ANISOTROPIC BIANCHI TYPE-III COSMOLOGICAL MODELS IN GENERAL RELATIVITY

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1 Intenational Jounal of Physics and Mathematical Sciences ISSN: 77- (Online) 0 Vol. () Octobe-Decembe, pp. -/Goyal et al. ANISOROPIC BIANCHI YPE-III COSMOLOGICAL MODELS IN GENERAL RELAIVIY Sumeet Goyal, *Hapeet and iwai R.K Chandigah Engineeing College, Landan, Mohali, Punjab, India Depatment of Applied Sciences, Sant Baba Bhag Singh Institute of Engineeing & echnology, Khiala, Padhiana, Jalandha- 00, Punjab, India Govt Engineeing College, Reva, M.P., India *Autho fo Coespondence ABSRAC In this pape we have obtained exact solutions of the field equations fo Bianchi type-iii space times with vaiable gavitational constant G(t) and cosmological constant (t) in the pesence of pefect fluid. We have discussed physical behavio of the model in detail. Also the model satisfies a Machian cosmological solution, i.e. G ~ H which follows fom the model of Kalligas et al., (99). Keywods: Bianchi ype-iii, Gavitational Constant, Cosmological Constant, Hubble paamete. INRODUCION he Einstein field equations have two paametes, the gavitational constant G and the cosmological constant Λ. he Newtonian constant of gavitation G plays the ole of a coupling constant between geomety and matte in the Einstein field equation. In an evolving univese, it is natual to loo at this constant as a function of time. Diac (97, 97, 98, 97) suggested a possible time vaying gavitational constant. he lage numbe hypothesis poposed by Diac leads to a cosmology whee G vaies with time. Many othe extensions of Einstein theoy with time-dependent G have also been poposed by Hoyle and Nalia (96), Canuto et al., (977a, 977b) Desaissian (98). he Λ tem aises natually in geneal elativistic quantum field theoy whee it is intepeted as the enegy density of the vacuum (Zeldovich, 967, 968; Ginzbug et al., 97; Fulling et al., 97). he Λ tem has also been intepeted in tems of the Higgs scala field by Begmann. 968 Deitlan97 suggested that the mass of the Higgs boson is connected with Λ, and Linde (97) poposed that Λ is a function of tempeatue and is elated to the pocess of boen symmeties. Recently, seveal models with the Fiedman-Robetson-Wale (FRW) metic whee G and Λ ae the functions of the time have been studied. Fo these models, the enegy-momentum tenso is descibed as a pefect fluid (Abdel-Rahman, 99; Beman, 99; Abdussatta and Vishwaama, 977, 99, 996). Also Abab (997) discussed the bul viscous models. Beesham (986a, 986b) and Kilinc (006) discussed the Bianchi type-i model with vaiables G and Λ. Wang (00, 00, 00, 006) studied the Bianchi type-iii model with bul viscosity. In this pape, we conside space-time of the Bianchi type III model in a geneal fom with vaiable G and Λ. We apply the equation of state p and scala of expansion popotional to the shea scala. Model and Field Equations We conside the Bianchi type-iii metic in the fom x ds dt A dx B e dy C dz, () Whee A, B and C ae the function of cosmic time t alone, and is a constant. Einstein s field equations with vaiables G and Λ in suitable units ae R Rg 8G g () he enegy momentum tenso fo a pefect fluid is Copyight 0 Cente fo Info Bio echnology (CIBech)

2 Intenational Jounal of Physics and Mathematical Sciences ISSN: 77- (Online) 0 Vol. () Octobe-Decembe, pp. -/Goyal et al. p v v pg, i j Whee is the enegy density of cosmic matte and p is its pessue. Einstein s field equation () fo the metic () leads to B C BC 8G( t) p ( t), B C BC A C AC 8Gp, A C AC A B AB 8Gp, A B AB A AB AC BC 8G, AB AC BC A A B 0, A B whee dots on A, B and C denote the odinay diffeentiation with espect to t. In view of the vanishing divegence of the Einstein tenso, Eq. () gives A B C G p 0 A B C G 8G ; We now assume that the law of consevation of enegy 0 j gives A B C p 0 A B C Using Eq. (9) yields G, 8 () indicating that G inceases o deceases as Λ decease o inceases. We also conside the pefect fluid equation of state, p () whee suggested by Wang (00) may be defined by m () with m, m and being the matte cest mass and adiation enegy densities. As the vaiation of (t) is slow as compaed with the expansion of the univese, except nea the time when matte and adiation enegy densities ae equal, we can appoximate (t) as a step function: /, intheadiationdomin ated( RD) univese, 0, inthemattedomin ated( MD) univese. Fom Eq. (9), we have Copyight 0 Cente fo Info Bio echnology (CIBech) 6 () () () (6) (7) (8) (9) (0) ()

