Latest supernova data in the framework of the generalized Chaplygin gas model

Monthly Notices of the Royal Astronomical Society, Sep 2004

We use the most recent Type Ia supernova data in order to study the dark energy–dark matter (XCDM) unification approach in the context of the generalized Chaplygin gas (GCG) model. Rather surprisingly, we find that data allow models with α > 1. We have studied how the GCG adjusts flat and non-flat models, and our results show that GCG is consistent with a flat case up to 68 per cent confidence level. Actually this holds even if one relaxes the flat prior assumption. We have also analysed what one should expect from a future experiment such as the Supernova/Acceleration Probe (SNAP). We find that there is a degeneracy between the GCG model and an XCDM model with a phantom-like dark energy component.

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Latest supernova data in the framework of the generalized Chaplygin gas model

Mon. Not. R. Astron. Soc. Latest supernova data in the framework of the generalized Chaplygin gas model O. Bertolami 1 A. A. Sen 1 S. Sen 0 P. T. Silva 1 0 CAAUL, Departamento de F ́ısica da FCUL , Campo Grande 1749-016 Lisboa , Portugal 1 Instituto Superior Te ́cnico, Departamento de F ́ısica , Av. Rovisco Pais, 1, 1049-001 Lisboa , Portugal A B S T R A C T We use the most recent Type Ia supernova data in order to study the dark energy-dark matter (XCDM) unification approach in the context of the generalized Chaplygin gas (GCG) model. Rather surprisingly, we find that data allow models with α > 1. We have studied how the GCG adjusts flat and non-flat models, and our results show that GCG is consistent with a flat case up to 68 per cent confidence level. Actually this holds even if one relaxes the flat prior assumption. We have also analysed what one should expect from a future experiment such as the Supernova/Acceleration Probe (SNAP). We find that there is a degeneracy between the GCG model and an XCDM model with a phantom-like dark energy component. methods; data analysis - supernovae; general - cosmological parameters - cosmology; observations - distance scale 1 I N T R O D U C T I O N Recent cosmological observations reveal that the Universe is dominated by two invisible components. Type Ia supernova (SN Ia) observations (Garnavich et al. 1998; Riess et al. 1998; Perlmutter et al. 1999), nucleosynthesis constraints (Burles, Nollet & Turner 2001), cosmic microwave background radiation (CMBR) power spectrum (Balbi et al. 2000; de Bernardis et al. 2000; Jaffe et al. 2001), largescale structure (Peacock et al. 2001) and determinations of the matter density (Bachall & Fan 1998; Carlberg et al. 1999; Turner 2000) allow for a model where the clumpy component that traces matter, dark matter, amounts for about 23 per cent of the cosmic energy budget, while an overall smoothly distributed component, dark energy, amounts for approximately 73 per cent of the cosmic energy budget. The most interesting feature of this dark energy component is that it has a negative pressure and drives the current accelerated expansion of the Universe (Garnavich et al. 1998; Riess et al. 1998; Perlmutter et al. 1999). On the theoretical side, great effort has been devoted to model dark energy. The most obvious candidate is the vacuum energy, an uncancelled cosmological constant (see, for example, Bento & Bertolami 1999; Bento, Bertolami & Silva 2001) for which ωx ≡ px/ρ x = −1. Another possibility is a dynamical vacuum (Bronstein 1933; Bertolami 1986a,b; Ozer & Taha 1987) or quintessence. Quintessence models most often involve a single scalar field (Ratra & Peebles 1988a,b; Wetterich 1988; Caldwell, Dave & Steinhardt 1998; Ferreira & Joyce 1998; E-mail: (OB); (AAS); (SS); (PTS) Amendola 1999; Bine´truy 1999; Kim 1999; Uzan 1999; Zlatev, Wang & Steinhardt 1999; Albrecht & Skordis 2000; Bertolami & Martins 2000; Banerjee & Pavo´n 2001a,b; Sen & Sen 2001; Sen, Sen & Sethi 2001) or two coupled fields (Fujii 2000; Masiero, Pietroni & Rosati 2000; Bento, Bertolami & Santos 2002a). In these models, the cosmic coincidence problem, i.e. why did the dark energy start to dominate the cosmological evolution only fairly recently, has no satisfactory solution and some fine tuning is required. More recently, it has been proposed that the evidence for a dark energy component might be explained