aa r X i v : . [ a s t r o - ph ] A p r Geometrical Diagnostic for Generalized Chaplygin Gas Model
Jianbo Lu ∗ and Lixin Xu † School of Physics and Optoelectronic Technology,Dalian University of Technology, Dalian, 116024, P. R. China
A new diagnostic method, Om is applied to generalized Chaplygin gas (GCG) model as theunification of dark matter and dark energy. On the basis of the recently observed data: the Unionsupernovae, the observational Hubble data, the SDSS baryon acoustic peak and the five-year WMAPshift parameter, we show the discriminations between GCG and ΛCDM model. Furthermore, it iscalculated that the current equation of state of dark energy w de = − .
964 according to GCG model.
PACS numbers: 98.80.-kKeywords: generalized Chaplygin gas; geometrical diagnostic.
1. Introduction
The type Ia supernovae (SNe Ia) investigations [1], the cosmic microwave background(CMB) results from WMAP[2] observations, and surveys of galaxies [3] all suggest that the expansion of present universe is speeding up ratherthan slowing down. The accelerated expansion of the present universe is usually attributed to the fact that dark energy(DE) is an exotic component with negative pressure. Many kinds of DE models have already been constructed such asΛCDM [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. In addition, model-independent method and modified gravity theories (such as scalar-tensor cosmology [18], braneworld models [19]) to interpret acceleratinguniverse have also been discussed. So, a general and model-independent manner to distinguish these models introducedby different theories or methods is necessary. Statefinder diagnostic method is presented in Refs. [20], and it has beenapplied to a large number of DE models [22]. Recently, another geometrical diagnostic Om is also introduced in Ref.[21] to differentiate ΛCDM with other models. An important property for Om diagnostic is that it can be used todistinguish DE models with small influence from density parameter Ω m , though the current observations suggest anuncertainties of at least 25% in the value of current matter density Ω m [23]. In this paper, we apply Om diagnosticto GCG model.The paper is organized as follows. In section 2, the GCG model as the unification of dark matter and dark energyis introduced briefly. Based on the recently observed data: the Union SNe Ia [24], the observational Hubble data(OHD) [25], the baryon acoustic oscillation (BAO) peak from Sloan Digital Sky Survey (SDSS) [26] and the five-year ∗ Electronic address: [email protected] † Electronic address: [email protected] For example, using mathematical fundament one expands equation of state of DE w de or deceleration parameter q with respect to scalefactor a or redshit z , such as w de ( z ) = w =const [12], w de ( z ) = w + w z [13], w de ( z ) = w + w ln(1 + z ) [14], w de ( z ) = w + w z z [15], q ( z ) = q + q z [12], q ( z ) = q + q z z [16], and so forth. Where w , w , or q , q are model parameters. For more information aboutmodel-independent method, please see review paper [17]. WMAP CMB shift parameter [27], Om diagnostic is used to GCG model in section 3. Section 4 is the conclusions.
2. generalized Chaplygin gas model
In the GCG approach, the most interesting property is that the unknown dark sections in universe–dark energyand dark matter, can be unified by using an exotic equation of state. The energy density ρ and pressure p are relatedby the equation of state (EOS) [8] p = − Aρ α , (1)where A and α are parameters in the model.By using the energy conservation equation: d ( ρa ) = − pd ( a ), the energy density of GCG is expressed as ρ GCG = ρ GCG [ A s + (1 − A s )(1 + z ) α ) ] α , (2)where a is the scale factor, A s = Aρ α . For the GCG model, as a scenario of the unification of dark matter anddark energy, the GCG fluid is decomposed into two components: the dark energy component and the dark mattercomponent, i.e., ρ GCG = ρ de + ρ dm , p GCG = p de . Then according to the general recognition about dark matter, ρ dm = ρ dm (1 + z ) , the energy density of the DE in the GCG model is given by ρ de = ρ GCG − ρ dm = ρ GCG [ A s + (1 − A s )(1 + z ) α ) ] α − ρ dm (1 + z ) . (3)Furthermore, considering spatially flat FRW (Friedmann-Robertson-Walker) universe with baryon matter ρ b and GCGfluid ρ GCG , the equation of state of DE can be derived as w de = p de ρ de = − (1 − Ω b ) A s [ A s + (1 − A s )(1 + z ) α ) ] − α α (1 − Ω b )[ A s + (1 − A s )(1 + z ) α ) ] α − Ω dm (1 + z ) , (4)where Ω dm and Ω b are present values of dimensionless dark matter density and baryon matter component. AndHubble parameter H is H = 8 πGρ t H E = H { (1 − Ω b )[ A s + (1 − A s )(1 + z ) α ) ] α + Ω b (1 + z ) } . (5) H denotes the current value of Hubble parameter. Om diagnostic for GCG model It is well known that model-independent quantity H ( z ) is very important for understanding the properties of DE,since its value can be directly obtained from cosmic observations ( for example, the relation between luminositydistance D L and Hubble parameter is H ( z ) = [ ddz ( D L ( z )1+ z )] − [28] for SNe investigations). Recently, a new diagnosticof dark energy Om is introduced to differentiate ΛCDM with other dynamical models. The starting point for Om diagnostic is Hubble parameter, and it is defined as [21] Om ( z ) ≡ E ( z ) − x − , x = 1 + z. (6) z H ( z ) (kms − Mpc) −
