Research Article | Open Access

# Statefinder Diagnostic for Variable Modified Chaplygin Gas in LRS Bianchi Type I Universe

**Academic Editor:**Remi Leandre

#### Abstract

The Locally Rotationally Symmetric (LRS) Bianchi type I cosmological model with variable modified Chaplygin gas having the equation of state , where , is a positive constant, and is a positive function of the average-scale factor of the universe (i.e., ) has been studied. It is shown that the equation of state of the variable modified Chaplygin gas interpolates from radiation-dominated era to quintessence-dominated era. The statefinder diagnostic pair (i.e., parameter) is adopted to characterize different phases of the universe.

#### 1. Introduction

Recent observations of type Ia supernovae team [1, 2] and the WMAP data [3–5] evidenced that the expansion of the universe is accelerating. While explaining these observations, two dark components known as CDM (the pressureless cold dark matter) and DE (the dark energy with negative pressure) are invoked. The CDM contributes which gives the theoretical interpretation of the galactic rotation curves and large-scale structure formation. The DE provides which causes the acceleration of the distant type Ia supernovae. Different models described this unknown dark sector of the energy content of the universe, starting from the inclusion of exotic components in the context of general relativity to the modifications of the gravitational theory itself, such as a tiny positive cosmological constant [6], quintessence [7, 8], DGP branes [9, 10], the non-linear F(R) models [11–13], and dark energy in brane worlds [14, 15], among many others [16–33] including the review articles [34, 35]. In order to explain anomalous cosmological observations in the cosmic microwave background (CMB) at the largest angles, some authors [36] have suggested cosmological model with anisotropic and viscous dark energy. The binary mixture of perfect fluid and dark energy has been studied for Bianchi type I by [37] and Bianchi type-V by [38]. The anisotropic dark energy has been studied for Bianchi type III in [39] and Bianchi type VIo in [40].

As per [41–43], a unified dark matter—dark energy scenario could be found out, in which these two components (CDM and DE) are different manifestations of a single fluid. Generalised Chaplygin gas is a candidate for such unification which is an exotic fluid with the equation of state , where *B* and are two parameters are to be determined. It was initially suggested in [44] with and then generalized in [45] for the case .

As per [46], the isotropic pressure of the cosmological fluid obeys a modified Chaplygin gas equation of state where , and is a positive constant.

When and the moving volume of the universe is small (i.e., ), this equation of state corresponds to a radiation-dominated era. When the density is small (i.e., ), this equation of state corresponds to a cosmological fluid with negative pressure (the dark energy). Generally, the modified Chaplygin gas equation of state corresponds to a mixture of ordinary matter and dark energy. For the matter content is pure dust with . The variable Chaplygin gas model was proposed by [47] and constrained using SNeIa 2 “gold” data [48].

Recently, another important form of EOS for variable modified Chaplygin gas [49, 50] is considered as where , is a positive constant, and is a positive function of the average-scale factor of the universe (i.e., ).

Since there are more and more models proposed to explain the cosmic acceleration, it is very desirable to find a way to discriminate between the various contenders in a model independent manner. Sahni et al. [51] proposed a cosmological diagnostic pair called statefinder, which is defined as to differentiate among different forms of dark energy. Here is the Hubble parameter and is the deceleration parameter. The two parameters are dimensionless and are geometrical since they are derived from the cosmic scale factor alone, though one can rewrite them in terms of the parameters of dark energy and dark matter. This pair gives information about dark energy in a model-independent way, that is, it categorizes dark energy in the context of back-ground geometry only which is not dependent on theory of gravity. Hence, geometrical variables are universal. Therefore, the statefinder is a “geometrical diagnostic” in the sense that it depends upon the expansion factor and hence upon the metric describing space and time. Also, this pair generalizes the well-known geometrical parameters like the Hubble parameter and the deceleration parameter. This pair is algebraically related to the equation of state of dark energy and its first time derivative. The statefinder parameters were introduced to characterize primarily flat universe models with cold dark matter (dust) and dark energy.

