BULK VISCOUS BIANCHI TYPE IX STRING DUST COSMOLOGICAL MODEL WITH TIME DEPENDENT TERM SWATI PARIKH Department of Mathematics and Statistics,

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1 UL VISCOUS INCHI YPE IX SRING DUS COSMOLOGICL MODEL WIH IME DEPENDEN ERM SWI PRIH Department of Mathematcs and Statstcs, Unversty College of Scence, MLSU, Udapur, 3300, Inda UL YGI Department of Mathematcs and Statstcs, Unversty College of Scence, MLSU, Udapur, 3300, Inda RH RNI RIPHI Department of Mathematcs and Statstcs, Unversty College of Scence, MLSU, Udapur, 3300, Inda SRC We have nvestgated homogeneous anch type IX strng dust cosmologcal model for vscous flud dstrbuton. We have assumed the condton =.e. strng tenson densty s equal to rest energy densty. We have also used a condton that the coeffcent of bulk vscosty s nversely proportonal to the expanson n the model. he physcal and geometrcal aspects of the model are also dscussed. EYWORDS: anch ype IX space-tme, vscous flud, cosmc strng and varable. INRODUCION anch type IX cosmologcal models are nterestng because these models allow not only expanson but also rotaton and shear and n general these are ansotropc. Many relatvsts have taken keen nterest n studyng anch type IX unverse because famlar solutons lke Robertson-Walker unverse, the De-stter unverse, the aub-nut solutons etc. are of anch type IX space tme. ulk vscosty s supposed to play a very mportant role n the early evoluton of the unverse. It s well known that n an early stage of unverse when the radaton n the form of photons as well as neutrnos decoupled from matter, t behaved lke a vscous flud. Msner [7, 8] have studed the effect of vscosty on the evoluton of cosmologcal models. Nghtangle [9] has nvestgated the role of vscosty n cosmology. hus, we should consder the presence of a materal dstrbuton other than a perfect flud to have realstc cosmologcal models. On the other hand, one outstandng problem n cosmology s the cosmologcal constant problem [0]. In modern cosmologcal theores, the cosmologcal constant remans a focal pont of nterest. wde range of observatons now suggests that the unverse possess a nonzero cosmologcal constant [0]. al et al. [, 5, 6] have obtaned anch type I, V strng cosmologcal models wth magnetc feld and bulk vscosty n general relatvty. Yadav et al. [] have studed some anch ype I vscous flud strng cosmologcal models wth magnetc feld. Wang [8, 9] has also dscussed anch type I and III strng cosmologcal models wth ulk vscosty and magnetc feld. LRS anch ype II magnetzed strng cosmologcal model wth bulk vscous flud n general relatvty was also studed by yag and Sharma [3]. yag and Sharma [6] have also studed anch ype - II bulk vscous strng cosmologcal model n General Relatvty. al and Dave [] nvestgated the anch ype III strng cosmologcal model wth bulk vscosty. 3 P a g e

2 al and Dave [3] have also nvestgated anch ype IX strng cosmologcal model n General Relatvty. al and umawat [] have nvestgated anch ype IX stff flud tlted cosmologcal model wth bulk vscosty. lso yag and Sharma [] nvestgated anch IX strng cosmologcal model for perfect flud dstrbuton n General Relatvty. Homogeneous nsotropc anch ype IX cosmologcal model for perfect flud dstrbuton wth electromagnetc feld s studed by yag and Chhaed [7]. yag and Sngh [5] nvestgated magnetzed bulk vscous anch type IX cosmologcal models wth varable. Some LRS anch type II strng cosmologcal models for vscous flud dstrbuton are nvestgated by Sharma and yag []. Son and Shrmal [] have also studed strng cosmologcal models n anch type III space tme wth bulk vscosty and term. In ths paper, we nvestgate bulk vscous anch ype IX strng dust cosmologcal model wth tme dependent term. o get determnate soluton we assume = and coffcent of bulk vscosty s nversely proportonal to the expanson of the model.e. =. he physcal and geometrcal aspects of the model are also dscussed. HE MERIC ND FIELD EQUIONS We consder homogeneous ansotropc anch type IX metrc n the form of ds dt dx dy ( sn y cos y)dz cosydxdz... () where and are functons of t alone. he Ensten feld equaton s gven by R Rg g... () (n geometrcal unt 8G = and C = ) he energy momentum tensor s gven by for a cloud of massve strng for vscous flud dstrbuton v v x x (v v g )... (3) Where s the proper energy densty for a cloud strng attached to them, s the strng tenson densty, v s the four velocty of the partcle and x s a unt space lke vector representng the drecton of strng. If the partcle densty s denoted by p then p... () In a co-movng coordnate system we have v (0,0, 0,), x (, 0, 0, 0)... (5) and v v x x, v x 0... (6) P a g e