69 83 70 87 117 168 177 140 2021 σ uncertainty ± ± ± ± ± ± ± ± ± H ( z ) data [36][37]. Since Om ( z ) only depends upon the scale factor a and its derivative, it is a ”geometrical” diagnostic. For ΛCDMmodel, Om ( z ) = Ω m is a constant, then it provides a null test of this model . The benefit of Om diagnostic is thatthe quantity Om ( z ) can distinguish DE models with less dependence on matter density Ω m relative to the EOS ofDE w de ( z ) [21].In what follows, we use a combination of the recent standard candle data (Union SNe Ia [24]) and the OHD toconstrain the evolutions of Om ( z ) and w de ( z ) for GCG model. The Union SNe data includes the SNe samples fromthe Supernova Legacy Survey [30], ESSENCE Surveys [31], distant SNe discovered by the Hubble Space Telescope[32], nearby SNe [33] and several other, small data sets [24]. The OHD are given by calculating the differential ages ofpassively evolving galaxies from the GDDS [34] and archival data [35]. According to the expression H ( z ) = − z dzdt ,one can see that the value of H ( z ) can be directly obtained by the determination of the differential age dz/dt . Ref.[25] get nine values of H ( z ) in the range of 0 < z < . O m H z L W b = O m H z L O m H z L W b = O m H z L O m H z L W b = O m H z L - - - - - - w d e H z L W m = - - - - - - w d e H z L - - - - - - w d e H z L W m = - - - - - - w d e H z L - - - - w d e H z L W m = - - - - w d e H z L FIG. 1: Evolutions of Om ( z ) and w de ( z ) by using a combination of Union SNe data and OHD for GCG model. Here threedifferent values Ω b =0.02, 0.042, 0.07 for Om ( z ) evolution diagram, and Ω m = Ω b + Ω dm =0.22, 0.27, 0.32 for w de ( z ) diagramare assumed. The shaded regions show the 1 σ confidence level. The dashed lines show the values of Om ( z ) and w de ( z ) forΛCDM model. From Eq. (4), it can be seen that both Ω b and Ω dm are included in the expression of w de ( z ) for GCG model. Given For null test of ΛCDM model, one can also see Ref. [29]. three different values of Ω m , the evolutions of w de ( z ) with 1 σ confidence level for GCG model are plotted in Fig. 1(lower) by using the Union SNe data and the OHD. Furthermore according to Eq. (5), we can see that the Hubbleparameter H ( z ) for GCG model is dependent on the baryon density Ω b and two model parameters ( A s , α ). It doesnot explicitly include current matter density Ω m . And one knows that the observational constraints on parameterΩ b is more stringent , i.e., it has a relative smaller variable range relative to Ω m . On the basis of Eq. (6), we plotthe evolutions of Om ( z ) for GCG model in Fig. 1 (upper). From Fig. 1, it can be found that the Om ( z ) diagram forGCG model as the unification of dark matter and dark energy is almost independent of the variation of Ω b , but theevolution of w de ( z ) is sensitive to the variation of matter density.In Ref. [39], Om diagnostic has been used to distinguish ΛCDM and Ricci DE model. Assuming the matter densityΩ m to be a free parameter, based on the recent cosmic observations Ref. [21] plots the evolution diagram of Om ( z )in a model-independent CPL scenario . In this paper, treating Ω b as a free parameter, we apply the Om diagnosticto GCG model. One knows that for the same dark energy model, the different evolutions of cosmological quantitycan be obtained from different observational datasets. This is the so-called data-dependent. And in order to diminishsystematic uncertainties and get the stringent constraint on cosmological parameters, people often combine manyobservations to constrain the evolutions of cosmological quantities. Next we use a combination of the recent standardcandle data, the standard ruler data (the BAO peak from SDSS and the five-year WMAP CMB shift parameter R )and the OHD to constrain the evolution of Om ( z ) for GCG model.Because the universe has a fraction of baryons, the acoustic oscillations in the relativistic plasma would be imprintedonto the late-time power spectrum of the non-relativistic matter [41]. Then the observations of acoustic signaturesin the large-scale clustering of galaxies can be used to constrain DE models with detection of a peak. The measureddata at z BAO = 0 .