The statefinder pair represents a cosmological constant with a fixed equation of state and a fixed Newton’s gravitational constant. The standard cold dark matter model containing no radiation has been represented by the pair . The Einstein static universe corresponds to pair [52]. The statefinder diagnostic pair is analyzed for various dark energy candidates including holographic dark energy [53], agegraphic dark energy [54], quintessence [55], dilation dark energy [56], Yang-Mills dark energy [57], viscous dark energy [58], interacting dark energy [59], tachyon [60], modified Chaplygin gas [61], and gravity [62].

Gorini et al. [63, 64] proved that the simple flat Friedmann model with Chaplygin gas can equivalently be described in terms of a homogeneous minimally coupled scalar field and a self-interacting potential with effective Lagrangian Barrow [65, 66] and Kamenshchik et al. [67, 68] have obtained homogeneous scalar field and a potential to describe Chaplygin cosmology.

The Bianchi type V cosmological model with modified Chaplygin gas has been investigated by [69], and Bianchi type-V cosmological model with variable modified Chaplygin gas has been also studied by [70]. In the present paper, the spatially homogeneous and anisotropic LRS Bianchi type I cosmological model with variable modified Chaplygin gas has been investigated. It is shown that the equation of state of this modified model is valid from the radiation era to the quintessence. The statefinder diagnostic pair, that is, parameter is adopted to characterize different phase of the universe.

#### 2. Metric and Field Equations

The spatially homogeneous and anisotropic LRS Bianchi type I line element can be written as where and are functions of cosmic time only.

In view of (2.1), the Einstein field equations are () where and are the energy density and pressure, respectively.

The energy conservation equation is

The spatial volume of the universe is defined by where is an average-scale factor of the universe.

Let us introduce the variable modified Chaplygin gas having equation of state where , is a positive constant, and is a positive function of the average scale factor of the universe (i.e., ).

Now, assume is in the form where and are positive constants.

Using (2.5), (2.7), and (2.8), one can obtain where is a constant of integration.

For expanding universe, must be positive, and for positivity of first term in (2.9), we must have .

*Case 1. * Now, for small values of the scale factors and (refer to [38, 71]), one may have
which is very large and corresponds to the universe dominated by an equation of state
From (2.2)–(2.4), one can get

Using (2.10) and , (2.12) yields
where and are constants of integration.

For , and , (2.13) leads to
where

Subtracting (2.3) from (2.4), we obtain

Solving (2.16) and then using (2.6), one may get the values of the scale factors and as
where is a constant of integration.

From (2.10), the value of the pressure and the energy density of the universe is given by
therefore

The Hubble parameter and the deceleration parameter [] are found as

The universe is decelerating.

From (1.3), the statefinder parameters are found as

*Case 2 2. *Now, for large values of the scale factors and (refer to [38, 71]), one may have
and the pressure is given by
Using (2.22) and (2.23) in (2.12), we get
where
Solving (2.16) and then using (2.24), one may get the values of the scale factors and as
where is a constant of integration.

The Hubble parameter and the deceleration parameter [] are found as
From (1.3), the statefinder parameters are found as

The relation (Figure 1) between the statefinder parameters and is

To describe the variable modified Chaplygin gas cosmology, consider the energy density and pressure corresponding to a scalar field having a self-interacting potential as
From (2.31), we have
When kinetic term is small compared to the potential, we obtain
therefore

We know that different possibilities can be distinguished for nature of dark energy by its equation of state characterized by . (The equation of state parameter for radiation is simply , whereas for matter, it is ). The equation (2.33) recovers the constant solution for dark energy with . This is consistent with the central value determined by WMAP as In both cases (Cases 1 and 2), from (2.17), (2.26), and (2.27), it is observed that, when , we get , which is also supported by recent observations of supernovae Ia [1, 2] and WMAP [5]. Therefore, the present model is free from finite time future singularity.

#### 3. Conclusion

The spatially homogeneous and anisotropic LRS Bianchi type I cosmological model with variable modified Chaplygin gas has been studied. It is noted that the equation of state for this model is valid from the radiation era to the quintessence. It reduces to dark energy for small kinetic energy. In first case, it is observed that initially the universe is decelerating and later on (in the second case) it is accelerating which is consistent with the present day astronomical observations. The present model is free from finite time future singularity. The statefinder diagnostic pair (i.e., parameter) is adopted to differentiate among different forms of dark energy.

