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Physics Letters B

www.elsevier.com/locate/physletb

Cosmology with a variable generalized Chaplygin gas

Jianbo Lu

School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian H6024, PR China

ARTICLE INFO

ABSTRACT

Article history:

Received 27 April 2009

Received in revised form 5 September 2009

Accepted 11 September 2009

Available online 17 September 2009

Editor: S. Dodelson

PACS: 98.80.-k

We investigate observational constraint on the variable generalized Chaplygin gas (VGCG) model as the unification of dark matter and dark energy by using the Union supernovae sample and the baryon acoustic oscillations data. Based on the best fit parameters for VGCG model it is shown that the current value of equation of state for dark energy is w0de = -1.08 < -1, and the universe will not end up with big rip in the future. In addition, we also discuss the evolution of several quantities in VGCG cosmology such as deceleration parameter, fractional density parameters, growth index and sound speed. Finally, the statefinder diagnostic is performed to discriminate the VGCG with other models.

© 2009 Elsevier B.V. All rights reserved.

Keywords:

Variable generalized Chaplygin gas (VGCG) Unification of dark matter and dark energy

1. Introduction

The recent 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 the baryon matter component is about 5% of the total energy density, and about 95% of the energy density in the universe is invisible. The accelerated expansion of the present universe is usually attributed to the fact that dark energy (DE) is an exotic component with negative pressure. It is shown that DE takes up about two-thirds of the total energy density from cosmic observations. The simplest candidate for the DE is a cosmological constant model [4]. However, it suffers from both the coincidence and the fine-tuning problem. In order to solve these problems, other candidates to represent DE have also been suggested such as quintessence [5], quintom [6], braneworld models [7] and so on.

It is well known that the Chaplygin gas (CG) [8] and several extended CG models have been widely studied for interpreting the accelerating universe, e.g. variable Chaplygin gas (VCG) [9] and generalized Chaplygin gas (GCG) model [10], etc. [11]. The most interesting property for these scenarios is that two unknown dark sections in universe—dark energy and dark matter—can be unified by using an exotic equation of state, since these CG fluids behave as dust at early stage and as dark energy at later stage. Furthermore, one knows that the actions of VCG and GCG are related to the Born-Infeld Lagrangian [9] and its generalized form [10], respectively.1 In this Letter, we consider a generalized case related to these two CG models, i.e. the variable generalized Chaplygin gas (VGCG) model [12], and apply the Union SNe Ia data and the baryon acoustic oscillations (BAO) data from the Sloan Digital Sky Survey (SDSS) to constrain this model.

The Letter is organized as follows. In Section 2, the VGCG model as the unification of dark matter and dark energy is introduced briefly. Section 3 presents the methods of data analysis. In Section 4 we constrain the VGCG model parameters, and according to the best fit values we show the evolution of several cosmological quantities in VGCG cosmology. The sound speed vs of VGCG fluid and statefinder diagnostic approach to VGCG model are discussed in Section 5. Section 6 is for the conclusions.

E-mail address: lvjianbo819@163.com.

1 According to Ref. [10], the GCG Lagrangian density can be expressed as a generalized Born-Infeld form with a scalar field 9, CgBI = —A i+w [1 — (gßv9yß9v) t+H •+«, which reduces to the Born-Infeld Lagrangian for a = 1. And for VCG, Ref. [9] shows that this model can be constructed by considering the parameter A as a function of the cosmic scale factor a and taking A = A0a—n in the Born-Infeld Lagrangian theory.

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2. Variable generalized Chaplygin gas model

The VGCG background fluid with its energy density pvgcg and pressure pvccc can be described by an equation of state (EOS) [12]

A0a-n m

pVGCG =--a-' (1)

where A0, n and a are parameters in the model. For n = 0, this model reduces to the GCG case, and for a = 1 it presents the VCG model. By using the energy conservation equation, d(pa3) = -pd(a3), the energy density of VGCG can be derived as

pVGCG = p0VGCG[As(1 + z)n + (1 - As)(1 + z)3(1+a)] ^ , (2)

where a is the scale factor, As = 3(31(?+))n —r+r". For an expanding universe, there should be n > 0 and 3(1 + a) > 0. If they are negative,