by a change in the equation of state of the background fluid, with an exotic equation of state, the generalized Chaplygin gas (GCG) model, rather than by a cosmological constant or the dynamics of a scalar field rolling down a potential (Kamenshchik, Moschella & Pasquier 2001; Bento, Bertolami & Sen 2002b; Bilic´, Tupper & Viollier 2002). In this proposal, we consider the evolution of the equation of state of the background fluid instead of a quintessence potential. The striking feature of this model is that it allows for a unification of dark energy and dark matter (Bento et al. 2002b). Moreover, it is shown that the GCG model may be accommodated within the standard structure formation scenario (Bilic´ et al. 2002; Bento et al. 2002b). Concerns about this point have been raised by Sandvik et al. (2002), however, in this analysis, the effect of baryons has not been taken into account, which was shown to be important and allowing compatibility with the 2DF mass power spectrum (Bec¸a et al. 2003). Also, the Sandvik et al. (2002) claim was based on the linear treatment of perturbations close to the present time, thus neglecting any non-linear effects. Thus, given its potentialities, the GCG model has been the subject of great interest, and various attempts have been made to constrain its parameters using the available observational data. Studies include supernova data and power spectrum (Avelino et al. 2002), age of the Universe and strong lensing statistics (Dev, Alcaniz & Jain 2002), age of the Universe and supernova data (Alcaniz, Jain & Dev 2003; Makler, Oliveira & Waga 2002). The tightest constraints were obtained by Bento, Bertolami & Sen (2003a) using the CMBR power spectrum measurements from BOOMERANG (de Bernardis et al. 2002) and Archeops (Benoit et al. 2003), together with the SNe Ia constraints. It is shown that 0.74 As 0.85 and α 0.6, ruling out the pure Chaplygin gas model. From the bound arising from the age of the APM 08279+5255 source, which is As 0.81 (Alcaniz et al. 2003), we can obtain tight constraints, namely 0.81 As 0.85 and 0.2 α 0.6, which also rules out the cold dark matter (CDM) model. These results were in agreement with the Wilkinson Microwave Anisotropy Probe (WMAP) data (Bento, Bertolami & Sen 2003b). It was also shown that the gravitational lensing statistics from future large surveys together with SN Ia data from the Supernova/Acceleration Probe (SNAP) will be able to place interesting constraints on the parameters of the GCG model (Silva & Bertolami 2003). As we shall see in Sections 3 and 4, all these constraints are consistent with supernova data at 95 per cent confidence level. Recently, Choudhury & Padmanabhan (2003) have analysed the supernova data with currently available 194 data points (see also Padmanabhan & Choudhury 2003) and have shown that it yields relevant constraints on some cosmological parameters. In particular, it shows that when one considers the full supernova data set, it rules out the decelerating model with significant confidence level. They have also shown that one can measure the current value of the dark energy equation of state with higher accuracy and the data prefer the phantom kind of equation of state, ω X < −1 (Caldwell 2002). Moreover, the most significant observation of their analysis is that, without a flat prior, the latest supernova data also rule out the preferred flat CDM model, which is consistent with other cosmological observations. In a previous paper, Alam et al. (2003) have reconstructed the equation of state of the dark energy component using the same set of supernova data and have found that the dark energy evolves rapidly from ωx 0 in the past to a strongly negative equation of state (ωx −1) in the present, suggesting that CDM may not be a good choice for dark energy. More recently, other groups have also analysed these recent supernova data in the context of different cosmological models for dark energy (Gong & Duan 2004; Nesseris & Perivolaropoulos 2004). In this paper, we analyse the GCG model in the light of the latest supernova data (Tonry et al. 2003; Barris et al. 2004). We consider both flat and non-flat models. Our analysis shows that the problem with the flat model, which has been discussed in Choudhury & Padmanabhan (2003), can be solved in the GCG model in the sense that the flat GCG model is consistent with the latest supernova data even without a flat prior. We have also analysed the confidence contours for a GCG model, which one expects from a future experiment such as SNAP. We find that there is a degeneracy between the GCG model and an XCDM model with a phantom-like dark energy component. This paper is organized as follows. In Section 2, we discuss various aspects of the GCG model and its theoretical underlying assumptions. In Section 3, we describe our best-fitting analysis of the most recent supernova data in the context of the GCG model. Section 4 contains our analysis for expected SNAP results. Finally, in Section 5 we present our conclusions. 