#### Acknowledgments

The author is thankful to honourable Referee for valuable comments which has improved the standard of the research article. The author also records his thanks to UGC, New Delhi for financial assistance through Major Research Project.

#### References

- S. Perlmutter, G. Aldering, G. Goldhaber et al., “Measurements of Ω and Λ from 42 high-redshift Supernovae,”
*Astrophysical Journal Letters*, vol. 517, no. 2, pp. 565–586, 1999. View at: Google Scholar - A. G. Riess, A. V. Filippenko, P. Challis et al., “Observational evidence from supernovae for an accelerating universe and a cosmological constant,”
*Astronomical Journal*, vol. 116, no. 3, pp. 1009–1038, 1998. View at: Google Scholar - P. De Bernardis, P. A. R. Ade, J. J. Bock et al., “A flat Universe from high-resolution maps of the cosmic microwave background radiation,”
*Nature*, vol. 404, no. 6781, pp. 955–959, 2000. View at: Publisher Site | Google Scholar - S. Hanany, P. Ade, A. Balbi et al., “MAXIMA-1: a measurement of the cosmic microwave background anisotropy on angular scales of $1{0}^{\prime}$-5°,”
*Astrophysical Journal Letters*, vol. 545, no. 1, pp. L5–L9, 2000. View at: Google Scholar - D. N. Spergel, R. Bean, O. Doré et al., “Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: implications for cosmology,”
*Astrophysical Journal*, vol. 170, no. 2, pp. 377–408, 2007. View at: Publisher Site | Google Scholar - R. R. Caldwell, R. Dave, and P. J. Steinhardt, “Cosmological imprint of an energy component with general equation of state,”
*Physical Review Letters*, vol. 80, no. 8, pp. 1582–1585, 1998. View at: Google Scholar - A. R. Liddle and R. J. Scherrer, “Classification of scalar field potentials with cosmological scaling solutions,”
*Physical Review D*, vol. 59, Article ID 023509, 1999. View at: Google Scholar - P. J. Steinhardt, L. Wang, and I. Zlatev, “Cosmological tracking solutions,”
*Physical Review D*, vol. 59, Article ID 123504, 1999. View at: Google Scholar - G. Dvali, G. Gabadadze, and M. Porrati, “Metastable gravitons and infinite volume extra dimensions,”
*Physics Letters, Section B*, vol. 484, no. 1-2, pp. 112–118, 2000. View at: Publisher Site | Google Scholar - C. Deffayet, “Cosmology on a brane in Minkowski bulk,”
*Physics Letters, Section B*, vol. 502, pp. 199–208, 2001. View at: Publisher Site | Google Scholar - S. Capozziello, S. Carloni, and A. Troisi, “Quintessence without scalar fields,” http://arxiv.org/abs/astro-ph/0303041. View at: Google Scholar
- S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, “Is cosmic speed-up due to new gravitational physics?”
*Physical Review D*, vol. 70, no. 4, Article ID 043528, 2004. View at: Publisher Site | Google Scholar - S. Nojiri and S. D. Odintsov, “Modified gravity with negative and positive powers of curvature: unification of inflation and cosmic acceleration,”
*Physical Review D*, vol. 68, Article ID 123512, 2003. View at: Publisher Site | Google Scholar - P. K. Townsend and M. N. R. Wohlfarth, “Accelerating cosmologies from compactification,”
*Physical Review Letters*, vol. 91, no. 6, Article ID 061302, 4 pages, 2003. View at: Google Scholar - G. W. Gibbons, “Aspects of supergravity theories,” in
*Supersymmetry, Supergravity and Related Topics*, F. del Aguila, J. A. de Azcarraga, and L. E. Ibanez, Eds., pp. 346–351, World Scientific, 1985. View at: Google Scholar - J. Maldacena and C. Nuñez, “Supergravity description of field theories on curved manifolds and a no go theorem,”
*International Journal of Modern Physics A*, vol. 16, no. 5, pp. 822–855, 2001. View at: Google Scholar - N. Ohta, “Accelerating cosmologies from spacelike branes,”