( ) P0VGCG

a implies p which is not the case for expanding universe. Here in order to study the evolution tendency of dark energy and

dark matter and explore the properties of DE, we decompose the VGCG fluid into two components, the dark energy and the dark matter, i.e., pVccG = pde + pdm, pVGCG = Pde. Then according to the general recognition about dark matter, pdm = podm(1 + z)3, the energy density of the DE in the VGCG model is given by

pde = pVGCG - pdm = p0VGCG[As(1 + z)n + (1 - As)(1 + z)3(1+a)]^ - p0dm(1 + z)3. (3)

Considering spatially flat FRW (Friedmann-Robertson-Walker) universe with baryon matter pb and VGCG fluid pvgcg, according to the Friedmann equation the Hubble parameter H can be written as

H2 = 8nG3ptota' = H2E2 = H2{(1 - tf0b)[As(1 + z)n + (1 - As)(1 + z)3(1+a)] + tf0b(1 + z)3}, (4)

where H0 is the present value of the Hubble parameter. Subscript "0" denotes the current value of the variable in this Letter. Next on the basis of expression (4), we apply the recently observed data to constrain VGCG model and discuss the evolution of universe.

3. Data and analysis method

3.1. Type Ia supernovae

It is necessary for the investigation of type Ia supernovae to explore DE properties 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, comparing with the present rate. Theoretical dark-energy model parameters are determined by minimizing the quantity [13]

2 i r>\ (f obs(zi) - fth(z,f0,0))2

xSNe(f0,6) =-2-, (5)

i=1 °obs;i

where N = 307 for the Union SNe Ia data [14], which include the SNe samples from the Supernova Legacy Survey [15], ESSENCE Surveys [16], distant SNe discovered by the Hubble Space Telescope [17], nearby SNe [18] and several other, small data sets. o^.t are errors due to the flux uncertainties, intrinsic dispersion of SNe Ia absolute magnitude and peculiar velocity dispersion, respectively. 6 denotes the model parameters. f obs is the observed value of distance modulus and can be given by the SNe dataset. f th is the theoretical distance modulus, which is related to the apparent magnitude of SNe at peak brightness m and the absolute magnitude M,

mth(z) = mth(z) - M = 5logw(Dl(z)) + fi0. (6)

Here the luminosity distance

f H 0 dz'

Dl(z) = H0dL(z) = (1 + z)j , (7)

f 0 = 5logjJ + 25 = 42.38 - 5logw h. (8)

It should be noted that f0 is independent of the data and the dataset. By expanding the x2 of Eq. (5) relative to the nuisance parameter f0, the minimization with respect to f0 can be made trivially [19,20]

xS2Ne(6) = A(6) - 2f0B(6) + f0C, (9)

m = J2

[mobs(z;) - mth(z; mo = 0,0)]2

B(0) = Z

Mobs(Zi) - Mth(Zi; Mo = 0,0)

N 1 c=N ^

Obviously, according to Eq. (9) xSNe has a minimum for = B/C. Thus, the expression of x2 for SNe constraint can be written as

XsNe) = A(0) - B (9)2/C.

(10) (11) (12)

Since xSNe mjn = x|ne min and xsNe is independent of the nuisance parameter here we utilize the expression (13) to displace (5) for

the SNe constraint.

3.2. Baryon acoustic oscillations

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 [21]. Then the observations of acoustic signatures in the large-scale clustering of galaxies are very important for constraining cosmological models. One can minimize the x|AO defined by [22]

Xbao (0) =

[A (9) - Aobs]2

A (9) = E (zbao )-1/3

zbao J E(z'; 9) o

For the matter density parameter Q0m is not explicitly included in the VGCG model, here we set Q0m = 0.279 according to a joint analysis of the five-year WMAP, SNe la and BAO observations [23]. The observed value Aobs with its 1a error aA is Aobs = 0.469(ns/0.98)-0 35 ± 0.017 measured from the SDSS at zBAO = 0.35, where ns is the scalar spectral index and its value is taken to be 0.96 as shown in Ref. [23].

4. Constraint on model parameters and the evolution of cosmological quantities in VGCG cosmology

We consider a combined constraint on VGCG model. The total x2 is xtotal = xSNe + xBAO,

where xSNe and x|AO are given by Eqs. (13) and (14). One knows that the current observational constraints on baryon density i20b are stringent.2 In this Letter we put the notice to VGCG model parameters (As, n, a), and take i20t,h2 = 0.02265 [23] as a prior in our numerical analysis. lt can be calculated that the values of VGCG model parameters are almost invariable for the different priors on the parameter ^0b at its confidence range. Here the renormalized quantity h is related to the current Hubble parameter by H0 = 100h kms-1 Mpc-1. Thus using the above two observational datasets, we obtain the best fit values (As, n,a) in the VGCG model (0.91, 0.75,1.53) with xmin = 310.46. Then the reduced x2 value, x^m/dof = 1.02. The value of dof (degrees of freedom) for the model equals the number of observational data points minus the number of parameters.

