princeton 250th Anniversary conference, June 1996, Critical Dialogues in Cosmology, ed. N.Turok (World Scientific), [arXiv:astro-ph/9610078].[4] A. G. Riess et al. , Astron. J. , 1009(1998); S. Perlmutter et al. , Nature, , 51(1998); J.L. Tonry etal. , Astrophys. J. , 1(2003).[5] T. Padmanabhan, Gen. Rel. Grav. , 529 (2008).[6] S. Capozziello, S. Nojiri, S. D. Odintsov and A. Troisi, Phys. Lett. B 639 , 135 (2006),[arXiv:astro-ph/0604431]; S. Nojiri and S. D. Odintsov, Phys. Rev.
D 74 , 086005 (2006),[arXiv:hep-th/0608008]; S. Nojiri and S. D. Odintsov, Int. J. Geom. Meth. Mod. Phys. , 115 (2007),[arXiv:hep-th/0601213]; L. Amendola, R. Gannouji, D. Polarski and Shinji Tsujikawa, Phys. Rev. D 75 ,083504 (2007), [arXiv:gr-qc/0612180].[7] T. Buchert, M. Kerscher and C. Sicka, Phys. Rev.
D62 , 043525 (2000), [arXiv:astro-ph/9912347]; S. Rasa-nen, JCAP , 003 (2004), [arXiv:astro-ph/0311257]. http://lambda.gsfc.nasa.gov/product/map/dr3/parameters.cfm .[9] E. P. S. Shellard and R. A. Battye, Phys. Rept. , 227 (1998), [arXiv:astro-ph/9808220].[10] A. Falvard et al. , Astropart. Phys. , 467 (2004), [arXiv:astro-ph/0210184].[11] A. Bottino, F. Donato, N. Fornengo and P. Salati, Phys. Rev. D 72 , 083518 (2005), [arXiv:hep-ph/0507086].[12] T. Appelquist, H. C. Cheng, and B. A. Dobrescu, Phys. Rev.
D 64 , 035002 (2001), [arXiv:hep-ph/0012100].[13] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999), [arXiv:hep-ph/9905221].[14] S. Weinberg, Rev. Mod. Phys. , 1 (1989); S. M. Carroll, Living Rev. Rel. , 1 (2001),[arXiv:astro-ph/0004075];[15] B. Ratra and P. J. E. Peebles, Phys. Rev. D 37 , 3406 (1988); R. R. Caldwell, R. Dave, and P. J. Steinhardt,Phys. Rev. Lett. , 1582 (1998), [arXiv:astro-ph/9708069].[16] C. Armendariz-Picon, V. Mukhanov, and P. J. Steinhardt, Phys. Rev. Lett. , 4438 (2000),[arXiv:astro-ph/0004134]; M. Malquarti, E. J. Copeland, A. R. Liddle, and M. Trodden, Phys. Rev. D67 , 123503 (2003), [arXiv:astro-ph/0302279].[17] A. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett.
B511 , 265 (2001), [arXiv:gr-qc/0103004].[18] M. Bordemann and J. Hoppe, Phys. Lett.
B317 , 315 (1993), [arXiv:hep-th/9307036]; N. Ogawa, Phys. Rev.
D 62 , 085023 (2000), [arXiv:hep-th/0003288]; R. Jackiw and A. P. Polychronakos: Supersymmetric fluidmechanics. Phys. Rev. D 62, 085019 (2000)[19] M. C. Bento, O. Bertolami and A. A. Sen, Phys. Rev.
D 66 , 043507 (2002), [arXiv:gr-qc/0202064].[20] L. Amendola, F. Finelli, C. Burigana and D. Carturan, JCAP, , 005 (2003).[21] Puxum Wu and Hongwei Yu, ApJ, , 663 (2007).[22] T. Giannantonio and A. Melchiorri, Classical and Quantum Gravity, , 12 (2006).[23] M. C. Bento, O. Bertolami and A. A. Sen, Phys. Rev. D 70 , 083519 (2004), [astro-ph/0407239]; N. Bilic,R. J. Lindebaum, G. B. Tupper and Raoul D. Viollier, JCAP , 008 (2004), [astro-ph/0307214]; J. C.Fabris, S. V. B. Gon¸calves and R. de S´a Ribeiro, Gen. Rel. Grav. , 211 (2004), [arXiv:astro-ph/0307028].[24] P. T. Silva and O. Bertolami, Astrophys. J. , 829 (2003), [arXiv:astro-ph/0303353].[25] J. V. Cunha, J. S. Alcaniz and J. A. S. Lima, Phys. Rev. D 69 , 083501 (2004), [arXiv:astro-ph/0306319].[26] J. S. Alcaniz, D. Jain and A. Dev, Phys. Rev.
D 67 , 043514 (2003), [arXiv:astro-ph/0210476].[27] S. del Campo and J. Villanueva, IJMPD , 2007 (2009); Ch-G. Park, Jai-chan Hwang, J. Park and H.Noh, arXiv:0910.4202.[28] A. V. Filippenko, White Dwarfs: Comsological and galactic probes vol. 332, Springer (2005)[arXiv:astro-ph/0410609].[29] J.C. Fabris, S. V. B. Gon¸calves and P. E. de Souza [arXiv:astro-ph/0207430]; R. Colistete, J. C. Fabris,S.V.B. Gon¸calves and P.E. de Souza, Int. J. Mod. Phys.