*Physical Review Letters*, vol. 91, no. 6, Article ID 061303, 4 pages, 2003. View at: Google Scholar - M. N. R. Wohlfarth, “Accelerating cosmologies and a phase transition in M-theory,”
*Physics Letters, Section B*, vol. 563, no. 1-2, pp. 1–5, 2003. View at: Publisher Site | Google Scholar - S. Roy, “Accelerating cosmologies from M/string theory compactifications,”
*Physics Letters, Section B*, vol. 567, no. 3-4, pp. 322–329, 2003. View at: Publisher Site | Google Scholar - J. K. Webb, M. T. Murphy, V. V. Flambaum et al., “Further evidence for cosmological evolution of the fine structure constant,”
*Physical Review Letters*, vol. 87, no. 9, Article ID 091301, pp. 1–4, 2001. View at: Google Scholar - J. M. Cline and J. Vinet, “Problems with time-varying extra dimensions or “Cardassian expansion” as alternatives to dark energy,”
*Physical Review D*, vol. 68, Article ID 025015, 2003. View at: Publisher Site | Google Scholar - N. Ohta, “A study of accelerating cosmologies from superstring/M theories,”
*Progress of Theoretical Physics*, vol. 110, no. 2, pp. 269–283, 2003. View at: Google Scholar - N. Ohta, “Accelerating cosmologies and inflation from M/superstring theories,”
*International Journal of Modern Physics A*, vol. 20, no. 1, pp. 1–40, 2005. View at: Publisher Site | Google Scholar - C. M. Chen et al., “Hyperbolic space cosmologies,”
*Journal of High Energy Physics*, vol. 10, article 058, 2003. View at: Publisher Site | Google Scholar - E. Bergshoeff, A. Collinucci, U. Gran, M. Nielsen, and D. Roest, “Transient quintessence from group manifold reductions or how all roads lead to Rome,”
*Classical and Quantum Gravity*, vol. 21, no. 8, pp. 1947–1969, 2004. View at: Publisher Site | Google Scholar - Y. Gong and A. Wang, “Acceleration from M theory and fine-tuning,”
*Classical and Quantum Gravity*, vol. 23, pp. 3419–3426, 2006. View at: Publisher Site | Google Scholar - I. P. Neupane and D. L. Wiltshire, “Accelerating cosmologies from compactification with a twist,”
*Physics Letters, Section B*, vol. 619, no. 3-4, pp. 201–207, 2005. View at: Publisher Site | Google Scholar - I. P. Neupane and D. L. Wiltshire, “Cosmic acceleration from M theory on twisted spaces,”
*Physical Review D*, vol. 72, Article ID 083509, 2005. View at: Publisher Site | Google Scholar - K.-I. Maeda and N. Ohta, “Inflation from superstring and M-theory compactification with higher order corrections,”
*Physical Review D*, vol. 71, no. 6, Article ID 063520, pp. 1–27, 2005. View at: Publisher Site | Google Scholar - I. P. Neupane, “Accelerating universes from compactification on a warped conifold,”
*Physical Review Letters*, vol. 98, no. 6, Article ID 061301, 2007. View at: Publisher Site | Google Scholar - Y. Gong, A. Wang, and Q. Wu, “Cosmological constant and late transient acceleration of the universe in the Horava-Witten heterotic M-theory on S
^{1}/ Z_{2},”*Physics Letters, Section B*, vol. 663, no. 3, pp. 147–151, 2008. View at: Publisher Site | Google Scholar - P. R. Pereira, M. F. A. Da Silva, and R. Chan, “Anisotropic self-similar cosmological model with dark energy,”
*International Journal of Modern Physics D*, vol. 15, no. 7, pp. 991–999, 2006. View at: Publisher Site | Google Scholar - C. F. C. Brandt, R. Chan, M. A. F. Da Silva, and J. F. V. Da Rocha, “Inhomogeneous dark energy and cosmological acceleration,”
*General Relativity and Gravitation*, vol. 39, no. 10, pp. 1675–1687, 2007. View at: Publisher Site | Google Scholar - E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,”
