Einstein Rosen inflationary Universe in general relativity

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1 PRAMANA c Indian Acadmy of Scincs Vol. 74, No. 4 journal of April 2010 physics pp Einstin Rosn inflationary Univrs in gnral rlativity S D KATORE 1, R S RANE 2, K S WANKHADE 2, and N K SARKATE 3 1 Dpartmnt of Mathmatics, SGB Amravati Univrsity, Amravati , India 2 Dpartmnt of Mathmatics, Y.C. Scinc and Arts Collg, Mangrulpir , India 3 Dpartmnt of Mathmatics, Adarsh Scinc Collg, Hingoli, India Corrsponding author. wankhad.kishor@rdiffmail.com MS rcivd 2 May 2009; rvisd 20 Octobr 2009; accptd 24 Novmbr 2009 Abstract. Einstin Rosn inflationary Univrs is invstigatd in th prsnc of masslss scalar fild with a flat potntial. To gt an inflationary Univrs, w hav considrd a flat rgion in which th potntial V is constant. Som physical proprtis of th modl ar discussd. Kywords. Einstin Rosn mtric; inflationary Univrs; gnral rlativity. PACS Nos Cq; 04.00; k 1. Introduction In rcnt yars, thr has bn a lot of intrst in cosmological modls of th Univrs which ar important in undrstanding th mystris of th arly stags of its volution. In particular, inflationary modls play an important rol in solving a numbr of outstanding problms in cosmology lik th homognity, th isotropy and th flatnss of th obsrvd Univrs. Th standard xplanation for th flatnss of th Univrs is that it has undrgon at an arly stag of th volution a priod of xponntial xpansion namd as inflation. It is wll-known that slf-intracting scalar filds play a vital rol in th study of inflationary cosmology. Guth [1], Lind [2] and La and Stinhardt [3] ar som of th authors who hav invstigatd svral aspcts of inflationary Univrs in gnral rlativity. Burd and Barrow [4], Wald [5], Barrow [6], Ellis and Mdsn [7] and Huslr [8] studid diffrnt aspcts of scalar filds in th volution of th Univrs. Th rol of slf-intracting scalar filds in inflationary cosmology has bn invstigatd by Bhattacharj and Baruah [9], Bali and Jain [10] and Rahaman t al [11]. Rddy t al [12], Rddy and Naidu [13] hav discussd inflationary Univrs in gnral rlativity in four and fiv dimnsions. Rcntly, Rddy t al [14] hav 669

2 S D Kator t al invstigatd a plan symmtric Bianchi typ-i inflationary Univrs in gnral rlativity. Vry rcntly, Rddy [15] has discussd Bianchi typ-v inflationary Univrs in gnral rlativity. Also Rddy t al [16], Kator and Ran [17] hav studid th Kantowski Sachs inflationary Univrs in gnral rlativity. In this papr, w hav invstigatd Einstin Rosn inflationary Univrs in gnral rlativity in th prsnc of masslss scalar fild with a flat potntial. To gt dtrminat solution, w hav considrd a flat rgion in which th potntial is constant. 2. Mtric and fild quations W hav considrd th cylindrically symmtric Einstin Rosn mtric in th form ds 2 = 2α 2β (dt 2 dρ 2 ) ρ 2 2β dψ 2 2β dz 2, (1) whr α and β ar functions of cosmic tim t only and x 1 = ρ, x 2 = ψ, x 3 = z, x 4 = t. Th non-vanishing componnts of th Einstin tnsor for th mtric (1) ar and G 1 1 = 2α+2β β 2 4, G 2 2 = 2α+2β (α 44 + β 2 4), G 3 3 = 2α+2β (2β 44 α 44 β 2 4), G 4 4 = 2α+2β β 2 4. (2) Hr th subscript 4 dnots th diffrntiation with rspct to t. In th cas of gravity minimally coupld to a scalar fild V (φ) th Lagrangian is [ L = R 1 gd 2 gij φ,i φ,j V (φ)] 4 x, (3) which by varying L with rspct to dynamical filds lads to Einstin fild quations G ij = R ij 1 2 g ijr = T ij (4) with [ ] 1 T ij = φ,i φ,j 2 φ,kφ,k + V (φ) g ij, (5) φ i dv (φ) ;i = dφ, (6) whr comma (,) and smicolon (;) indicat ordinary and covariant diffrntiation rspctivly. Othr symbols hav thir usual manings and units ar takn so that 8πG = c = Pramana J. Phys., Vol. 74, No. 4, April 2010

