Kantowski-Sachs Cosmological Model in f(r,t) Theory of Gravity

Transcription

1 he Afrca Revew of Physcs (05 0:009 9 Katows-Sachs Cosologcal Model f(r, heory of Gravty V. U. M. Rao,* ad G. Suryaarayaa Deartet of Aled Matheatcs, Adhra Uversty, Vsahaata, Ida Deartet of Matheatcs, ANIS, Vsahaata, Ida A ew class of satally hoogeeous Katows-Sachs cosologcal odels flled wth erfect flud the fraewor of f gravty roosed by Haro et al. (0 have bee studed wth a arorate choce of a fucto f R, = f ( R f (. he odels obtaed ad reseted here are asotroc, exadg ad free fro sgulartes for ( > 0 ad t s establshed that the addtoal codto, secal law of varato of Hubble araeter roosed by Bera (98, tae by Saata (0 s suerfluous. Also soe ortat features of the odels, thus obtaed, have bee dscussed.. Itroducto Several theores of gravtato have bee roosed as dfferet alteratve to Este s theory of gravtato sce there have bee ay crtcss of geeral relatvty due to the lac of certa desrable features the theory. For exale, Mach s rcle s ot fully cororated geeral relatvty ad aearace of sgulartes s aother roble. Recet observatoal data suggest that our uverse s acceleratg. hs accelerato s exlaed ters of late te accelerato of the uverse ad the exstece of the dar atter ad dar eergy. Earler, Haro et al. [] develoed a geeralzed f(r, gravty where the gravtatoal Lagraga s gve by a arbtrary fucto of the Rcc scalar R ad of the trace of the stress eergy tesor. hey have obtaed feld equatos etrc forals. he equatos of oto for test artcles, whch follow fro covarat dvergece of the stress eergy tesor, are also reseted. hey have obtaed several odels ths theory corresodg to soe exlct fors of the fucto f (R,. I f(r, gravty, the feld equatos are obtaed fro the Hlbert-Este tye varatoal rcle. Usg gravtatoal uts (by tag G ad c as uty the corresodg feld equatos of f(r, gravty are obtaed by varyg the acto rcle wth resect to g as * urao57@hotal.co R f g ( g µ ( µ = 8π ( Θ δ Where, αβ Θ = g αβ. δg Here s the covarat dervatve ad s usual atter eergy-oetu tesor derved fro the Lagraga L. Reddy et al. [,] have obtaed Kaluza-Kle cosologcal odel the resece of erfect flud source ad Bach tye-iii cosologcal odel ths theory assug the law of varato of Hubble araeter that was roosed by Bera [4]. Houdo [5] has develoed the cosologcal recostructo of f(r, gravty as f(r, = f ( R f ( ad dscussed the trasto of atter doated hase to a accelerated hase. Reddy ad Sath Kuar [6] have dscussed soe asotroc cosologcal odels a odfed theory of gravty by tag dfferet relatos betwee ressure ad eergy desty. Sharf ad Zubar [7] have dscussed eergy codtos costrats ad stablty of ower law solutos f gravty. Saata ad Dhal [8] have dscussed hgher desoal cosologcal odels flled wth erfect flud f(r, theory. Shara ad Sgh [9] have obtaed Bach tye-ii strg