2 G E N E R A L I Z E D C H A P LY G I N G A S M O D E L The GCG is characterized by the equation of state A where A and α are positive constant. For α = 1 the equation of state is reduced to the so-called Chaplygin gas scenario first studied in a cosmological context by Kamenshchik et al. (2001). Inserting the above equation of state into the energy conservation equation, we can integrate it to obtain (Bento et al. 2002b) where ρ ch0 is the present energy density of GCG and As ≡ A/ρ(c1h+0α). One of the most striking features of this expression is that the energy density of this GCG, ρ ch, interpolates between a dustdominated phase, ρ ch ∝ a−3, in the past and a de Sitter phase, ρ ch = − pch, at late times. This property makes the GCG model an interesting candidate for the unification of dark matter and dark energy. Indeed, it can be shown that the GCG model admits inhomogeneities and that, under the Zel’dovich approximation, they evolve in a qualitatively similar fashion like the CDM model (Bento et al. 2002b). Furthermore, this evolution is controlled by the homogeneous parameters of the model, namely, α and A. There are several important aspects of the above equation which we should discuss before constraining the relevant parameters using supernova data. First, we can see from the above equation that As must lie in the range 0 As 1. For As = 0, GCG behaves always as matter, whereas for As = 1 it behaves always as a cosmological constant. Hence, to use it as a unified candidate for dark matter and dark energy, we have to exclude these two possibilities resulting in the range for As as 0 < As < 1. To have an idea about the possible range for α, we have to consider the propagation of sound through this fluid. Given any Lagrangian L(X , φ) for a field φ, where X = (1/2) gµν φ ,µ φ ,ν , the effective speed of sound entering the equations for the evolution of small fluctuations is given by Thus, for a standard scalar field model which has a canonical kinetic energy term like L = X − V (φ), the speed of sound is always equal to 1, irrespective of the equation of state. However, for a Lagrangian containing a non-canonical kinetic energy term, one can have a sound speed quite different from 1. Actually, even cs2 > 1 is possible, which physically means that the perturbations of the background fluid can travel faster than light as measured in the preferred frame where the background is homogeneous. For a time-dependent background field, this does not lead to any violation of causality as the underlying theory is manifestly Lorentz invariant (Erickson et al. 2002). For GCG it has been shown that the equation of state (1) can be obtained from a generalized version of the Born–Infeld action (Bento et al. 2002b) L = − A1/(1+α) 1 − (gµν φ,µ φ,ν )(1+α)/2α α/(1+α) , which, for α = 1, leads to the Born–Infeld action. If we compute cs2 for this action, we can obtain using equation (2) the present value as cs20 = α As. As As is always positive, it restricts α to only positive values. In all previous work, α has been restricted to a value up to 1. However, we can see from the above expression for cs20 that, as 0 < As < 1, the maximum allowed value for α can be surely greater than 1 and that also depends on the value of As, e.g. for As = 0.5 the allowed range for α is 0 α 2. Notice also that the dominant energy condition ρ + p 0 is always valid in this case. Furthermore, there is no big rip in the future and asymptotically the Universe goes toward a de Sitter phase. Hence, on general grounds, restricting α up to 1 is not a very justified assumption. Moreover, this restriction arises mainly by considering the present-day value of cs2, which is not an important epoch for structure formation. In