35 from SDSS is [26] A = q Ω eff m E ( z BAO ) − / [ 1 z BAO Z z dz ′ E ( z ′ ) ] / = 0 . ± . , (7)where Ω eff m is the effective matter density parameter [42].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 using the shift parameter [43] R = q Ω eff m Z z rec H dz ′ H ( z ′ ) = 1 . ± . , (8)where z rec = 1089 is the redshift of recombination, and the value of R is given by five-year WMAP data [27][21].We plot the evolution of Om ( z ) for GCG model by using the single standard candle data in Fig. 2 (a). Fromthis figure, it is easy to see that the difference between GCG and ΛCDM model is obvious. Since the Om diagnosticis relatively insensitive to the density parameter, the difference between this figure and Fig. 1 (upper) is caused byusing the different datasets to constrain the quantity Om ( z ), i.e. figure 1 is determined from a combination of UnionSNe Ia and OHD data, but figure 2 (a) is plotted by means of Union SNe Ia data alone. Furthermore based on abovefour observational datasets, the combined constraint on Om ( z ) is presented in Fig. 2 (b). According to Fig. 2 (b), Such as Ω b h = 0 . ± . b h = 0 . ± . h = H / It is an expansion for EOS of DE relative to scale factor a , w de ( a ) = w + w (1 − a ), or w de ( z ) = w + w z z [15][40]. O m H z L SNe0 0.5 1 1.5z0.10.20.30.4 O m H z L O m H z L SNe + OHD + CMB + BAO0 0.5 1 1.5z0.10.20.30.4 O m H z L (a) (b)FIG. 2: Om ( z ) diagnostic for GCG model from Union SNe data and a combination of Union SNe, OHD, CMB and BAO data.Here Ω b is treated as a free parameter. The shaded regions show the 1 σ confidence level. The dashed lines show the values of Om ( z ) for ΛCDM model obtained from the corresponding observational constraint. we can see that the best fit evolution of Om ( z ) for GCG model is near to ΛCDM case, and Om (0) ≡ Om ( z = 0) =0 . +0 . − . (1 σ ) for GCG model. In addition, by using above four datasets to ΛCDM model, it is obtained that thebest fit value of Ω m with confidence level is Ω m = 0 . +0 . − . (1 σ ). We know Om ( z ) = Ω m for ΛCDM, then itsbest fit evolution is included in the 1 σ confidence level of Om ( z ) in GCG scenario. And it can be seen that at 1 σ confidence level, these two models can not be clearly distinguished by current observed data according to the Om ( z )diagram.At last, according to the expression Om ( z ) − Ω m − Ω m ≃ w de ( z ≪
1) [21], it can be calculated that the current EOSof DE w de ≃ − .
964 by taking Ω m = 0 .
273 and the best fit value Om (0) = 0 .
4. Conclusion
On the basis of the recently observed data: the Union SNe Ia data, the nine observational Hubble data, the SDSSbaryon acoustic peak and the five-year WMAP result, we apply a geometrical diagnostic Om to distinguish GCGmodel and ΛCDM model. From Fig. 1, it is shown that the larger error for the evolution of w de ( z ) may be produced bythe erroneous estimation of matter density Ω m . And the Om ( z ) is a better quantity than w de ( z ) to truly distinguishDE models and to show the properties of DE. According to the Om diagram, it is easy to see that for the constraintfrom the single standard candle data, the difference between GCG model and ΛCDM model is obvious, while for thecombined constraint, the best fit evolutions of Om ( z ) for them are similar and the difference between these two modelsis not clear at 1 σ confidence level. In addition, we also calculate the value of current EOS of DE, w de = − . Om (0) for GCG model. Here the Om ( z ) diagram is not sensitive to the variation of densityparameter. Acknowledgments
The research work is supported by NSF (10703001) of PR China. [1] A.G. Riess et al , 1998
Astron. J. et al , 1999
Astrophys. J. et al , 2003
Astrophys. J. Suppl.
175 [arXiv:astro-ph/0302209][3] A.C. Pope et al , 2004
Astrophys. J.