*International Journal of Modern Physics D*, vol. 15, no. 11, pp. 1753–1935, 2006. View at: Publisher Site | Google Scholar - T. Padmanabhan, “Dark energy and gravity,”
*General Relativity and Gravitation*, vol. 40, no. 2-3, pp. 529–564, 2008. View at: Publisher Site | Google Scholar - T. Koivisto and D. F. Mota, “Anisotropic dark energy: dynamics of the background and perturbations,”
*Journal of Cosmology and Astroparticle Physics*, vol. 06, no. 6, article 018, 2008. View at: Google Scholar - S. Bijan,
*Chinese Journal of Physics*, vol. 43, p. 1035, 2005. - T. Singh and R. Chaubey, “Bianchi type-V cosmological models with perfect fluid and dark energy,”
*Astrophysics and Space Science*, vol. 319, no. 2–4, pp. 149–154, 2009. View at: Publisher Site | Google Scholar - O. Akarsu and C. B. Kilinç, “Bianchi type III models with anisotropic dark energy,”
*General Relativity and Gravitation*, vol. 42, no. 4, pp. 763–775, 2010. View at: Publisher Site | Google Scholar - K. S. Adhav, A. S. Bansod, S. L. Munde, and R. G. Nakwal, “Bianchi type-VI0 cosmological models with anisotropic dark energy,”
*Astrophysics and Space Science*, vol. 332, no. 2, pp. 497–502, 2011. View at: Publisher Site | Google Scholar - T. Matos and L. A. Ureña-López, “Quintessence and scalar dark matter in the Universe,”
*Classical and Quantum Gravity*, vol. 17, pp. L75–L81, 2000. View at: Publisher Site | Google Scholar - C. Wetterich, “Cosmon dark matter?”
*Physical Review D*, vol. 65, no. 12, Article ID 123512, 2002. View at: Publisher Site | Google Scholar - T. Padmanabhan and T. R. Choudhury, “Can the clustered dark matter and the smooth dark energy arise from the same scalar field?”
*Physical Review D*, vol. 66, Article ID 081301, 2002. View at: Google Scholar - A. Kamenshchik, U. Moschella, and V. Pasquier, “An alternative to quintessence,”
*Physics Letters, Section B*, vol. 511, no. 2–4, pp. 265–268, 2001. View at: Publisher Site | Google Scholar - M. C. Bento, O. Bertolami, and A. A. Sen, “Generalized Chaplygin gas, accelerated expansion, and dark-energy-matter unification,”
*Physical Review D*, vol. 66, Article ID 043507, 2002. View at: Publisher Site | Google Scholar - U. Debnath, A. Banerjee, and S. Chakraborty, “Role of modified Chaplygin gas in accelerated universe,”
*Classical and Quantum Gravity*, vol. 21, no. 23, pp. 5609–5617, 2004. View at: Publisher Site | Google Scholar - Z.-K. Guo and Y.-Z. Zhang, “Cosmology with a variable Chaplygin gas,”
*Physics Letters, Section B*, vol. 645, no. 4, pp. 326–329, 2007. View at: Publisher Site | Google Scholar - A. G. Riess, L.-G. Sirolger, J. Tonry et al., “Type Ia supernova discoveries at z > 1 from the hubble space telescope: evidence for past deceleration and constraints on dark energy evolution,”
*Astrophysical Journal Letters*, vol. 607, no. 2 I, pp. 665–687, 2004. View at: Publisher Site | Google Scholar - U. Debnath, “Variable modified Chaplygin gas and accelerating universe,”
*Astrophysics and Space Science*, vol. 312, no. 3-4, pp. 295–299, 2007. View at: Publisher Site | Google Scholar - M. Jamil and M. A. Rashid, “Interacting modified variable Chaplygin gas in a non-flat universe,”
*European Physical Journal C*, vol. 58, pp. 111–114, 2008. View at: Publisher Site | Google Scholar - V. Sahni, T. D. Saini, A. A. Starobinsky, and U. Alam, “Statefinder—a new geometrical diagnostic of dark energy,”
*JETP Letters*, vol. 77, no. 5, pp. 201–206, 2003. View at: Publisher Site | Google Scholar - U. Debnath, “Emergent universe and the phantom tachyon model,”
*Classical and Quantum Gravity*, vol. 25, Article ID 205019, 2008. View at: Publisher Site | Google Scholar - X. Zhang, “Statefinder diagnostic for holographic dark energy model,”
