Observational constraint on generalized Chaplygin gas model

We investigate observational constraints on the generalized Chaplygin gas (GCG) model as the unification of dark matter and dark energy from the latest observational data: the Union SNe Ia data, the observational Hubble data, the SDSS baryon acoustic peak and the five-year WMAP shift parameter. It is obtained that the best fit values of the GCG model parameters with their confidence level are $A_{s}=0.73^{+0.06}_{-0.06}$ ($1\sigma$) $^{+0.09}_{-0.09}$ $(2\sigma)$, $\alpha=-0.09^{+0.15}_{-0.12}$ ($1\sigma$) $^{+0.26}_{-0.19}$ $(2\sigma)$. Furthermore in this model, we can see that the evolution of equation of state (EOS) for dark energy is similar to quiessence, and its current best-fit value is $w_{0de}=-0.96$ with the $1\sigma$ confidence level $-0.91\geq w_{0de}\geq-1.00$.


Introduction
The recently cosmic observations from the type Ia supernovae (SNe Ia) [1], the cosmic microwave background (CMB) [2], the clusters of galaxies [3] etc., all suggest that the expansion of present universe is speeding up rather than slowing down. And it indicates that baryon matter component is about 5% for total energy density, and about 95% energy density in universe is invisible. Considering the four-dimensional standard cosmology, the accelerated expansion of the present universe is usually attributed to the fact that dark energy (DE) is an exotic component with negative pressure. And it is shown that DE takes up about two-thirds of the total energy density from cosmic observations. Many kinds of DE models have already been constructed such as ΛCDM [4], quintessence [5], phantom [6], quintom [7], generalized Chaplygin gas (GCG) [8], modified Chaplygin gas [9], holographic dark energy [10], agegraphic dark energy [11], and so forth. Furthermore, model-independent method 1 and modified gravity theories (such as scalar-tensor cosmology [17], braneworld models [18]) to interpret accelerating universe have also been discussed.
It is well known that the GCG model have been widely studied for interpreting the accelerating universe [19].
The most interesting property for this scenario is that, two unknown dark sections in universe-dark energy and dark matter can be unified by using an exotic equation of state (EOS). In this paper, we use the latest observational data: the Union SNe Ia data [20], the observational Hubble data (OHD) [21], the baryon acoustic oscillation (BAO) peak from Sloan Digital Sky Survey (SDSS) [22] and the five-year WMAP CMB shift parameter [23] to constrain the GCG * Electronic address: lvjianbo819@163.com 1 Using mathematical fundament, one expands equation of state of DE w de or deceleration parameter q with respect to scale factor a or redshit z. For example, w de (z) = w 0 =const [12], w de (z) = w 0 + w 1 z [13], w de (z) = w 0 + w 1 ln(1 + z) [14], w de (z) = w 0 + w 1 z 1+z [15], q(z) = q 0 + q 1 z [12], q(z) = q 0 + q 1 z 1+z [16], where w 0 , w 1 , or q 0 , q 1 are model parameters.
model. And we discuss whether the parameter degeneration [24] [25] for the GCG model can be broken by the latest observed data, since it is always expected that the model degeneration problem can be solved by the more accurate observational data.
The paper is organized as follows. In section 2, the GCG model as the unification of dark matter and dark energy is introduced briefly. Based on the observational data, we constrain the GCG model parameter in section 3. The evolutions of EOS of DE and deceleration parameter for GCG model are presented in section 4. Section 5 is the conclusions.