655 [arXiv:astro-ph/0401249][4] S. Weinberg, 1989
Mod. Phys. Rev. Phys. Rev. D. Phys. Rev. Lett. Eur. Phys. J. C Phys. Lett. B
35 [arXiv:astro-ph/0404224][8] A.Y. Kamenshchik, U. Moschella and V. Pasquier, 2001
Phys. Lett. B
265 [arXiv:gr-qc/0103004]M.C. Bento, O. Bertolami and A.A. Sen, Phys. Rev. D 66 (2002) 043507 [arXiv:gr-qc/0202064]P.X. Wu and H.W. Yu 2007
Phys. Lett. B et al , 2008
Phys. Lett. B , 87[10] M. Li, 2004
Phys. Lett. B
Phys. Lett. B
34M R Setare, 2007
Phys.Lett.B
329 [arXiv:hep-th/0704.3679][11] R.G. Cai, 2007
Phys. Lett. B
228 [arXiv:hep-th/0707.4049][12] A.G. Riess et al , 2004
Astrophys. J.
665 [arXiv:astro-ph/0402512][13] A.R. Cooray and D. Huterer, 1999
Astrophys. J.
L95 [arXiv:astro-ph/9901097]J.V. Cunha, L. Marassi and R.C. Santos, 2007
Int. J. Mod. Phys. D Mon. Not. R. Astron. Soc.
33 [arXiv:astro-ph/0201336][15] E.V. Linder, 2003
Phys. Rev. Lett. Int. J. Mod. Phys. D
213 [arXiv:gr-qc/0009008][16] L.X. Xu and J.B. Lu, 2009
Mod. Phys. Lett. A Int. J. Mod. Phys. D Phys. Rev. Lett. Int. J. Mod. Phys. D Phys. Lett. B
208 [arXiv:hep-th/0005016]V. Sahni and Yu. Shtanov, 2003
J. Cosmol. Astropart. Phys.
014 [arXiv:astro-ph/0202346]V. Sahni, Yu. Shtanov and A. Viznyuk, 2005
J. Cosmol. Astropart. Phys.
005 [arXiv:astro-ph/0505004]I. Brevik, 2008
Eur. Phys. J. C JETP Lett. Phys. Rev. D Int. J. Mod. Phys. D J. Cosmol. Astropart. Phys.
003 [arXiv:astro-ph/0411221][23] E. Komatsu et al , [arXiv:astro-ph/0803.0547] [24] D. Rubin et al , [arXiv:astro-ph/0807.1108][25] J. Simon et al , 2005
Phys. Rev. D , 123001[26] D.J. Eisenstein et al , 2005 Astrophys. J. , 560 [arXiv:astro-ph/0501171][27] J. Dunkley et al , [astro-ph/0803.0586][28] T. Nakamura and T. Chiba, 1999
Mon. Not. R. Astron. Soc.
696 [arXiv:astro-ph/9810447]A.A. Starobinsky, 1998
JETP Lett.
757 [arXiv:astro-ph/9810431][29] C. Zunckel and C. Clarkson, 2008
Phys. Rev. Lett.
Prog. Theor. Phys.
815 [arXiv:astro-ph/0708.3877][30] P. Astier et al , 2006
Astron. Astrophys.
31 [arXiv:astro-ph/0510447][31] W.M. Wood-Vasey et al , [arXiv:astro-ph/0701041][32] A.G. Riess et al , [arXiv:astro-ph/0611572][33] M. Hamuy, M.M. Phillips, N.B. Suntzeff, R.A. Schommer and J. Maza, 1996
Astron. J.
Astrophys. J.
122 [arXiv:astro-ph/0612666][34] R.G. Abraham et al , 2003
Astron. J. et al , 1999
Mon. Not. R. Astron. Soc. et al , 2001
Mon. Not. R. Astron. Soc.
Astrophys. J.
L5 [astro-ph/0607301]R. Jimenez, L. Verde, T. Treu and D. Stern, 2003
Astrophys. J.
622 [astro-ph/0302560][37] Z.L. Yi and T.J. Zhang, 2007
Mod. Phys. Lett. A Phys. Lett. B
Eur. Phys. J. C
311 [arXiv:astro-ph/0812.3209]Hui Lin et al , [arXiv:astro-ph/0804.3135][38] D. Kirkman et al , 2003
Astrophys. J. Suppl.
Phys. Lett. B
Astrophys. J.
605 [arXiv:astro-ph/9709112][42] M. Makler, S.Q. Oliveira and I. Waga, 2003
Phys. Rev. D Astron. Astrophys.
Mon. Not. R. Astron. Soc.291