general, cs2 in this model is a time-dependent quantity and in such cases it is not very proper to constrain α with the present-day value of cs2. There is another reason for not restricting α up to 1. It has been shown by Kamenshchik et al. (2001) that one can also model the Chaplygin gas with a minimally coupled scalar field with canonical kinetic energy term in the Lagrangian density. Performing this exercise for the GCG leads to a potential for this scalar field of the form In our analysis, we take the 230 data points listed in Tonry et al. (2003) along with the more recent 23 points from Barris et al. (2004). Also, as discussed by Choudhury & Padmanabhan (2003), for low redshifts, data might be affected by the peculiar motions, making the measurements of the cosmological redshifts uncertain; hence we shall consider only those points with redshifts z > 0.01. Moreover, because it is difficult to be sure about the host galaxy extinction, Av, we do not consider points which have Av > 0.5. Hence in our final analysis, we consider only 194 points, which are similar to those considered by Choudhury & Padmanabhan (2003). The supernova data points given by Tonry et al. (2003) and Barris et al. (2004) are listed in terms of luminosity distance log10 dL(z) together with the corresponding error σlog10 dL (z). These distances are obtained assuming some value of M, which may not be the true value. Hence, in our analysis we shall keep it as a free parameter while fitting the data. The best-fitting model is obtained by minimizing the quantity i=1 log10 dLobs(zi ) − 0.2M − log10 dLth(zi ; cα) 2 where M = M − Mobs is a free parameter denoting the difference between the actual M and the assumed value Mobs in the data. As discussed by Choudhury & Padmanabhan (2003), we have also taken into account the uncertainty arising because of the peculiar motion at low redshift by adding an uncertainty of v = 500 km 2 2 σlog10 DL (z) → σlog10 DL (z) + ln 10 DL c This correction is more effective at low redshifts, i.e. for small values of DL. In our subsequent best-fitting analysis, the minimization of equation (8) is performed with respect to M , α, As and k. The parameter M is a model-independent parameter and hence its best-fitting value should not depend on a specific model. We have checked that when minimizing equation (8) with respect to M , the best-fitting value for M for all of the models considered here is −0.033, which is also consistent with that obtained by Choudhury & Padmanabhan (2003). Hence, in our subsequent analysis, we shall use always this best-fitting value, M = −0.033. 3.1 Flat case For this case, we assume k = 0 and consider only two parameters, α and As. We first restrict α to be 1. In Fig. 1, we have shown the 68 and 95 per cent confidence contours in α– As parameter space. The best-fitting values for [α, As] are given by [0.999, 0.79]. The best-fitting value of α is very close to its upper limit because the actual best-fitting value lies in the region beyond α = 1. Also, up to 68 per cent confidence level, the α = 0, i.e. the CDM case, is excluded although it is consistent at 95 per cent confidence level. Next we allow α to vary for a wider range. Fig. 2 represents the same as Fig. 1 but with α taking a wider range. The best-fitting values for [α, As] are now [3.75, 0.936]. This high value of α might just be a statistical fluke though, as the confidence regions exhibit a very shallow valley along the α direction. Here, also at 68 per cent confidence level, the α = 0, CDM case, is excluded, although it is consistent at 95 per cent confidence level. It should be noted that both Choudhury & Padmanabhan (2003) and Tonry et al. (2003) also found some conflict between the CDM model and the SN Ia data; namely, that when imposing a flat universe prior, the data ruled where V 0 is a constant and m = 3(α + 1). For α = 1, we recover the potential obtained by Kamenshchik et al. (2001). Now, as we have discussed earlier, for a minimally coupled scalar field with a canonical kinetic energy term, the value of cs2 is always 1 irrespective of the equation of state. Hence, if we consider this type of scalar field to model GCG, there is no such restriction on α coming from the sound speed. In what follows, we shall consider that As lies in the range 0 < As < 1 and the only constraint on α that we shall consider is that it takes