*International Journal of Modern Physics D*, vol. 14, no. 9, pp. 1597–1606, 2005. View at: Publisher Site | Google Scholar - H. Wei and R.-G. Cai, “Statefinder diagnostic and w-w′ analysis for the agegraphic dark energy models without and with interaction,”
*Physics Letters, Section B*, vol. 655, pp. 1–6, 2007. View at: Publisher Site | Google Scholar - X. Zhang, “Statefinder diagnostic for coupled quintessence,”
*Physics Letters, Section B*, vol. 611, no. 1-2, pp. 1–7, 2005. View at: Publisher Site | Google Scholar - Z. G. Huang, X. M. Song, H. Q. Lu, and W. Fang, “Statefinder diagnostic for dilaton dark energy,”
*Astrophysics and Space Science*, vol. 315, no. 1–4, pp. 175–179, 2008. View at: Publisher Site | Google Scholar - W. Zhao, “Statefinder diagnostic for the yang-mills dark energy model,”
*International Journal of Modern Physics D*, vol. 17, pp. 1245–1254, 2008. View at: Publisher Site | Google Scholar - M.-G. Hu and X.-H. Meng, “Bulk viscous cosmology: statefinder and entropy,”
*Physics Letters, Section B*, vol. 635, no. 4, pp. 186–194, 2006. View at: Publisher Site | Google Scholar - W. Zimdahl and D. Pavón, “Statefinder parameters for interacting dark energy,”
*General Relativity and Gravitation*, vol. 36, no. 6, pp. 1483–1491, 2004. View at: Publisher Site | Google Scholar - Y. Shao and Y. Gui, “Statefinder parameters for tachyon dark energy model,”
*Modern Physics Letters A*, vol. 23, pp. 65–71, 2008. View at: Publisher Site | Google Scholar - W. Chakraborty and U. Debnath, “Is modified Chaplygin gas along with barotropic fluid responsible for acceleration of the universe?”
*Modern Physics Letters A*, vol. 22, no. 24, pp. 1805–1812, 2007. View at: Publisher Site | Google Scholar - S. Li, “Statefinder diagnosis for the Palatini f(R) gravity theories,” http://arxiv.org/abs/1002.3867. View at: Google Scholar
- V. Gorini, “The Chaplygin gas as a model for dark energy,” http://arxiv.org/abs/gr-qc/0403062. View at: Google Scholar
- V. Gorini, A. Kamenshchik, and U. Moschella, “Can the Chaplygin gas be a plausible model for dark energy?”
*Physical Review D*, vol. 67, Article ID 063509, 2003. View at: Publisher Site | Google Scholar - J. D. Barrow, “String-driven inflationary and deflationary cosmological models,”
*Nuclear Physics, Section B*, vol. 310, no. 3-4, pp. 743–763, 1988. View at: Google Scholar - J. D. Barrow, “Graduated inflationary universes,”
*Physics Letters, Section B*, vol. 235, pp. 40–43, 1990. View at: Google Scholar - A. Kamenshchik, U. Moschella, and V. Pasquier, “An alternative to quintessence,”
*Physics Letters, Section B*, vol. 511, no. 2–4, pp. 265–268, 2001. View at: Publisher Site | Google Scholar - V. Gorini, A. Kamenshchik, U. Moschella, and V. Pasquier, “Tachyons, scalar fields, and cosmology,”
*Physical Review D*, vol. 69, no. 12, Article ID 123512, 2004. View at: Publisher Site | Google Scholar - K. S. Adhav, “Statefinder diagnostic for modified Chaplygin gas in Bianchi type-V universe,”
*European Physical Journal Plus*, vol. 126, no. 5, pp. 1–9, 2011. View at: Publisher Site | Google Scholar - K. S. Adhav, “Statefinder diagnostic for variable modified Chaplygin gas in Bianchi type-V universe,”
*Astrophysics and Space Science*, vol. 335, no. 2, pp. 611–617, 2011. View at: Publisher Site | Google Scholar - S. Ram, M. Zeyauddin, and C. P. Singh, “Generalized chaplygin gas dominated anisotropic bianchi type-i cosmological models,”
*International Journal of Theoretical Physics*, vol. 48, no. 1, pp. 50–60, 2009. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2012 K. S. Adhav. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.