Hypersurface-Homogeneous Universe with Λ in f(r,t) Gravity by Hybrid Expansion Law

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1 4 Theoretcal Physcs, Vol., No., March 7 Hypersurface-Hoogeneous Unverse wth Λ n f(r,t) Gravty by Hybrd Expanson Law A.Y.Shakh, K.S.Wankhade * Departent of Matheatcs, Indra Gandh Mahavdyalaya, Ralegaon-4454.Inda. *Departent of Matheatcs, Y.C.Scence College, Mangrulpr. Inda. Eal: shakh_4ay@yahoo.co Abstract. In the present study we nvestgate, Hypersurface-Hoogeneous cosologcal odel n f(r,t) theory of gravty wth a ter Λ. We obtan the gravtatonal feld equatons n the etrc forals, whch follow fro the covarant dvergence of the stress-energy tensor. The feld equatons correspond for a specfc choce of f(r,t)=f (R)+f (T), wth the ndvdual superor functons f (R)=λR and f (T)=λT. In ths paper, we consder a sple for of expanson hstory of Unverse referred to as the hybrd expanson law a product of power-law and exponental type of functons. Ensten s feld equatons have been solved by takng nto account the hybrd expanson law for scale factor that yelds te dependent deceleraton paraeter (DP). Soe physcal and geoetrc propertes of the odel along wth physcal acceptablty of the solutons have also been dscussed n detal. Keywords: Hypersurface-Hoogeneous unverse, varable cosologcal constant, odfed theory of gravty. PACS nubers: 4.5.kd. Introducton The analyss of observatonal data shows that our Unverse for later stages of evoluton ndcates accelerated expanson. Ths concluson s based on the observatons of hgh redshft type SN Ia supernovae [ 4]. Recent astronocal observatons ndcate that about 7 % of the Unverse conssts of dark energy wth negatve pressure.in recent years, odfed gravtes have recently been verfed to explan the late-te accelerated expanson of the Unverse. Dfferent odfed theores of gravtaton are fr ( ) gravty [5-8] and Gauss Bonnet gravty or fg ( ) gravty [9-]. Another approach to odfed gravty s so-called ft ( ) gravty [-5], where T s the scalar torson. Recently, Harko et al. [6] proposed f( RT, ) gravty theory by takng nto account the gravtatonal Lagrangan as the functon of Rcc scalar R and of the trace of energy-stress tensor T. They have obtaned the equaton of oton of test partcle and the gravtatonal feld equaton n etrc forals both. The f( RT, ) gravty odels could justfy the late te cosc accelerated enlargeent of the Unverse. Many Authors [7-46] studed copletely dfferent cosologcal odels n f( RT, ) theory of gravty. Motvated by the above observatonal facts, n ths paper, we propose to study cosologcal odel represented by Hypersurface -Hoogenous reference syste for perfect flud dstrbuton wthn the fraework of (, ) f R λr wth ( ) f RT gravty. We choose a specfc choce of the functonal f( RT, ) = f( R) + f ( T) = and f ( T) = λt. Gravtatonal Feld Equatons of Modfed Gravty Theory The f( RT, ) theory of gravty s the odfcaton of General Relatvty (GR). In ths theory the odfed gravty acton s gven by