D13 , 669 (2004); P. T. Silva and O. Bertolami,Astrophys. J. , 829 (2003).[30] J. Berian James, T. M. Davis, B. P. Schmidt and A. G. Kim, Mon. Not .Roy. Astron. Soc. , 933 (2006),[arXiv:astro-ph/0605147].[31] D. Q. Lamb and D. E. Reichart, Proceed. Rome Workshop on Gamma-ray Bursts in the Afterglow Era[arXiv:astro-ph/0108099]; V. Bromm and A. Loeb, Apj 575, 11 (2002) [arXiv: astro-ph 0201400].[32] R. W. Klebesadel, I. B. Strong and R. A. Olson, Ap. J. Lett. , L85 (1973).[33] C. A. Meegan et al. , Nature , 143 (1992); E. Costa et al. , Nature , 783 (1997).
34] Z. G. Dai, E. W. Liang and D. Xu, Astrophys. J. , L101 (2004), [arXiv:astro-ph/0407497]; D. Xu, Z. G.Dai and E. W. Liang, Astrophys. J. , 603 (2005), [arXiv:astro-ph/0501458].[35] B. Schaefer et. al., Astrophys. J. , 102 (2003); C. Firmani, G. Ghisellini, G. Ghirlanda and V. Avila-Reese, Mon. Not. Roy. Astron. Soc. , L1 (2005); B. E. Schaefer, Astrophys. J. , 16 (2007); L. Amatiet. al., Mon. Not. Roy. Astron. Soc. , 577 (2008).[36] C. Firmani, G. Ghisellini, G. Ghirlanda and V. Avila-Reese, Mon. Not. Roy. Astron. Soc. , L1 (2005),[arXiv:astro-ph/0501395].[37] Vahe Petrosian, Aurelien Bouvier, Felix Ryde, [arXiv:0909.5051].[38] N. Liang, W. K. Xiao, Y. Liu and S. N. Zhang, Astrophys. J. , 354 (2008), [arXiv:0802.4262].[39] N. Liang, P. Wu and Z. H. Zhu, (2010), [arXiv:1006.1105v1].[40] N. Liang, L. Xu and ZH. Zhu, Astronomy & Astrophysics , A11 (2011), [arXiv:1009.6059v3].[41] T. M. Davies et al. Astrophys. J., , 716 (2007).[42] L. Amanullah et al, Astrophys. J., , 712 (2010) [arXiv:1004.1711].[43] S. Capozziello and L. Izzo, Astron. Astrophys. , 31 (2008); L. Izzo, S. Capozziello, G. Covone and M.Capaccioli, Astron. Astrophys. , 63 (2009); V.F. Cardone, S. Capozziello and M.G. Dainotti, Mon. Not.R. Astron. Soc. 400, (2009); N. Liang, P. Wu, S. N. Zhang, Phys. Rev. D , 083518 (2010); H. Wei,JCAP , 020 (2010); M. Demianski, E. Piedipalumbo and C. Rubano, (2010)[arXiv:1010.0855v1].[44] F. Y. Wang, Z. G. Dai and S. Qi, Astron. Astrophys. , 53 (2009); N. Liang, L. Xu, Z. H. Zhu, Astron.Astrophys. , (2011).[45] O. Bertolami, P. T. Silva, Mon. Not. Roy. Astron. Soc. , 1149 (2006), [arXiv:astro-ph/0507192v1].[46] H. J. Mosquera Cuesta, M. H. Dumet, R. Turcati, C. A. Bonilla Quintero, C. Furlanetto and J. Morais,[arXiv:astro-ph/0610796v1].[47] J. C. Fabris and J. Martin, Phys. Rev. D 55 , 5205 (1997).[48] M. Hicken et al ., Astrophys. J. , 1097 (2009). [arXiv:astro-ph/0901.4804].[49] L. Xu and J. Lu, JCAP 1003, 25 (2010) [arXiv:1004.3344].[50] V. Gorini, A. Y. Kamenshchik, U. Moschella, O. F. Piattella and A. A. Starobinsky, JCAP , 016 (2008),[arXiv:0711.4242].[51] L. Brillouin, Wave Propagation and Group Velocity , 1960 (Academic Press).[52] R. Colistete Jr. and J. C. Fabris, Class. Quant. Grav , 2813 (2005), [arXiv:astro-ph/0501519].[53] Verkhodanov, O. V., Parijskij, Yu. N. and Starobinsky, A. A., Bull. Spec. Astrophys. obs., , 5-15 (2005);Arp, H., Astrophys J. , 615-618 (2002).[54] J. C. Fabris, S. V. B. Gon¸calves, H. E. S. Velten and W. Zimdahl, Phys. Rev. D , 103523 (2008); O.F.Piattella, JCAP 1003:012 (2010); J. C. Fabris, H. E. S. Velten and W. Zimdahl, Phys.Rev.D81:087303(2010)., 103523 (2008); O.F.Piattella, JCAP 1003:012 (2010); J. C. Fabris, H. E. S. Velten and W. Zimdahl, Phys.Rev.D81:087303(2010).