2 he Afrca Revew of Physcs (05 0: cosologcal odel wth agetc feld, Reddy et al. [0] have obtaed Katows-Sachs bul vscous strg cosologcal odels, Sharf ad Zubar [] have dscussed Bach-I asotroc odel, Rao ad Neela [,] have dscussed erfect flud Este-Rose, Bach tye-vi 0 ad o-statc lae syetrc Uverses, ad Rao et al. [4] have obtaed Bach tye III, V ad VI 0 bul vscous strg cosologcal odels f(r, gravty. Recetly, Rao ad Suryaarayaa [5] have dscussed hgher desoal erfect flud cosologcal odels ths theory, resectvely. Saata [6] has dscussed Katows-Sachs uverse flled wth erfect flud ths theory by usg a secal law of varato for the Hubble araeter that was roosed by Bera [4]. war ad Dwved [7], Adhav et al. [8], Katore ad Rae [9], Chaubey [0] are soe of the authors who have studed Katows-Sachs cosologcal odels varous theores. I ths aer, we wll vestgate satally hoogeeous ad asotroc Katows-Sachs cosologcal odel flled wth erfect flud the fraewor of f(r, gravty roosed by Haro et al. [] wth a arorate geeral choce of a fucto f(r, = f R f (. (. Metrc ad Eergy Moetu esor We cosder a satally hoogeeous Katows- Sachs etrc of the for ds dt A dr B ( dθ S θ dφ = ( Where, A ad B are the fuctos of te t oly. Besde Bach tye etrcs, the Katows- Sachs [] odels are also descrbg satally hoogeeous Uverses. For a revew of Katows-Sachs etrcs oe ca refer to MacCallu []. hese etrcs rereset hoogeeous but asotrocally exadg (or cotractg cosologes ad rovde odels where the effects of asotroy ca be estated ad coared wth all well ow Freda- Robersto-Waler class of cosologes. he feld equatos f (R, gravty for the fucto f (R,= f R f ( ( hs s so whe the atter source s erfect flud as gve by Haro et al. [] are ( R R f R g = ( 8π ( ( he atter tesor for erfect flud s where Θ δ = ( ρ u u f ( g ( δ (4 We cosder a artcular for of the fucto f ( R = R ad f ( = where ad are ay araeters, so that f = R. he the feld equatos ( wll reduce to ( 8π g = R Rg (5 he feld equatos (5 xed for ca be tae as G R δ R 8π. Solutos of Feld Equatos δ Now wth the hel of Eqs. (-(5, the feld equatos (6 for the etrc Eq. ( ca be wrtte as 6π B && B& = ρ B B B A && B&& AB & & 6π = ρ A B AB A && B&& AB & & 6π = ρ A B AB B& B AB & & AB B (7 (8 (9 (6 6π = ρ (0

3 he Afrca Revew of Physcs (05 0:009 4 Here the over head dot deotes dfferetato wth resect to t. Eqs. (9 to (0 s a syste of three deedet equatos four uows A, B, ad ρ. I order to get a deterstc soluto, we tae the followg lausble hyscal codto: the shear scalar σ s roortoal to scalar exasoθ leadg to a lear relatosh betwee the etrc otetals A ad B,.e., B = A ( Where, 0 s a costat. Fro Eqs. (7, (8 ad (, we get A& A& = ( A ( A Fro Eq. (, we get [ ( t ] A = ( Where, =, ± costat of tegrato. Fro Eqs. ( ad (, we get [ ( t ] ad s a B = (4 he etrc ca ow be wrtte as ds = dt [ ( t ] [ ( t ] ( dθ S θ dφ dr (5 Fro Eqs. (, (4 ad Eqs. (7 to (0, we get the eergy desty as ( ( ( ( 8π 8π ρ = ( ( t (6 Ad, the total ressure as hus the etrc Eq. (5 together wth Eqs. (6 ad (7 costtutes Katows-Sachs erfect flud cosologcal odel f (R, gravty, where f = R. For artcular values of = ad =, f = R, the the eergy desty ad the total ressure wll becoe ρ = 4 = 4 ( ( t ( ( t ( ( ( π ( (8 ( ( ( π (9 For -<<0 ad for < < wth >, > 0 ad ρ > 0. Hece, Eqs. (8 ad (9 satsfy the eergy codtos. Also, whe <0 for < < ad < < wth < show that the erfect flud behaves le a hato-tye dar eergy. So, we ca coclude that the erfect flud ay be a source of early dar eergy due to the egatve ressure, sce eergy codtos are volated. For the above artcular values of ad, the etrc Eq. (5 together wth Eqs. (8 ad (9 reresets Katows-Sachs erfect flud cosologcal odel f(r, gravty wth f = R, whch s ore geeral. he addtoal codto, secal law of varato of Hubble araeter roosed by Bera [4], tae by Saata [6] s suerfluous. 4. Soe Other Iortat Features of the Model he volue eleet of the odel Eq. (5 s gve by ( V = g = [ ( t ] Sθ (0 = ( 8π ( ( 8π ( ( t (7 We ca observe that the satal volue s creasg wth te. he exresso for the exaso scalar θ s gve by