positive values. We should also point out that the α = 0 case corresponds to the CDM model. The Friedmann equation for a non-flat unified GCG model, in general, is given by H 2 = H02 ch As + (1 − As)(1 + z)3(1+α) 1/(1+α) + ch = 1 − where H 0 is the present-day value of the Hubble constant, and ch and k are the present-day density parameters for GCG and the curvature. For a flat universe k = 0 and ch = 1, whereas for the non-flat case k. 3 R E C E N T S U P E R N OVA D ATA A N D T H E B E S T- F I T T I N G A N A LY S I S To perform the best-fitting analysis of our GCG model with the recent supernova data, we follow the method discussed by Choudhury & Padmanabhan (2003). As far as the supernova observation is concerned, the cosmologically relevant quantity is the apparent magnitude, m, given by m(z) = M + 5 log10 DL (z), where DL = (H 0/c)dL(z) is the dimensionless luminosity distance, and the luminosity distance dL(z) is given by dL(z) = r (z)(1 + z) where r(z) is the comoving distance. Also M = M + 5 log10[(c/H0)/1 Mpc] + 25 where M is the absolute magnitude for the supernova which is believed to be constant for all SNe Ia. Using best fit value As = 0.89 −0.08 −0.06 −0.04 −0.02 our Universe deviates slightly from the flat model assuming k to vary between [ −0.1, 0.1]. In this case the best-fitting values for α, As and k are [2.87, 0.89, −0.099], which suggests that the data prefer a negative curvature. In Fig. 3, we have shown the 68 and 95 per cent confidence contours in the k–α plane assuming the best-fitting value for As, whereas in Fig. 4, we have shown the same contours in the k– As plane assuming the best-fitting value for α. Both α and As are constrained significantly and 68 per cent confidence limits on α and As are [1.6 3.625)] and [0.856 0.946)], respectively. This shows that the data prefer a higher value for α and the CDM limit (α = 0) is excluded, both for 68 and 95 per cent confidence limits. Also, we can see from both figures that the flat case k = 0 is consistent with the data for both 68 and 95 per cent confidence levels, but for a higher value of α. We allow now more curvature in our model and consider the range for k to vary as [−1, 1]. In this case, the best-fitting values for α, As and k are [0.73, 0.65, −0.999] and the resulting χ 2min is 197.99, showing that there is an improvement in the quality of fit. The 68 per cent confidence limits on α and As are [0.052 1.056] and [0.62 0.81], respectively. Also, the allowed range for α shifts more significantly −0.08 −0.06 −0.04 −0.02 0 Ω k Figure 1. Confidence contours in the α– As parameter space for flat unified GCG model. The solid and dashed lines represent the 68 and 95 per cent confidence regions, respectively. The best-fitting value used for M is −0.033. Figure 3. Confidence contours in the k–α parameter space for a non-flat unified GCG model. The solid and dashed lines represent the 68 and 95 per cent confidence regions, respectively. The best-fitting value used for M is −0.033. Best fit values: out the vacuum energy as an allowed dark energy component, and actually favoured a phantom energy component. Here we see that the GCG model fits the data well, and, as mentioned previously, without the theoretical complications that plague the phantom energy model; namely, the dominant energy condition is not broken, and there is no big rip singularity in the future. It is also clear from the minimum value for χ 2 obtained in these two cases that when we allow α to vary beyond 1, we obtain a better fit to the supernova data. 3.2 Non-flat case Another problem that the CDM has with the new SN Ia data is that without a flat prior, a flat CDM universe has also been ruled out at 68 per cent confidence (Choudhury & Padmanabhan 2003). Tonry et al. (2003) argued that this was probably due to some overlooked systematic error, because a small systematic error of 0.04 mag was able make the flat CDM consistent with the data. Here we attempt to find if the GCG model might alleviate this problem. For this, we allow a non-vanishing curvature in our model. We now have three parameters in our model: α, As and the density parameter for the curvature at present, k. First, we assume that As = 0.65 0 Ω k 0.6 s A As = 0.81 0 Ω k towards smaller values and data do allow the model to get closer to CDM (α = 0.052) as well as to the Chaplygin gas model (α = 1). In Fig. 5, we