generalized Chaplygin gas model
The GCG background fluid with its energy density ρ GCG and pressure p GCG are related by the EOS [8] where A and α are parameters in the model. When α = 1, it is reduced to the CG scenario.
Considering the Friedmann-Robertson-Walker (FRW) cosmology, by using the energy conservation equation: , the energy density of GCG can be derived as where a is the scale factor, . For the GCG model, as a scenario of the unification of dark matter and dark energy, the GCG fluid is decomposed into two components: the dark energy component and the dark matter component, i.e., ρ GCG = ρ de + ρ dm , p GCG = p de . Then according to the general recognition about dark matter the energy density of the DE in the GCG model is given by Next, we assume the universe is filled with two components, one is the GCG component, and the other is baryon matter component, ie., ρ t = ρ GCG + ρ b . In a flat universe, making use of the Friedmann equation, the Hubble parameter H is expressed as Where H 0 = 100h km S −1 M pc −1 is the present Hubble constant, h = 0.72 ± 0.08 is given by Hubble Space Telescope key projects [26]. Ω 0b is the present value of dimensionless baryon matter density, and a joint analysis of five-year WMPA, SNe Ia and BAO data gives Ω 0b h 2 = 0.02265 ± 0.00059 [27]. In the following section, we will use the cosmic observations to constrain the GCG model parameter (A s , α).
It is necessary for the investigation of type Ia supernovae to explore dark energy and constrain the models. Since SNe Ia behave as excellent standard candles, they can be used to directly measure the expansion rate of the universe up to high redshift with comparing with the present rate. Theoretical dark-energy model parameters are determined by minimizing the quantity [28] where N = 307 for the Union SNe Ia data [20], which includes the SNe samples from the Supernova Legacy Survey [29], ESSENCE Surveys [30], distant SNe discovered by the Hubble Space Telescope [31], nearby SNe [32] and several other, small data sets. The 1σ error σ obs;i are from flux uncertainty, intrinsic dispersion of SNe Ia absolute magnitude and peculiar velocity dispersion, which are assumed to be Gaussian and uncorrelated. θ denotes the model parameters.
µ obs is the observed value of distance modulus and can be given by the SNe dataset. The theoretical distance modulus µ th is defined as Here m th (z) is the apparent magnitude of the SNe at peak brightness and absolute magnitude M can be given by relating to the magnitude zero point offset M , with µ 0 = 5log 10 ( Mpc ) + 25 = 42.38 − 5log 10 h. Thus according to Eqs. (7), (8) and (9), the theoretical distance modulus can be written as where D L (z) is the Hubble free luminosity distance Since the nuisance parameter µ 0 is independent of the data and the dataset, from above equations one can see that the distance modulus of different SNe (i.e. at different redshift z), µ(z i ) and µ(z j ) are uncorrelated. So, the covariance matrix included in the χ 2 SN e (Eq. (6)) is diagonal with entries σ i . Furthermore, by expanding the χ 2 SN e of expression (6) relative to µ 0 , the minimization with respect to µ 0 can be made trivially [28][33] [34] where Evidently, Eq. (6) has a minimum for µ 0 = B/C at Since χ 2 SN e,min = χ 2 SN e,min and χ 2 SN e is independent of nuisance parameter µ 0 [34], here we utilize expression (16) to displace (6) for SNe constraint.
Since the Hubble parameter H(z) depends on the differential age of the universe, the value of H(z) can directly be measured through a determination of dz/dt. By using the differential ages of passively evolving galaxies from the GDDS [35] and archival data [36], Ref. [21] got nine values of H(z) in the range of 0 < z < 1.8 (see Table 1). Here the observed Hubble data H(z i ) and H(z j ) are uncorrelated, for they are obtained by the observations of galaxies at different redshift. Using these nine observational Hubble data one can constrain DE models by minimizing [37][38] where H th is the predicted value of the Hubble parameter, H obs is the observed value, σ obs;i is the 1σ uncertainty of the measurement of standard deviation. Here H 0 contained in the χ 2 Hub (H 0 , θ) as a nuisance parameter is marginalized by integrating the likelihood L(θ) = dH 0 P (H 0 ) exp (−χ 2 (H 0 , θ)/2). P (H 0 ) is the prior distribution function of the present Hubble constant, and a Gaussian prior H 0 = 72 ± 8 km S −1 M pc −1 [26] is adopted in this paper.
Using a joint analysis of Union SNe Ia data and OHD (i.e., χ 2 total = χ 2 SN e + χ 2 Hub ), Fig. 1 shows the constraint on GCG parameter space A s -α at the 1σ (68.3%) and 2σ (95.4%) confidence levels. For this analysis the best fit parameters are A s = 0.80 and α = 0.42. It is obvious that two model parameters, A s and α, are degenerate. And it can be seen that model parameter α has the lager variable range. Then in order to get the stringent constraint and diminish systematic uncertainties, in what follows we combine the standard ruler data (the BAO peak from SDSS and the five-year WMAP CMB shift parameter R) with the Union SNe Ia data and the OHD to constrain the GCG model.
Because the universe has a fraction of baryon, the acoustic oscillations in the relativistic plasma would be imprinted onto the late-time power spectrum of the nonrelativistic matter [39]. Then the observations of acoustic signatures in the large-scale clustering of galaxies are very important for constraining cosmological models. From the BAO constraint, the best fit values of parameters in the DE models can be determined by constructing [40] Where Ω 0m is the effective matter density parameter given by 1+α [24][25] [41]. The observed value A obs with its 1σ error σ A is A obs = 0.469(n s /0.98) −0.35 ± 0.017 measured from the SDSS at z BAO = 0.35, here n s is the scalar spectral index [42] and its value is taken to be 0.96 as shown in Ref. [27].
The structure of the anisotropies of the cosmic microwave background radiation depends on two eras in cosmology, i.e., the last scattering era and today. They can also be applied to limit DE models by minimizing [43] Where the shift parameter [44] R(θ) = Ω 0m z rec = 1089 is the redshift of recombination. The observed value R obs = 1.710, and its corresponding 1σ error is σ R = 0.019 according to the five-year WMAP result [23].
Above four observational data are uncorrelated for each other, since they are given by different experiments and methods. Then the total likelihood χ 2 total can be constructed as are plotted in Fig. 2 (b). From this figure, we obtain the values of model parameters with the confidence levels, . It can be seen that parameters A s and α are also degenerate, and at the 1σ confidence level these results are consistent with the standard dark energy plus dark matter scenario (i.e., the case of α = 0). Furthermore, one can see that this constraint on parameter α is more stringent than the results in Refs. [24] [25], where the constraint results for the GCG model parameters are A s = 0.70 +0. 16 −0.17 and α = −0.09 +0.54 −0.33 at 2σ confidence level by using the X-ray gas mass fractions of galaxy clusters and the dimensionless coordinate distance of SNe Ia and FRIIb radio galaxies [24], and A s = 0.75 +0.08 −0.08 , α = 0.05 +0.37 −0.26 at 2σ confidence level by means of the observational Hubble data, the 115 SNLS SNe Ia data and the SDSS baryonic acoustic oscillations peak [25].
At last, we also consider the constraint on the GCG model parameter from a combination of Union SNe Ia and BAO data, the best fit happens at A s = 0.75 and α = 0, which can be reduced to the standard dark energy plus dark matter scenario. But at their confidence levels, the two parameters are also highly degenerate. In Fig. 2 (a), we display the constraint result for this analysis.