3 Einstin Rosn inflationary Univrs in gnral rlativity Now th Einstin fild quations (4) for th mtric (1) with th hlp of q. (5) ar givn by ( β 2α+2β ) 2 φ2 4 = V (φ), (7) ( α 2α+2β 44 β4 2 1 ) 2 φ2 4 = V (φ), (8) (2β 2α+2β 44 α 44 β4 2 1 ) 2 φ2 4 = V (φ), (9) (β 2α+2β ) 2 φ2 4 = V (φ) (10) and q. (6) for th scalar fild taks th form 2α+2β φ 44 = dv dφ. (11) Hr th subscript 4 dnots diffrntiation with rspct to t. 3. Solutions of th fild quations and th modl Hr, w ar intrstd in inflationary solutions of th fild quations (7) (11). Stin-Schabs [18] has shown that Higg s fild φ with potntial V (φ) has a flat rgion and th fild volvs slowly but th Univrs xpands in an xponntial way du to vacuum fild nrgy. It is assumd that th scalar fild will tak sufficint tim to cross th flat rgion so that th Univrs xpands sufficintly to bcom homognous and isotropic on th scal of th ordr of th horizon siz. Thus w ar intrstd, hr, in inflationary solutions of th fild quations, th flat rgion is considrd whr th potntial is constant, i.. V (φ) = constant = V 0 (say). (12) Using q. (12), th fild quations (7) (11) admit th xact solutions and 2α = 2(a 1t+a 2 ), (13) 2β = 2(a3t+a4) (14) φ = a 5 t + a 6, (15) whr a 1, a 2, a 3, a 4, a 5 and a 6 ar constants of intgration. Aftr suitabl choic of coordinats and constants of intgration, th Einstin Rosn inflationary cosmological modl can b writtn as ds 2 = (dt 2 dρ 2 ) ρ 2 2T dψ 2 2T dz 2. (16) It is intrsting to not that th modl (16) is isotropic and has no initial singularity. Pramana J. Phys., Vol. 74, No. 4, April

4 S D Kator t al 4. Physical proprtis of th modl Th modl (16) rprsnts an xact Einstin Rosn inflationary cosmological modl in gnral rlativity, whn th scalar fild φ is minimally coupld to th gravitational fild. Th modl has no initial singularity at T = 0. Th physical paramtrs for th modl (16) hav th following xprssions: Expansion scalar: θ = 2 3 (a 3 a 5 ). Shar scalar: σ 2 = 2 27 (a 3 a 5 ) 2. Dclration paramtr: q = 1 < 0. W obsrvd that th xpansion scalar (θ) and shar scalar (σ) ar constants. Th rol of th dclration paramtr (q) sms to spcify th xpansion of th Univrs. Th positiv valu of th dclration paramtr q indicats that th modl dclrats in th standard way. But in th prsnt obsrvation th modl inflats bcaus th dclration paramtr q is ngativ. 5. Conclusions In this papr, w hav obtaind cylindrically symmtric Einstin Rosn inflationary Univrs in th prsnc of masslss scalar fild with flat potntial in gnral rlativity. It is obsrvd that th modl is non-singular and xpanding. Th inflationary modl obtaind hr has considrabl astrophysical significanc. For xampl, classical scalar filds ar ssntial in th study of th prsnt day cosmological modls. In viw of th fact that thr is an incrasing intrst, in rcnt yars, in scalar filds in gnral rlativity and altrnativ thoris of gravitation in th contxt of inflationary Univrs and thy hlp us to dscrib th arly stags of volution of th Univrs. W hop that our modl obtaind hr will b usful for a bttr undrstanding of inflationary cosmology in Einstin Rosn spac-tim. Acknowldgmnts Th authors ar thankful to UGC, Nw Dlhi, for sanctioning projct and financial support. Also th authors xprss thir sincr gratitud to th anonymous rfr for th constructiv commnts to improv this papr. Rfrncs [1] A H Guth, Phys. Rv. D23, 347 (1981) [2] A D Lind, Phys. Ltt. B108, 389 (1982) [3] D La and P J Stinhardt, Phys. Rv. Ltt. 62, 376 (1981) [4] A B Burd and J D Barrow, Nucl. Phys. B308, 923 (1988) [5] R Wald, Phys. Rv. D28, 2818 (1983) [6] J D Barrow, Phys. Ltt. B187, 12 (1987) 672 Pramana J. Phys., Vol. 74, No. 4, April 2010

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