3 Intenational Jounal of Physics and Mathematical Sciences ISSN: 77- (Online) 0 Vol. () Octobe-Decembe, pp. -/Goyal et al. A B (since 0), A B () which leads to A B, (6) whee is a constant of integation. Fom Eqs. () and (6), using Eq. (6), we have B C B B C B C B B C B (7) Solution of the Field Equations hee ae only six independent equations in the seven unnowns A, B, C, p,, G and Λ, an exta equation is needed to solve the system completely. We assume that scala of expansion is popotional to the shea scala, which leads to a elation between metic potential B C (8).Using Eqs. (8) and (7), we have C C, 0, C C C (9) Integating Eq. (9), we obtain C c (0) whee is constant of integation. With the help of Eqs. (6), (8) and (0), the line element () educes to C ds dc C dx x C e dy C dz By suitable tansfomation of coodinates, the line element () educes to ds d x e dy dz dx 0 () Fo the model () the physical and geometical paametes can be easily obtained. he expessions fo the enegy density, gavitational constant G(t), and cosmological constant Λ(t) ae given by, const, w () G(t) 8 ( t). [ 8 n 6 Copyight 0 Cente fo Info Bio echnology (CIBech) 7 () ] ()

4 Intenational Jounal of Physics and Mathematical Sciences ISSN: 77- (Online) 0 Vol. () Octobe-Decembe, pp. -/Goyal et al. he expansion scala and shea fo the model () ae / / (7) Fo 0, we equie 0. he model has singulaity at = 0. he model stats with,,, all being infinite and continues to expand till. Fo this model, the scale factos ae zeo at = 0, which shows that the space time exhibits point type singulaity. Gavitational constant G (t) is zeo initially and gadually inceases and tends to infinity at late times. Since = constant, the model does not appoach isotopy fo lage values of. heefoe, the model descibes a continuously expanding, sheaing, nonotating univese with the big-bang stat. In this model we obseve that the cosmological tem is infinite initially, gadually deceases, and becomes zeo at late times. In the special case of = 0, fom Eq. (0) the line element () educes to ds d dx / n x e dy dz Afte suitable tansfomation Eq. (8) educes to x ds d dx e dy dz he physical and gavitational paametes of the model (9) ae, const, G(t) ( ) (t) he shea and expansion scala ae given by Copyight 0 Cente fo Info Bio echnology (CIBech) 8 () (6) (8) (9) (0) () () () ()

5 Intenational Jounal of Physics and Mathematical Sciences ISSN: 77- (Online) 0 Vol. () Octobe-Decembe, pp. -/Goyal et al. const, Since the model does not appoach isotopy. We can obtain the deceleation paamete q, which shows that q is constant. he model of constant deceleation paamete has been consideed by Beman and Som (990). he Hubble paamete H() eads H(), () Which can be ewitten as H 6q Fo the pesent phase p, p 6q H p p (7) It is evident that negative q p would inceases the pesent age of the univese. Fom Eq. (), we obtain G G and the pesent value is G 6n q G n p p H p G We can find that the quantity satisfies the condition fo a Machian cosmological solution i.e. G ~ H, which follows fom the model of Kalligas et al., (99). Fo the enegy density to be positive definite, we must have 0. he enegy density deceases as time inceases and tends to zeo as tends to infinity. We also obseve that the spatial volume is zeo at = 0. hus, the singulaity exists at = 0 in the model. he gavitational constant G(t) is zeo initially and gadually inceases and tends to infinity at late times povided n0, whee as cosmological tem (t) vaies as squae of the age of univese and tends to zeo as. n,. Deceleation paamete is constant fo all time. Fo 0H 0 his is within the cuent 0.8. limits fo the univese age 0H 0 H and in good ageement with the best estimation 0 0. (Abdussatta and Vishwaama, 997). CONCLUSION In summay, we have obtained exact solutions of the field equations fo Bianchi type-iii space times with vaiable gavitational constant G (t) and cosmological constant (t) in the pesence of pefect fluid. In geneal, the space time exhibits point type singulaity at initial stage and gavitational constant is zeo but cosmological tem vaies as squae of the age of univese. Cosmological tem is infinite at the beginning of the model and it deceases to become zeo at late times. Deceleation paamete is constant fo all time. Also the model satisfies a Machian cosmological solution, i.e. G ~ H which follows fom the model of Kalligas et al., (99). Copyight 0 Cente fo Info Bio echnology (CIBech) 9 (6) (8) (9)

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7 Intenational Jounal of Physics and Mathematical Sciences ISSN: 77- (Online) 0 Vol. () Octobe-Decembe, pp. -/Goyal et al. Wang XX (006). Bianchi ype-iii Sting Cosmological Model with Bul Viscosity and Magnetic Field. Chinese Physics Lettes 70. Zel dovich YAB (968). he Cosmological constant and the theoy of elementay paticles. Soviet Physics Uspehi 8. Zeldovich YB (967). Cosmological Constant and Elementay Paticles. Jounal of Expeimental and heoetical Physics Lettes Copyight 0 Cente fo Info Bio echnology (CIBech)