have shown the 68 and 95 per cent confidence contours in the k–α plane assuming the best-fitting value for As, as well as for its values in the wings of 68 per cent confidence limit. This shows that the allowed range is quite sensitive to the parameter As. For smaller values of As (but within its 68 per cent confidence limit) the flat model ( k = 0) is more inconsistent with the data and negative curvature is preferred. However, for higher values of As, e.g. As = 0.81, which still falls within the 68 per cent confidence limit, it shows that flat models as well as models with small but both positive and negative curvature are allowed (consistent with WMAP bound on total, 0.96 < total < 1.08; Spergel et al. 2003) but for non-zero values of α. This suggests that, assuming a GCG model, we can alleviate the problem of consistency with a flat universe as pointed out by Choudhury & Padmanabhan (2003) for a CDM model. In Fig. 6, we have shown the same contours but now in the k– As plane assuming the best values for α as well as its values at the wings of the 68 per cent confidence limit. Fig. 6(c) is for α = 0.052 which is almost a CDM model, and it is quite similar to that obtained by Choudhury & Padmanabhan (2003) for a CDM model. It shows that model that behaves more like CDM (α = 0.052) is inconsistent with a flat universe at 68 per cent confidence level. On the other hand, a model that deviates more from the CDM model, Figs 6(a) and (b), is consistent with the flat universe with better confidence level. Like Fig. 5, it again shows that even if we do not take a flat prior, unlike the CDM model, the flat GCG model is consistent with the supernova data up to 68 per cent confidence level. 4 E X P E C T E D S N A P C O N F I D E N C E R E G I O N S 4.1 Method To find the expected precision of a future experiment such as SNAP, one must assume a fiducial model, and then simulate the experiment assuming it as a reference model. This allows for estimates of the precision that the experiment might reach (see Silva & Bertolami 2003 and references therein for a more detailed description of the method employed here). Let us then assume a fiducial model and functions χ 2 based on hypothetical magnitude measurements at the various redshifts. In this case, zi =0 zmax [mmodel(zi ) − mfid(zi )]2 where the sum is made over all redshift bins and m(z) is as specified in Section 3. Here, together with the GCG model we also analyse a flat XCDM model (i.e. a cold dark matter model together with a dark energy with an equation of state px = ωx ρ x ). We aim to show that if the 0.6 s A As = 0.62 0 Ω k 0 Ω k Redshift interval Universe is indeed described by the GCG model, the fitting of an XCDM model to the data will reveal that the cosmic expansion is drawn by a phantom-like dark energy component. To fully determine the χ 2 functions, the error estimates for SNAP must be defined. Following Weller & Albrecht (2002), we assume that the systematic errors for the apparent magnitude, m, are given by which are measured in magnitudes such that at z = 1.5 the systematic error is 0.02 mag, while the statistical errors for m are estimated to be σ sta = 0.15 mag. We place the supernovae in bins of width z ≈ 0.05. We add both types of errors quadratically where ni is the number of supernovae in the ith redshift bin. The distribution of supernovae in each redshift bin is, as before, taken from Weller & Albrecht (2002), and shown in Table 1. Summarizing, for each fiducial model, the method used, consists of the following: (i) choose a fiducial model; (ii) fit the XCDM model to the mock data, and obtain the respective confidence regions; (iii) repeat the previous step to the GCG. 4.2 Results In Figs 7–9 we show the confidence contours for the GCG and XCDM models for future SNAP observations taking different fiducial models. In all these figures, β ≡ −ωx . Also, along the x-axis in all of these figures, we have plotted 1 − As (instead of As as 1 − As represents m when α = 0) for the GCG and m for the XCDM. As mentioned above, our main aim is to explicitly show that a GCG universe might appear as an XCDM universe with a dark energy component that has a phantom-type equation of state. To do so, we have considered two fiducial models. The first corresponds to a Chaplygin model (α = 1), with 1 − As = 0.25. If we attempt to fit an XCDM model to the data (Fig. 