A Cosmological Model with Varying and in General Relativity—Part III
We are investigating Bianchi type-I cosmological model in perfect fluid. The cosmological model is obtained by assuming proportional to . We also observed some physical properties of the model and discussed them.
Cosmology is the study of the largest-scale structures and dynamics of our universe, and it deals with subjects regarding their origin and evolution. Cosmology involves itself in studying the motions of the celestial bodies. At the present state of evolution, the universe is isotropic and homogeneous. The cosmological constant problem is very interesting. And the simplest way out of the problem is to consider a varying cosmological term. This can be done by considering different values for cosmological constant .
As we are aware that the expansion of the universe is undergoing time acceleration Perlmutter et al. [1–3], Riess et al. [4, 5], Allen et al. , Peebles and Ratra , Padmanabhan , and Lima . In the literature cosmological models with proportional to scale factor have been studied by Chen and Wu , Pavn , Carvalho et al. , Lima and Maria , Lima and Trodden , Arbab and Abdel-Rahman , Cunha and Santos , and Carneiro and Lima . A number of authors investigated Blanches models, using the approach that there is a link between variation of gravitational constant and cosmological constant (see [18–22]). A Lot of work has been done by Saha [23–26] in studying the anisotropic Bianchi type-I cosmological model in general relativity with varying and .
The cosmological constant is small because the universe is old. Models with dynamically decaying cosmological term representing the energy density of vacuum have been studied by Vishwakarma [27–29], Arbab , and Berman [31, 32]. In this paper, we study homogeneous Bianchi type-I space time with variable and containing matter in the form of a perfect fluid. We obtain solution of the Einstein field equations by assuming that cosmological term is proportional to . ( is scale factor.)
2. The Metric and Field Equations
We consider the Bianchi type-I metric in the orthogonal form as follows:
We assume that the cosmic matter is taken to be perfect fluid given by the energy-momentum tensor as the following: where is Ricci tensor, and and are the isotropic pressure and energy density of the fluid. We take equation of state is the four velocity vector of the fluid satisfying Einstein’s field equations with time dependent and are where For the metric (1) and energy-momentum tensor (2) in commoving system of coordinates, the field equation (5) yields In view of vanishing of the divergence of Einstein tensor, we have The usual energy conservation equation of general relativity quantities is Equation (8) together with (9) puts and in some sort of coupled field given by implying that is a constant whenever is constant. Using (3) in (9) and then integrating, we get ; in particular we are assuming .
Consider We define as the average scale factor of Bianchi type-I universe.
Consider The Hubble parameter , volume expansion , shear , and deceleration parameter are given by Einstein’s field equations (7) can be also written in terms of Hubble parameter , shear , and deceleration parameter as
On integrating (7), we obtain where and are constants of integration. From (15), we obtain implying that Thus, the presence of positive lowers the upper limit of anisotropy whereas a negative contributes to the anisotropy.
Equation (17) can also be written as where is the critical density and is the vacuum density.
From (14) and (15), we get Thus, the universe will be in decelerating phase for negative , and for positive the universe will slow down the rate of decrease, showing that the rate of volume expansion decreases during time evolution, and presence of positive slows down the rate of this decrease whereas a negative would promote it.
3. Solution of the Field Equations
The system of (7) and (10) supplies only five equations in seven unknown parameters (, , , , , , and ). Two extra equations are needed to solve the system completely. For this purpose, we take cosmological term to be proportional to . [33, 34]. That is, Using (11) and (21) in (10) we get From (14), (15), and (21), where is a constant of integration, we get By using (23) in (16) and the metric (1), we get where and , , and are constants.
For the model (24), the spatial , density , gravitational constant , and cosmological constant are Expansion scalar and shear are
4. Observations and Conclusion
(1)We observe that the spatial volume at and expansion scalar is infinite, which shows that universe starts evolving with zero volume at with an infinite rate of expansion. Hence, the model has a point type singularity at initial epoch. (2)Initially at , the energy density “,” pressures “,” shear , and cosmological term tend all to be infinite. (3)As increases the spatial volume increases, but the expansion rate decreases. Thus, the rate of expansion slows down with increase in time and tends to be zero.(4)As , the spatial volume becomes infinitely large. All parameters , , , , , and asymptotically but is decreasing. Therefore, at large value of the model gives empty universe. The cosmic scenario starts from a big bang at and continues until .(5)From (28), we observed that when the model is decelerating as is positive, and the model is accelerating when . (6)The ratio as . So the model approaches isotropy for a large value of .
The possibility of increasing with time, at least in some stages of the development of the universe, has been investigated by Abdel-Rahman , Chow , Levitt , and Milne . include Berman , Berman and Som , Berman et al. , and Bertolami [41, 42]. This form of is physically reasonable as observations suggest that is very small in the present universe. A decreasing functional form permits to be large in the early universe.
In summary, we have investigated the Bianchi type-I cosmological model with variable and in presence of perfect fluid where the cosmological term is proportional to . ( is scale factor) as suggested by Silveira and Waga [43, 44] and others. Initially the model has a point type singularity, gravitational constant is decreasing, and cosmological constant is infinite at this time, when time increases decreases. The model approaches isotropy for a large value of “,” the model is quasi-isotropic that is, .
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
J. A. S. Lima, “Alternative dark energy models: an overview,” Brazilian Journal of Physics, vol. 34, no. 1 A, pp. 194–200, 2004.View at: Google Scholar
J. A. S. Lima and J. M. F. Maria, “Deflationary cosmology with decaying vacuum energy density,” Physical Review D, vol. 49, no. 10, pp. 5597–5600, 1994.View at: Google Scholar
J. A. S. Lima and M. Trodden, “Decaying vacuum energy and deflationary cosmology in open and closed universes,” Physical Review D, vol. 53, no. 8, pp. 4280–4286, 1996.View at: Google Scholar
J. P. Singh, R. K. Tiwari, and S. Pratibha, “Bianchi type-III cosmological models with gravitational constant G and the cosmological constant Λ,” Chinese Physics Letters, vol. 24, no. 12, pp. 3325–3327.View at: Google Scholar
B. Saha, “Anisotropic cosmological models with perfect fluid and dark energy,” Chinese Journal of Physics, vol. 43, no. 6, pp. 1035–1043, 2005.View at: Google Scholar
A. I. Arbab, “Bianchi type I universe with variable G and Λ,” General Relativity and Gravitation, vol. 1401, 1998.View at: Google Scholar
J. M. Overduin and F. I. Cooperstock, “Evolution of the scale factor with a variable cosmological term,” Physical Review D, vol. 58, no. 4, p. 043506, 1998.View at: Google Scholar
E. A. Milne, Relativity, Gravitation and World Structure, Oxford University Press, 1935.
O. Bertolami, “Brans-Dicke cosmology with a scalar field dependent cosmological term,” Fortsch Physics, vol. 34, p. 829, 1986.View at: Google Scholar
V. Silveira and I. Waga, “Cosmological properties of a class of Λ decaying cosmologies,” Physical Review D, vol. 56, no. 8, pp. 4625–4632, 1997.View at: Google Scholar