3 he Ensten feld equaton () for metrc () lead to followng system of equatons: 3... (7)... (8) SOLUION OF FIELD EQUIONS... (9) We assume that strng tenson densty s equal to rest densty..e. =... (0) Usng (0) n equatons (7) and (9) we get... () o get determnate soluton, we assume that the coeffcent of bulk vscosty s nversely proportonal to expanson (). hs condton leads to k... () Usng () and = n n equaton () we obtan n n... (3) Hence equaton (3) leads to n... () n Let f ()... (5) So, equaton () becomes df d f n... (6) 3n From equaton (6) we have f. n n ( n) L... (7) 5 P a g e

4 where L s ntegratng constant, and f ( n) n L n... (8) Equaton (8) leads to ( n) d n L n t M... (9) where M s constant of ntegraton. he value of can be determned by equaton (9). he metrc () reduce to ds ( n) d n L n n dx dy ( sn n n Y cos Y)dZ cosy dx dz... (0) Where =, x = X, y = Y and z = Z. SOME PHYSICL ND GEOMERICL FEURES he rest energy densty () and the strng tenson densty () for model (0) are gven by ( n) n n (n ) L(n ) n... () (n ) ( n) n n n ) L(n )... () he scalar of expanson () and the shear () for the model (0) are gven by L n )... (3) ( n) n n (n ) L... () ( n) n n 3 he cosmologcal term () for the model (0) s gven by ( 3n n ( n) ) L ( n ) n(n )... (5) n n 6 P a g e

5 SOLUION IN SENCE OF VISCOUS FLUID In the absence of vscous flud the geometry of space-tme s gven by ds d L n n n dx dy ( sn Y n cos Y) dz n cosy dx dz... (6) he rest energy densty (), the strng tenson densty (), scalar expanson () and the shear () for the model (6) are gven by n (n ) L(n ) n... (7) (n )... (8) n n (n )... (9) n n 3 he cosmologcal term () for the model (6) s gven by Ln (n ) (n )... (30) n n CONCLUSION We have obtaned a new class of ansotropc cosmologcal model wth bulk vscous flud as a source. he realty condton ρ 0 [usng equaton (..)] leads to ( n) n n (n ) L(n ) 0 n he densty ρ do not vansh when n ( n) due to presence of vscous flud. he strng tenson densty (λ) also does not vansh when due to presence of vscous flud. he model starts expandng wth bg-bang at = 0 and expanson n the model decreases as tme ncreases. lso expanson of the model stops when n = - from equaton (3). Snce 7 P a g e

6 , constant so the model does not approach sotropy for large values of. However, the model sotropzed for n =. he cosmologcal constant from equaton (5) of the model s found to be decreasng functon of tme and approaches to zero at late tme whch s supported by present observatons. In general the model represents expandng, shearng and non-rotatng unverse. REFERENCES ) al, R. and nal,. (006). anch type I magnetzed strng cosmologcal model n general relatvty. strophyscs and Space Scence, 30, ) al, R. and Dave, S. (00). anch type-iii strng cosmologcal model wth bulk vscous flud n general relatvty. strophyscs and Space Scence, 8, ) al, R. and Dave, S. (00). anch type-ix strng cosmologcal model n general relatvty. Pramana Journal of Physcs, 56, ) al,r. and umawat, P. (00). Some anch type IX stff flud tlted cosmologcal models wth bulk vscosty n general relatvty. Electronc Journal of heortcal Physcs, 7, 3, ) al, R. and Sngh, D.. (005). anch type-v bulk vscous flud strng dust cosmologcal model n general relatvty. strophyscs and Space Scence, 300, ) al, R. and Upadhyay, R.D. (003). LRS anch ype I strng dust magnetzed cosmologcal models. strophyscs and Space Scence, 83, ) Msner, C.W. (967). ransport Processes n the Prmordal Freball. Nature, 0-. 8) Msner, C.W. (968). he Isotropy of the Unverse. strophyscal Journal, 5, ) Nghtangle, J.P. (973). strophyscal Journal, 85, 05. 0) Ress,.G. et al. (00). ype Ia Supernova Dscoveres at z > From the Hubble Space elescope Evdence for Past Deceleraton and Constrants on Dark Energy Evoluton. strophyscal Journal, ) Sharma,. and yag,.(0). LRS anch type II magnetzed strng cosmologcal model wth bulk vscous flud n general relatvty. Internatonal Journal of Mathematcal rchve, 3(8), ) Son, P. and Shrmal, S. (05). Strng cosmologcal models n anch type-iii spacetme wth bulk vscosty and Ʌ term. Internatonal Journal of Engneerng and ppled Scences, 6,, ) yag,. and Sharma,. (00). anch type-ii bulk vscous strng cosmologcal models n general relatvty. Internatonal Journal of heoretcal Physcs, 9, P a g e

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