Next according to the best fit parameters for VGCG model, we present the evolution of several cosmological quantities. By using Eqs. (1) and (3), the equation of state of DE can be derived as

(1 - tfob)[-1 + 3Ô+S) ][As(1 + z)n + (1 - As)(1 + z)3(1+a)]

-a 1+a

pde (1 - tfob)[As(1 + z)n + (1 - As)(1 + z)3(1+a)]TO - ^odm(1 + z)3

And according to Eq. (4), the deceleration parameter q can be obtained,

« ,1 dH 1 3 ,

q = (1 + z) — — - 1 = - + - As H dz 2 2

3(1 + a)

(1 + z)n

(1 - ^ob)[As(1 + z)n + (1 - As)(1 + z)3(1+a)]1

(1 - Oob)[As(1 + z)n + (1 - As)(1 + z)3(1+a)]^ + ^ob(1 + z)3

Furthermore, based on Eqs. (3) and (4), the dimensionless dark energy, dark matter and baryon matter density can be expressed as

For example, Q0bh2 = 0.02273 ± 0.00062 from the five-year WMPA data [24], Q0bh2 = 0.022-0'003 according to the DASl results [25] for the observation of CMB.

Fig. 1. The evolution of deceleration parameter q(z) and EOS of DE wje(z) for VGCG model.

Fig. 2. The evolution of density parameters and quantity R relative to the redshift z for VGCG model.

0 (1 - gQb)[As(1 + z)n + (1 - As)(1 + z)3(1+a)]T+b - ^0dm(1 + z)3 ,.Q. ^de = -E2(7)-, (19)

^ ^0dm(1 + z)3

^dm = -E2(z)-, (20)

^ W + z)3 (21)

= E 2 (z) . (21)

Thus by using the best-fit model parameters from the above combined constraint, the evolution of q(z) and wde(z) for VGCG model are plotted in Fig. 1. From Fig. 1(a), the transition redshift and current deceleration parameter are estimated to be zT = 0.47 and q0 = -0.67, which are consistent with the constraint results in Ref. [26], where zT = 0.49+°and q0 = -0.73+°° 20 are obtained from the Union SNe Ia data by using a model, q(z) = q0 + q1 z, a linear expansion for the deceleration parameter. From Fig. 1(b), we notice that for the VGCG model the best fit value w0de = wde(z = 0) = -1.08 < -1, and the evolution of wde(z) at z ~ 0.27 cross over the cosmological constant borderline (wde (z) = -1). Fig. 2 shows the evolution of density parameters and quantity R relative to the redshift z, where R = Q is the ratio of dark energy density and matter density. From this figure, we can see that the evolution of baryon, dark matter and dark energy in this model are consistent with what is recognized, and the matter density is equal to the dark energy density at z ~ 0.34. In addition, we also consider the evolution of growth index f in VGCG cosmology. It is defined as

d ln & adS , ,

f = = 17-, (22)

d ln a & da

where S = P- is the matter density contrast. Based on the expression (22) and the perturbation equation S + 2HS - 4nPmS = 0, the evolution equation for the growth index can be obtained with changing the variable from t to scale factor a

f'+ - + a

2 (ln E 2)r a2

f - = 0, (23)

J 2E2 a4 , ^ ;

where the prime denotes the derivative with respect to the scale factor a. The combination of the function H (z) and f can provide insight into the properties of dark energy and interpret the structure formation, so the parameter f is an important quantity to test a model. By using Eqs. (4) and (23) the evolution of f relative to the scale factor a is shown in Fig. 3, with the initial condition fjnj = 1 at the last scattering surface. From Fig. 3, we can see that when the redshift z = 0.15 (a ~ 0.87), the value of the growth index is about f ~ 0.57,

Fig. 3. The evolution of growth index f as a function of the scale factor a.

Fig. 4. The evolution of squared sound speed v2 with respect to the redshift z for VGCG, VCG and GCG fluid.

which is much near to the central value f = 0.58 ± 0.11 constrained from the 2dFGRS Collaboration for the measurement of over 220 000 galaxies [27]. Furthermore comparing with the ACDM model, it can be found that the trajectory of fVGCG is almost consistent with fACDM for 0.7 < a < 1 (0.43 > z > 0). Therefore, the formula fA — [^0m(1 + z)]0 545 for the ACDM model can be used in the VGCG model for the recent evolution. But for the time of about a < 0.5, the difference between the evolution of fVGCG and fACDM is obvious.

5. Sound speed of CG fluid and statefinder diagnostic

A negative sound speed will induce a serious classical instability to the system, the perturbations on small scales will increase quickly with time and the late time history of the structure formations will be significantly modified [28]. For the constraint from sound speed, model parameter a should be 0 < a < 1 for GCG fluid. In the following, we discuss the evolution of sound speed for VGCG fluid. The adiabatic squared sound speed for any fluid is defined as

V2 = dp. (24) And the concrete expressions can be derived,

2 w23p/3z w

vs =-= w--, (25)

where the equation of state w is defined by the ratio of pressure and density for the fluid, and "dot" denotes the derivative relative to the cosmic time t.

For VGCG fluid its equation of state wVGCG as a function of the redshift z is

= PVGCG _ (-1 + 30+0))As(1 + z)n

WVGCG " pVGCG = As (1 + Z)n + (1 - As )(1 + z)3(1+a). ( )

We present the evolution of squared sound speed of VGCG fluid in Fig. 4, here the values of model parameters are adopted according to the above combined constraint. It is obvious that the sound speed for this fluid is in the range from zero to one, which is not exceed the speed of light (unity). Here for clear, we only plot the evolution of sound speed relative to z from z = 0 to 3. And for the case of z > 3, it can be shown that v2s — 0. Furthermore, we also consider the VCG and GCG fluid. Using the same observational data as VGCG model, we get the best-fit model parameters As = 0.86 and n = 0.25 for VCG model, As = 0.84 and a = 0.87 for GCG case. And the evolution of sound speed for these two fluids are also in the interesting region, 0 < v2s < 1.

Fig. 5. Statefinder diagnostic for three CG models and ACDM. The bigger dot locates the ACDM fixed point (1,0), the smaller dots locate the current values of the statefinder pair {r, s} for three CG models, and arrows denote the evolution directions of the statefinder trajectories r(s).

Next by means of the best-fit model parameters, we apply statefinder parameters to discriminate these three CG models. lt is well known that many kinds of DE models have been constructed to interpret accelerating universe. So, it is necessary to distinguish these models by using a general and model-independent manner. According to the document [29], a diagnostic of dark energy called statefinder is defined as follows

a r — 1

r =—r-, s =-t-. (27)

aH3, 3(q — 1) ( )

The statefinder is a "geometrical" diagnostic, since the statefinder parameters {r, s} only depend on the scale factor a and its derivative. Trajectories in the r-s plane corresponding to different cosmological models exhibit qualitatively different behaviors. And for ACDM model, {r, s} = {1, 0} is a fixed point [29]. Furthermore, the concrete expressions of the statefinder parameters can be written as

9 3 w de w de

r = 1 + - w de^deO + Wde) — - , s = 1 + w de — ---. (28)

2 2 H 3 w de H

Based on Eq. (28), Fig. 5 shows the statefinder diagnostic for three CG models and ACDM model. According to this diagnostic we can easily find that three dynamical models are significantly different from ACDM, and the r(s) trajectories of VCG and GCG model are similar. ln addition, from the r-s planes it is shown that though the VGCG model is similar to other two CG models around the present time, the difference for them is obvious in the past.

6. Conclusion

To sum up, we apply the Union SNe la data and the SDSS baryon acoustic peak to constrain the VGCG model as the unification of dark matter and dark energy. Using the best-fit model parameters, we present the evolution of VGCG cosmology. It is shown that the evolution of growth index for this model is consistent with the current cosmic observations. And for the VGCG fluid, it has a normal evolution of sound speed, then it will not induce the instability of system or the trouble to structure formations for the effect from sound speed. Furthermore, it indicates that the evolution of EOS of DE should cross over the boundary of Wde = —1, and the values of transition redshift and current deceleration parameter are zT = 0.47 and q0 = —0.67. In addition, considering the evolution of universe in future in VGCG cosmological model, we find that though the expansion of the universe will be still accelerated, the evolution of EOS of DE indicates that the universe will not end up with a big rip, because the evolution of Wde will come back to the case of wde > —1 in the future. Finally, we discriminate VGCG with other models by using a geometrical diagnostic method. lt can be seen that the difference between this model and other two CG models is obvious in the past.

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