2 Theoretcal Physcs, Vol., No., March s= frt (, ) gdx+ L gdx 6π () where f( RT, ) s an arbtrary functon of the Rcc scalar R and the trace T of the stress energy tensor T of the atter, L s the atter Lagrangan densty. If f( RT, ) s replaced by fr, ( ) we get f RT by R leads to the acton of general relatvty. the acton for fr ( ) gravty and replaceent of (, ) Varyng the acton S about etrc tensor where g, the feld equatons of f( RT, ) gravty are obtaned as f ( RT, ) R f( RT, ) g + f ( RT, ) ( g ) = 8 πt f ( RT, ) T - f ( RT, R R j T T ) θ () (, ) θ = T + gl g ( ) δ f RT δ f RT, Here f =, f =, s the co-varant dervatve and T s the standard R T δr δt atter energy oentu tensor derved fro the Lagrangan L. If the atter s regarded as a perfect flud the stress energy tensor of the atter Lagrangan s gven T = ρ + p u u pg. Here u = (,,,) s the velocty vector n co-ovng coordnates that by ( ) j satsfes the condton uu = and u u =. Here ρ, p are energy densty and pressure of the j flud respectvely. For perfect flud, the atter Lagrangan densty ay have two choces ether L = p or L = ρ that has wdely been studed n lterature [47-49]. Here we have assued the atter Lagrangan as L = p. Now θ n equaton () can be reduced to αβ L g αβ g θ = T gp (4) It s entoned here that ths feld equaton depends on the physcal nature of the atter feld. There are three classes of these odels R + f ( T) f( RT, ) = f( R) + f ( T) (5) f ( R) + f ( R) f ( T) We consder t n the for f( RT, ) = f( R) + f( T) wth f ( R) = λr and f ( T) = λt, where λ s an arbtrary constant. The feld equatons (), for the specfc choce of f( RT, ) = λ( R+ T), reduces to () 8π + λ G = T p T g + + λ (6) We choose a sall negatve value for the arbtrary λ to draw a better analogy wth the usual Ensten feld equatons and we ntend to keep ths choce of λ throughout. We have the Ensten feld equaton wth cosologcal constant as G Λ g = 8π T (7) Coparng equatons (6) and (7), we yeld Λ=Λ ( T) = p + T (8) and 8π + λ 8π = (9) λ The dependence of the cosologcal constant Λ on the trace of the energy oentu tensor T has been proposed before by Poplawsk [5] where the cosologcal constant n the gravtatonal

3 6 Theoretcal Physcs, Vol., No., March 7 Lagrangan s consdered as a functon of the trace of the energy-oentu tensor. Snce we have consdered the perfect flud as the source, accordng to Poplawsk [5], the trace of energy-oentu tensor s functon of sotropc pressure and energy densty.e. T = p + ρ () Metrc and Feld Equatons We consder the Hypersurface-hoogeneous space te of the for, ds = dt A () t dx B () t dy + ( y, K ) dz () where At ( ) and B( t ) are the cosc scale functons, ( yk, ) = sn yy,,snhy respectvely when K =,,. Stewart and Ells[5] obtaned general solutons of Ensten s feld equatons for a perfect flud dstrbuton satsfyng a barotropc equaton of state for the Hypersurface-hoogeneous space te. Hajj- Boutros [5] developed a ethod to fnd exact solutons of feld equatons n case of the etrc () n presence of perfect flud and obtaned exact solutons of the feld equatons whch add to the rare solutons not satsfyng the barotropc equaton of state. Hypersurface-hoogeneous bulk vscous flud cosologcal odels wth te-dependent cosologcal ter have been dscussed by Chandel et al. [54]. Katore and Shakh [55] obtaned the exact solutons of the feld equatons for Hypersurface-hoogeneous space te under the assupton on the ansotropy of the flud (dark energy), whch are obtaned for exponental and power-law voluetrc expansons n a scalar-tensor theory of gravtaton. Katore and Shakh [56] presented a class of solutons of Ensten s feld equatons descrbng two-flud odels of the unverse n Hypersurface-Hoogenous space te. In a co-ovng coordnate syste, the feld equatons (8), for the etrc (), can be explctly wrtten as 8 B B K π + λ + + = p Λ () B B B λ A B AB 8π + λ + + = p Λ () A B AB λ 8 AB B K π + λ + + = ρ Λ (4) AB B B λ where an overhead dot hereafter, denotes ordnary dfferentaton wth respect to cosc te t only. The trace n our odel s gven by equaton ().e. T = p + ρ, so that the effectve cosologcal constant n equaton (8) reduces to Λ= ( ρ p) (5) The spatal volue s gven by V = a = AB (6) where a s the ean scale factor. 4 Soluton of Feld Equatons The ansotropy of the expanson can be paraeterzed after defnng the drectonal Hubble paraeters and the ean Hubble paraeter of the expanson. The drectonal Hubble paraeters, whch deterne the unverse expanson rates n the drectons of the xyz,, axes, are defned as A B H =, H = H = (7) x y z A B and for the average scale factor a = ( AB ), the Hubble paraeter H, that deternes the volue expanson rate of the unverse, can be generalzed to ansotropc cosologcal odels:

4 Theoretcal Physcs, Vol., No., March 7 7 a V H A B = = = + (8) a V A B The physcal quanttes of observatonal nterest are the expanson scalar θ, the average ansotropy paraeter A and the shear scalar σ. These are defned as A θ = u = + A A : B B (9) H H = () = H σ = AH () Now we have a set of three equatons wth fve unknown functons AB,, p, ρ, Λ. To get a deternate soluton of feld equatons, we need extra condtons. One can ntroduce ore condtons ether by an assupton correspondng to soe physcal stuaton of an arbtrary atheatcal k t n supposton. Pradhan et. al. [57] dscussed the law of varaton of scale factor a= ( te) whch yelds a te-dependent deceleraton paraeter (DP). We consder the generalzed hybrd expanson law for scale factor as followng n kt a= ( te ) () where,, n and k are non-negatve constants. The proposed law gves the te dependent deceleraton paraeter (DP) whch descrbes the transtonng unverse. The scale factor gven by Eq. () yelds a te-dependent deceleraton paraeter whch exhbts a transton of the unverse fro the early deceleratng phase to the present acceleratng phase. We also assue that the Expanson Scalar θ s proportonal to the Shear Scalar σ whch gves us A = B α () where α s a constant. The work of Thorne [58] otvates us to consder such assupton gven by Eq. (). Accordng to h, observatons of velocty red shft relaton for extragalactc sources suggest that Hubble expanson of the unverse s sotropc wthn about % range approxately (Kantowsk and Sachs [59]; Krstan and σ Sachs [6]) and red shft studes place the lt. where σ and H are shear scalar and Hubble H paraeter respectvely. The physcal sgnfcance of ths condton for perfect flud and barotropc EoS n a ore general case has been dscussed by Collns et. al.[6]. The condton () s used by any researchers [6-65] to fnd exact solutons of cosologcal odels. Now usng equatons (6), (), and () the expanson for the etrc coeffcent are α n kt ( α + ) A= ( te ) (4) n kt ( α + ) B= ( te ) (5) Wth the sutable choce of coordnates and constants, the etrc () wth the help of (4) and (5) can be 6α 6 n kt ( α ) n kt ( α ) + + ds = dt ( t e ) dx ( t e ) dy + ( y, K ) dz 5 Soe Physcal Propertes of the Model (6) The Spatal Volue s obtaned as V n kt = ( te ) (7)

5 8 Theoretcal Physcs, Vol., No., March 7.5 x 5 Spatal Volue Te Fgure. Spatal Volue vs te. Fro Eq. (7), we observe that the spatal volue s zero at t =, whch shows that the unverse starts evolvng wth zero volue at t = whch s bg bang scenaro. The Hubble paraeter s obtaned as n H = k + (8) t It shows that Hubble paraeter s a decreasng functon of te. The deceleraton paraeter yelds as d q = = dt H ( kt + n) (9) Deceleraton Paraeter Te Fgure. Deceleraton Paraeter vs te. Recent observatons of SNe Ia, expose that the present unverse s acceleratng and the value of Deceleraton Paraeter les to soe place n the range < q <. It follows that n our derved odel, one can choose the value of Deceleraton Paraeter consstent wth the observaton. The ean ansotropc paraeter becoes A ( α ) = ( α + ) Snce A s constant, the ean ansotropc paraeter s unfor throughout the evoluton of the unverse. The expanson scalar yelds n θ = k + t () ()