Constraint on EOS of dark energy and deceleration parameter
According the Eq. (5), deceleration parameter q in the GCG model can be obtained by The equation of state of dark energy is derived as where Ω 0dm is the present value of dimensionless dark matter density. Based on Eqs. (24) and (25), the confidence levels of the best fit w de (z) and q(z) calculated by using the covariance matrix are plotted in Fig. 3. From Fig. 3 (a), it is easy to see that the best fit value w 0de ≡ w de (z = 0) = −0.96 > −1, and the 1σ confidence level of w 0de is −0.91 ≥ w 0de ≥ −1.00. In addition, it can be found that the best fit evolution of w de (z) for GCG is similar to the quiessence model (w de (z) = const = −1). From Fig. 3 (b), we can see that the best fit values of transition redshift and current deceleration parameter with confidence levels are z T = 0.74 +0.04 −0.05 (1σ), q 0 = −0.55 +0.05 −0.06 (1σ). One knows that z T describes the expansion of universe from deceleration to acceleration, and q 0 indicates the expansion rhythm of current universe. Comparing our results with Ref. [45], where z T = 0.49 +0.14 −0.07 (1σ) and q 0 = −0.73 +0.21 −0.20 (1σ) are obtained from Union SNe Ia data by using a linear two-parameter expansion for the decelerating parameter, q(z) = q 0 + q 1 z, it is clear for our constraint, that the universe tends to an earlier time to acceleration and a milder expansion rhythm at present.

Conclusion
The constraints on the GCG model as the unification of dark matter and dark energy are studied in this paper by using the latest observational data: the Union SNe Ia data, the observational Hubble data, the SDSS baryon acoustic peak and the five-year WMAP shift parameter. We find that the model parameters A s and α are degenerate, and their values are constrained to A s = 0.73 +0.06 −0.06 (1σ) +0.09 −0.09 (2σ) and α = −0.09 +0.15 −0.12 (1σ) +0.26 −0.19 (2σ). This constraint on parameter α is more stringent than the results in Refs. [24] [25]. Furthermore, it is shown that the evolution of EOS of dark energy for the GCG model is similar to quiessence, and the best fit value of current EOS of DE