4 he Afrca Revew of Physcs (05 0:009 4 t θ = u, = ( Ad, the shear σ s gve by he average scale factor gve as [ ( t ] θ a( t V = S = (7 7 ( σ = σ σ = ( 8 ( t he decelerato araeter q s gve by ( q = θ ( θ, u θ = ( he Hubble s araeter H s gve by he red shft as Z = = a Ad the er araeter as &&& a J = = H a [ ( t ] S θ ( 4 ( ( (8 (9 θ H = = t he overall desty araeter Ω s gve by ρ H ( 8 π 8 π Ω = = he ea asotroy araeter A s gve by (4 (5 Fro Eq. (9 t ca be observed that for = the er araeter value overla wth the value.6, whch s obtaed fro the three eatcal data sets: the gold sale of tye Ia suerovae, the SNIa data fro the SNLS roect ad the X-ray galaxy cluster dstace easureets. Luosty araeter d L = r ( z where r = t 0 t dt a ( t H H A = = = H (6 Where, H = H H ( =,,. ( ( ( ( d L = 0 ( S θ ( t ( t ( t (0 w = u, u, s detcally zero ad hece ths Uverse s o-rotatoal. 5. Coclusos I ths aer, we have reseted satally hoogeeous ad asotroc Katows-Sachs cosologcal odel flled wth erfect flud the fraewor of f gravty roosed by Haro et al. [5] ad whch have bee obtaed wth a arorate choce of a fucto f R, = f ( R f (. We observe that at t ( =, the satal volue vashes ad creases cotuously wth te for >. hs shows that at the tal eoch the uverse starts wth zero volue ad exads cotuously wth te. Also the odel has o sgularty for > 0. he exaso scalarθ, the shear scalar σ ad the Hubble araeter H decreases wth the crease te ad dverges at t =. Fro Eqs. (6 ad (7, we ca see that eergy desty ad atter ressure wll vash wth the crease of te. Fro Eq. (, we ca observe that the decelerato araeter s egatve for < < ad hece t reresets a acceleratg uverse. Sce

5 he Afrca Revew of Physcs (05 0:009 4 A 0, ths dcates that ths odel s always asotroc. he odel reseted here s asotroc, o-rotatg, exadg ad also acceleratg. Iterestgly, for artcular values of ad, the etrc Eq. (5 together wth Eqs. (8 ad (9 reresets Katows-Sachs erfect flud cosologcal odel f(r, gravty wth f = R, whch s ore geeral tha the odel vestgated by Saata [0]. Refereces []. Haro, F. S. N. Lobo, S. Nor ad S. D. Odtsov, Phys. Rev. D 84, 0400 (0. [] D. R. K. Reddy, R. L. Nadu ad B. Satyaarayaa, It. J. heor. Phys. 5, (0. [] D. R. K. Reddy, R. Sat Kuar ad R. L. Nadu, Astrohys. Sace Sc. 4, 49 (0. [4] M. S. Bera, Nuovo Ceto B 74, 8 (98. [5] M. J. S. Houdo, It. J. Mod. Phys. D, 5000 (0. [6] D. R. K. Reddy ad R. Sat Kuar, Astrohys. Sace Sc. 44, 5 (0. [7] M. Sharf ad M. Zubar, J. Phys. Soc. J. 8, 0400 (0. [8] G. C. Saata ad S. N. Dhal, It. J. heor. Phys. 5, 4 (0. [9] N. K. Shara ad J. K. Sgh, It. J. heor. Phys. 5, 9 (04. [0] D. R. K. Reddy, S. Atha ad S. Uadev, Eur. Phys. J. Plus 9, 96 (04. [] M. Sharf ad M. Zubar, Astrohys. Sace Sc. 49, 457 (04. [] V. U. M. Rao ad D. Neela, Eur. Phys. J. Plus 8, 5 (0. [] V. U. M. Rao ad D. Neela, Astrohys. Sace Sc. 45, 47 (0. [4] V. U. M. Rao, M. Vaya Sath ad B. J. M. Rao, Presacete Joural 5, 758 (04. [5] V. U. M. Rao ad G. Suryaarayaa, Presacete Joural 5, 89 (04. [6] G. C. Saata, It. J. heor. Phys. 5, 647 (0. [7] Rsh Kuar war ad Utta Kuar Dwved, Astrohys. Sace Sc. 8, 49 (008. [8] K. S. Adhav, V. G. Mete, A. S. Nar ad A. M. Pud, It. J. heor. Phys. 47, 4 (008. [9] S. D. Katore ad R. S. Rae, Astrohys. Sace Sc., 9 (009. [0] R. Chaubey, It. J. heor. Phys. 5, 9(0. [] R. K. Katows-Sachs, J. Math. Phys. 7, 44 (966. [] MacCallu, Nature 0, 589 (97. Receved: 7 Arl, 05 Acceted: 8 August, 05