6 Theoretcal Physcs, Vol., No., March Scalar Expanson Te Fgure. Scalar Expanson vs te. Fro Eqs. (7) and (), we observe that the spatal volue s zero at t = and the expanson scalar s nfnte, whch show that the unverse starts evolvng wth zero volue at t = whch s bg bang scenaro whch resebles wth the nvestgatons of Pradhan et. al. [66], Katore and Shakh [67]. The Shear Scalar s found to be σ ( α ) n = k + ( α + ) t () Shear Scalar Te Fgure 4. Shear Scalar vs te. For large value of t, the shear scalar vanshes; hence the shape of the unverse reans unchanged durng evoluton whch resebles wth the nvestgatons of Katore et. al. [68]. The Pressure can be obtaned as α 4β n β α n K p = + k + + () β ( + β) ( α + ) t ( + β) ( α + ) t B 8π + λ Here β =. λ The Energy Densty s gven by n( α + ) 9 n ( α + ) K ρ = + k α + + (4) β ( + β) ( α + ) t ( α + ) t ( + β) B ( + β)

7 4 Theoretcal Physcs, Vol., No., March Energy Densty Te Fgure 5. Energy Densty vs te. Fro Eqs. (4), we observe that the energy densty ρ s always postve and decreasng functon of te and approaches to zero as t. Fgures 5 depct ρ versus te t showng the postve decreasng functon of t and approachng to zero at t. The cosologcal constant n( α + ) 9( α + ) n K Λ= k + 6 (5) ( + β) ( α + ) t ( α + ) t n kt ( α + ) ( te ) 5 5 Labda Te Fgure 6. Cosologcal Constant vs te ( K = ). A negatve effectve ass densty (repulson) s corresponded by a postve value of Λ. Hence, we expect that n the unverse wth a postve value of Λ the expanson wll tend to accelerate whereas n the unverse wth negatve value of Λ the expanson wll slow down, stop and reverse. Fgure 6,7,8 are the plots of cosologcal ter Λ versus te for K =,,.In all three fgures, we observe that Λ s decreasng functon of te t and t approaches a sall postve value at late te (.e. at present epoch).recent cosologcal observatons suggest the exstence of a postve cosologcal constant Λ wth the agntude Λ( Għ / c ).These observatons on agntude and red-shft of type Ia supernova suggest that our unverse ay be an acceleratng one wth nduced cosologcal densty through the cosologcal Λ -ter. Thus, the nature of Λ n our derved odel s supported by recent observatons.

8 Theoretcal Physcs, Vol., No., March Labda Te Fgure 7. Cosologcal Constant vs te ( K = ). 5 4 Labda Te Fgure 8. Cosologcal Constant vs te ( K = ). 6 Concluson To deal wth the probles of late te acceleraton of the unverse, Harko et al. [6] proposed a new theory known as f( RT, ) theory of gravty by odfyng general theory of relatvty. We chose a specfc choce of the functonal f( RT, ) = f( R) + f ( T) wth f ( R) = λr and ( ) f T = λt and nvestgated the exact solutons of Hypersurface Hoogenous cosologcal odel. We observed that the hybrd expanson law gves te dependent DP, representng a odel whch generates a transton of unverse fro an early deceleratng phase to a recent acceleratng phase. We observe that the energy densty ρ s always postve and decreasng functon of te and approaches to zero as t. The ean ansotropc paraeter s unfor throughout the evoluton of the unverse. The spatal scale factor and volue scalar vansh at t =. The nature of Λ n our derved odel s supported by recent observatons. Chrstan Corda [69] has shown that, by assung that advanced projects on the detecton of Gravtatonal Waves (GWs) wll prove ther senstvty allowng to perfor a GWs astronoy, accurate angular and frequency dependent response functons of nterferoeters for gravtatonal waves arsng fro varous Theores of Gravty,.e. General Relatvty and Extended Theores of Gravty, wll be the defntve test for General Relatvty. Acknowledgeents. The authors would lke to place on record ther sncere thanks for the Edtor and referee for ther valuable coents whch have helped n provng the qualty of the paper

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