7), we find that the data favour a larger amount of matter than expected and a phantom-type dark energy component. This is fully consistent with fig. 13 of Tonry et al. (2003). To further examine the degeneracy between models, in Fig. 8 we have repeated the procedure assuming an XCDM fiducial model, with m = 0.49 and β = 1.55 (ωx = −1.55). By examining Figs 7 and 8, we can see that both models appear essentially identical. Also, for GCG, the confidence regions for both fiducial models are quite identical to what we have shown earlier in Fig. 2 for the current supernova data. In Fig. 9 we used a fiducial CDM model, with m = 0.3. As can be seen, the confidence regions are completely different from Number of SNe Ω m;1−As Ω m;1−As Figure 8. Same as Fig. 7 but for an XCDM fiducial model with m = 0.49 and β = −ωx = 1.55. those found in the two previous cases. Also, for the GCG model the confidence regions are quite different from what we have earlier in Fig. 2 for the current supernova data, further hinting that indeed the CDM model is not a good description of the Universe. To illustrate the degeneracy between the GCG model and the XCDM model with a phantom-like equation of state (ωx < −1), we Taylor expand the luminosity distance as c dL = H0 1 − q0 − 3q02 + j0 z3 + .... Ω m;1−As where q0 is the deceleration parameter related to the second derivative of the scalefactor and j0 is the so-called jerk parameter (Visser 2004) related to the third derivative of the scalefactor. This is also one of the statefinder variables r proposed in Sahni et al. (2003). The subscript ‘0’ means that quantities are evaluated at present. The jerk parameter is related to the deceleration parameter q0 as For the GCG model, we can calculate q0 and (dq/ dz)|0 to obtain whereas for the XCDM model they turn out to be For the previous supernova data obtained by Perlmutter et al. (1999) and Riess et al. (1998) for low redshifts (z < 1), it is sufficient to consider the first two terms in the series expansion of the luminosity distance dL(z) given above. In that case, we can see from the expression of q0 for GCG that supernova data can only constrain As, as q0, for the GCG model, is independent of α. Moreover, in order to have degeneracy between the GCG and XCDM models, the q0 parameter of these two models must be equal, with the result that Thus, if we obtain a bound on As by fitting the GCG model with low-redshift supernova data, then the same data can be fitted by a host of XCDM models (including the CDM model) provided the above equation is satisfied. Hence, for low redshifts (z < 1), the GCG model is degenerate with all types of different dark energy models with a constant equation of state, including the CDM model. ωx = −α(1 − As) − 1 (1 + α)(1 − As) m = 1 + α(1 − As) Now, as we go to higher redshifts, which is the case for the current data that we are considering in this paper, we also have to consider the higher-order terms in the series expansion of dL(z). As far the data we are studying in our paper, it is enough to consider terms up to order z3 in the series expansion of dL. Hence, in order to have a degeneracy between GCG and XCDM even for the high redshifts, the jerk parameter j0 also has to be equal for the two models, which effectively means (dq/dz)|0 has to be equal for the two models. Using this together with equation (17), we find that In the above equation, ω x and m are the equation of state and the density parameter of the XCDM model which is degenerate with a GCG model with the corresponding α and As parameters, for higher redshifts. Now we can see that for any GCG model (α > 0), the corresponding equation state has to be always phantom type (ωx < −1) as 0 < As < 1. This shows that, although for the low-redshift data the GCG model is degenerate with all kinds of constant equation of state dark energy model including the CDM model, for higher redshift the GCG is degenerate with an XCDM model but only with a phantom type of equation of state. Finally, aiming to further illustrate the degeneracy of the GCG model with the XCDM model for large redshifts, we plot in Fig. 10 the behaviour of the dimensionless luminosity distance DL as a function of redshift for four different best-fitting models: CDM (w x = −1, m = 0.3), XCDM (w x =−2.07, m = 0.51), Chaplygin model (α = 1, As = 0.77) and GCG (α = 3.75, As = 0.936). We have actually plotted the stretched DL/z as a function of redshift in order to graphically show the degeneracy. 5 C O N C L U S I O N S We have analysed the currently available 194 supernova data points within the framework of the GCG model, regarding GCG as a unified candidate for dark matter and dark energy. We have considered both flat and non-flat cases, and we have used the best-fitting value for M = −0.033 which is independent of a specific model, throughout our analysis. For the first time, we have crossed the α = 1 limit for the GCG model and try to see whether the data actually allow it or not. For the flat case, we have studied both cases, restricting α to the range 0 α 1 and also without any restriction on the upper limit α. From Figs 1 and 2, it is quite clear that data favour α > 1, although there is a strong degeneracy in α. Also, the quality of fit improves substantially as we relax the α = 1 restriction. Moreover, the minimum values allowed for α and As at 68 per cent confidence level are [0.78, 0.778], which excludes the α = 0, CDM case, although there is no constraint on α at 95 per cent confidence level. Moreover, if we do not assume a flat prior for the analysis, our study shows that the flat GCG model for α values sufficiently different from zero is consistent with the supernova data up to 68 per cent confidence level. It also allows small, both positive and negative, curvature, making the GCG a somewhat better description than the CDM model. This is consistent with a recent result that shows that without a flat prior a flat CDM model, which is otherwise consistent with different cosmological observations, is not a good fit to the supernova data (Choudhury & Padmanabhan 2003). Moreover, the fact that GCG is a better fit to the supernova data than CDM is consistent with the result of Alam et al. (2003), who also have reconstructed a similar type of evolving equation of state for the dark energy from the latest supernova data. We have also studied the confidence contours for a GCG and XCDM model expected from the future SNAP observation assuming different fiducial universes. In this regard, the degeneracy between the GCG model and a phantom-like dark energy scenario has been made obvious in Section 4, where we have shown that when fitting an XCDM model to a GCG universe, the data will favour a phantom energy component, and vice versa. This degeneracy is also illustrated analytically through the expression for the luminosity distance dL as a function of redshift. We have shown that, for higher redshifts, the GCG model is completely degenerate with an XCDM model with a phantom type of constant equation of state. We mention that it has already been noted in Maor et al. (2002) that time-varying equations of state might be confused with phantom energy, and here we show that this is true for the GCG, without breaking the dominant energy condition and without a big rip singularity in the future. It should also be noted that, with the exception of a cosmological constant, most dark energy models predict a time-varying equation of state, therefore a constant dark energy equation of state might not be the best parametrization for dark energy. We have also shown that for a CDM fiducial model, confidence regions for a GCG model, which are expected from the future SNAP experiment, are quite different from what we have shown in Section 3.1, suggesting that SN Ia data do not favour the CDM model. Thus, our study shows that the GCG model is a very good fit to the latest supernova data, both with or without a flat prior. With future data, we expect the error bars to be reduced considerably, but we still expect that supernova data will favour a GCG model with high confidence. AC K N OW L E D G M E N T S OB acknowledges the partial support of Fundac¸a˜o para a Cieˆncia e a Tecnologia (Portugal) under the grant POCTI/FIS/36285/2000. The work of AAS is financed by Fundac¸ a˜o para a Cieˆncia e a Tecnologia (Portugal) under the grant SFRH/BPD/12365/2003. The work of SS is financed by Fundac¸a˜o para a Cieˆncia e a Tecnologia (Portugal) through CAAUL. 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O. Bertolami, A. A. Sen, S. Sen, P. T. Silva. Latest supernova data in the framework of the generalized Chaplygin gas model, Monthly Notices of the Royal Astronomical Society, 2004, 329-337, DOI: 10.1111/j.1365-2966.2004.08079.x