Bianchi Type V String Cosmological Model with Variable Deceleration Parameter

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1 September 013 Volume 4 Issue 8 pp Banch Type V Strng Cosmologcal Model wth Varable Deceleraton Parameter Kanka Das * &Tazmn Sultana Department of Mathematcs, Gauhat Unversty, Guwahat , Assam, Inda Artcle Abstract We consder n ths paper a four dmensonal Banch Type-V strng cosmologcal model. The exact solutons of Ensten s feld equatons have been obtaned by consderng tme dependent deceleraton parameter and by choosng the scale factor a( t) [snh( t)], where n s a postve constant. The physcal behavor of the Unverse s studed and t s observed that our model s evolvng from deceleratng phase to acceleratng phase. Also t s found that cosmc strngs do not exst n Banch type-v cosmology. Keywords: Cosmc Strng, Banch type-v unverse, Deceleraton Parameter, Cosmology. 1. Introducton Recently, Consderable work has been done n strng cosmology. Ensten s theory of gravty has been the subect of ntense study for ts success n explanng the observed accelerated expanson of the unverse at late tmes. Banch type cosmologcal models are mportant because these are homogeneous and ansotropc. The orgn of the unverse s one of the greatest cosmologcal mysteres even today. The exact physcal stuaton at early stage of the formaton of our unverse s stll unknown. The concept of strng theory was developed to descrbe events of the early stage of the evoluton of the unverse. The present day observatons of the unverse ndcate the exstence of a large scale network of strngs n the early unverse (Kbble 1976, 1980). In recent years there has been a lot of nterest n the study of cosmc strngs. Cosmc strngs have receved consderable attenton as they are beleved to have served n the structure formaton n the early stages of the unverse. Kbble (1976) showed that cosmc strngs may have been created durng phase transtons n the early era and they act as a source of gravtatonal feld (Leteler 1983).The study of cosmc strngs n relatvstc framework was ntated by Stachel (1990) and Leteler (1979). Kror et.al (1990, 1994), Ra Bal and Shuch Dave (001), Bhattacharee and Baruah (001), Rahaman et.al.(003), Reddy (003) are some of the authors who have studed varous aspects of strng cosmologes n general relatvstc theory as well as alternatve theores of gravtaton. Kror et al. (1994) have shown that n the * Correspondence Author: Kanka Das, Department of Mathematcs, Gauhat Unversty, Inda. E-mal: daskanka@gmal.com ISSN:

2 September 013 Volume 4 Issue 8 pp context of general relatvty cosmc strngs do not occur n Banch type V cosmology. Also Adhav et al. (009) have obtaned the same fndngs as Kroretal.about Banch type V cosmology. We present, n ths paper, an exact soluton of Banch type V strng cosmologcal model by assumng a specal type of scale factor and a varable deceleraton parameter. It s observed that cosmc strng do not occur n Banch type V model and the unverse showng a transton from an early deceleratng phase to a recent acceleratng phase. Ths paper s organzed as: In secton, the metrc and the feld equatons are presented. In secton 3, we deal wth an exact soluton of the feld equatons wth cloud of strngs. Secton 4, descrbes some physcal and geometrcal propertes of the models. Fnally conclusons are presented n secton 5.. The Metrc and the Feld Equatons The lne element for the spatally homogeneous and ansotropc Banch-V space-tme s gven by ds dt A dx e x B dy C dz. (1) where A( t), B( t) and C (t) are the scale factors n dfferent spatal drectons and s a constant. We defne 1/3 a (ABC) Hubble s parameter read as as the average scale factor of the space-tme (1) so that the average a H. () a where the overhead dot denotes dervatves wth respect to cosmc tme t. The energy momentum tensor flud s taken as T for a cloud of massve strngs and the dstrbuton of perfect T pv v pg x x. (3) where s the sotropc pressure; s the proper energy densty for a cloud of strngs wth partcle attached to them; s the strng tenson densty, v (0,0,0,1) s the four velocty of the partcles and s a unt space-lke vector representng the drecton of the strng. The vectors and satsfy the condtons ISSN:

3 September 013 Volume 4 Issue 8 pp v v x x 1, v 0. (4) x Choosng x parallel to x we have 1 x ( A,0,0,0). (5) If the partcle densty of the confguraton s denoted by, then p (6) The Ensten s feld equatons (n gravtatonal unts c 1,8 G 1) are as follows R 1 g R T, (7) The Ensten s feld equatons (7) for the lne-element (1) lead to the followng system of equatons B C BC B C BC A A C AC A C AC A A B AB A B AB A p, (8) p, (9) p, (10) A B AB AC BC 3, (11) AC BC A A B C 0. (1) A B C The energy conservaton equaton T 0leads to, A B C A p 0. (13) A B C A ISSN:

4 September 013 Volume 4 Issue 8 pp whch s obtaned from the feld equaton. The dot (.) denotes ordnary dfferentatng wth respect to t. 3. Soluton of Feld Equatons Integratng Equaton (1) and takng the constant of ntegraton as unty n B or C, wthout loss of generalty, we obtan A BC (14) Subtractng Equaton (9) from Equaton (10) and takng the second ntegral, we get the followng relaton where and are constants of ntegraton. B dt d1 exp k1 C (15) ABC Equatons (8)-(1) are fve ndependent equatons n sx unknowns complete determnaton of the system, we need one extra condton. A, B, C, p, and. For the Followng Pradhan et.al (01), we assume the law of varaton of scale factor as ncreasng functon of tme a (snh( t)) where n s a postve constant and s an arbtrary constant. Now the spatal volume V of the model s read as V (16) 3 3/ n a (snh( t)) (17) Equatons (14), (16) and (17) lead to (snh( t)) (18) A( t) Insertng equaton (18) nto (14) and (15), we get k1 dt B snh( t) d1 exp (19) 3/ n snh( t) ISSN:

5 September 013 Volume 4 Issue 8 pp k 1 dt C snh( t) exp (0) 3/ n d1 snh( t) 4. Some Physcal and Geometrcal Propertes The sotropc pressure are gven by, proper energy densty, strng tenson and partcle densty p / n k1 6 / n p snh( t) 3 coth( t) cosech( t) snh( t) (1) n n 4 k1 6/ n / n 3 coth( t) snh( t) 3 snh( t) () n 4 0 p (3) (4) Equaton (3) shows that cosmc strngs do not occur n Banch type-v space-tme wth average scale factor a (snh( t)). The average Hubble s parameter (H), expanson scalar ( ) ansotropy parameter ( A m ) and shear scalar( )of the model are gven by H a coth( t) (5) a n 3H 3 coth( t) (6) n ISSN:

6 September 013 Volume 4 Issue 8 pp A m 1 9H 1 n 6 A B A B k 1 B C B C 6 / n tanh( t) snh( t) C A C A 797 (7) The value of DP (q) s found to be 1 6 / n 3 Am H k1 snh t (8) 4 a q ah 1 tanh( t) 1 n (9) We observe that q 0 for n 1 and q 0 for n 1. Thus t s evdent that for 0 n 1, our model s n acceleratng phase but for n 1, our model s evolvng from deceleratng phase to acceleratng phase. Fg.1 The plot of DP (q) vs. tme (t) Fgure 1, shows the varaton of the deceleraton parameter q aganst tme t whch gves the behavor of q for dfferent values ofn. ISSN:

7 September 013 Volume 4 Issue 8 pp Fg. The plot of ansotropc parameter Am vs. tme (t) Fgure shows the varaton of parameter Am versus cosmc tme. It shows that Am decreases wth tme and tends to zero for suffcently large tmes. Thus the ansotropc behavor of the unverse des out at later tmes and the observed sotropy of the unverse can be derved by the model at the present epoch. Fg.3 The plot of proper energy densty vs. tme t Fgure 3 shows the varaton of proper densty versus cosmc tme. It shows that the unverse starts wth fnte values of proper energy densty. ISSN:

8 September 013 Volume 4 Issue 8 pp Fg.4 The energy condtons vs. tme t From Fgure 4, we can conclude that Therefore, the weak energy condton (WEC) as well as the domnant energy condton (DEC) are satsfed n our model. We can also observed that at ntal tme and at later tme whch n turn mply that the strong energy condton (SEC) volates n the present model on later tme. The volaton of SEC gves ant-gravtatonal effect. Due to ths effect, the unverse gets erk and the transton from the earler decelerated phase to the present acceleratng phase take place (Caldwell et al.006), hence the present model s turnng out as a sutable model for descrbng the late tme acceleraton of the unverse. It s observed that the above set of solutons satsfy the energy conservaton equaton (13) dentcally. Thus, the above solutons are exact solutons of Ensten s feld equatons (8)-(1). From equatons (17) and (6), we can conclude that the spatal volume s zero at and the expanson scalar s nfnte, whch shows that the unverse starts evolvng wth zero volume at whch s bg bang scenaro. From equatons (18)-(0), we see that the spatal scale factors are zero at the ntal epoch and hence the model has a pont type sngularty (MacCallum, 1971). All the physcal quanttes sotropc pressure, proper energy densty, Hubble s parameter and shear scalar dverge at. Thus we may conclude that the model represents an expandng unverse, whch starts wth a bg bang and approaches to sotropy at present epoch. ISSN:

9 September 013 Volume 4 Issue 8 pp Conclusons In ths paper, we have obtaned an exact soluton of Ensten s feld equatons for the ansotropc Banch type-v space-tme wth varable deceleraton parameter (DP). Interestngly, cosmc strngs do not occur n ths Banch type-v cosmologcal model. Also the DP yeld two phases of the unverse. Intally snce the sgn of the DP s postve that yelds the deceleratng phase of the unverse. At later tmes, the DP becomes negatve whch descrbes the present phase of acceleratng unverse. The physcal propertes are satsfed. Acknowledgments: The authors express ther profound grattude to Prof.K.D.Kror for hs constant encouragement and advce. Also the authors acknowledge the fnancal support of UGC, New Delh and the Department of Mathematcs, Gauhat Unversty for provdng all facltes for dong ths work. References Kbble, T. W. B.: J. Phys. A 9, 1387 (1976) Leteler, P. S.: Phys. Rev. D 8, 414 (1983) Stachel, J.: Phys. Rev. D 8, 171 (1990) Leteler, P. S.: Phys. Rev. D 0, 194 (1979) Kror, K.D., Choudhury, T. Mahanta, C. R.: Gen. Relatvty gravt., 13 (1990) Kror, K.D., Choudhury, T. Mahanta, C. R.: Gen. Relatvty gravt. 6, 65 (1994) Bal, R., Dave, S.: Pramana, J. Phys. 56, 4 (001) Bhattacharee, R., Baruah, K. K.: Indan J. Pure Appl. Math. 3, 47 (001) Rahaman, F., Chakravorty, S., das, S., Hossan, M., Bera, J.: Pramana, J. Phys. 60, 453 (003) Reddy, D. R. K.: Astrophys. Space Sc. 86, 359 ( Adhav, K. S., Nmkar, A. S.,Dawande, M. V., Ugale, M. A., Rom. Journ. Phys. Vol54, Nos.1-, 07-1 (009) Caldwell, R. R., Komp, W., Parker, L., &Vanezella, D. A. T.006, Phys. Rev. D, 73, (006) Pradhan, A., Jaswal, R., Jotana, K., Khare, R. K.: Astrophys Space Sc. 337, 401 (01) MacCallum, M. A. H.: Commun. Math. Phys. 0, 57 (1971) ISSN: