Canadian Journal of Physics. Kantowski-Sachs modified holographic Ricci dark energy model in Saez-Ballester theory of gravitation

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1 Cndin Journl of Physics Knowsi-Schs modifid hologrphic Ricci dr nrgy modl in Sz-Bllsr hory of grviion Journl: Cndin Journl of Physics Mnuscrip ID cjp--7.r Mnuscrip Typ: Aricl D Submid by h Auhor: -Jul-7 Compl Lis of Auhors: Bhsr Ro, M.P.V.V.; Vignn s Insiu of Informion Tchnology, REDDY, DRK; Andhr Univrsiy, Wlir, Mhmics Bbu, K. Sobhn; JNTU K Collg of Engg. Nrsrop, Kyword: Is h invid mnuscrip for considrion in Spcil Issu? : Knowsi-Schs modl, Ricci dr nrgy, Sz Bllsr hory, Hologrphic dr nrgy, Hybrid xpnsion lw N/A hps://mc.mnuscripcnrl.com/cjp-pubs

2 Pg of Cndin Journl of Physics Knowsi-Schs modifid hologrphic Ricci dr nrgy modl in Sz-Bllsr hory of grviion M.P.V.V. Bhsr Ro, D.R.K. Rddy, K. Sobhn Bbu Dprmn of Bsic Scincs nd Humniis, Vignn s Insiu of Informion Tchnology, Duvvd, Vishpnm, Indi. Dprmn of Applid Mhmics, Andhr Univrsiy, Vishpnm, Indi. Dprmn of Mhmics, JNTU Collg of Engg., Nrsrop, Indi. rddy_insin@yhoo.com Absrc: W hv considrd Knowsi-Schs spc-im in h prsnc of mr nd nisoropic modifid hologrphic Ricci dr nrgy componns in h sclr nsor hory of grviion formuld by Sz nd Bllsr Phys. L. A ; 47, 98 nd drivd h fild quions of h hory. W hv usd i Hybrid xpnsion lw proposd by Arsu l. JCAP,, 4, ii rlion bwn mric ponils nd iii modifid hologrphic Ricci dr nrgy dnsiy givn by Chn nd Jing Phys. L. B 79, 44, 9 o obin n xc soluion of h fild quions which dscribs Knowsi-Schs hologrphic modifid Ricci dr nrgy univrs in his hory. Physicl nd Kinmicl prmrs r lso compud nd hir physicl bhvior is discussd. Kywords: Knowsi-Schs modl, Ricci dr nrgy, Sz Bllsr hory.. Inroducion Th obsrvions from disn yp I suprnov confirm h h univrs prsn, is in n cclrd phs of xpnsion []-[]. Th cclrd xpnsion cn b ribud o n xoic form of nrgy, nown s dr nrgy. I provids ngiv prssur h givs r nigrviy ffc driving h cclrion. Howvr, h xc nur of dr nrgy sill rmins s mysry [4]. Cosmologicl consn in h clssicl FRW modl cn b simpl cndid for dr nrgy. Bu his hs fin uning problm nd coincidnc problm. Som ohr cndids proposd o consruc dr nrgy modls r quinssnc modls [5], phnom modls [], K-ssnc [7] nd so on. Thr r wo mjor pprochs o ddrss his problm of cosmic cclrion ihr bu inroducing dr nrgy componn in h univrs nd sudy is dynmics or modifying hps://mc.mnuscripcnrl.com/cjp-pubs

3 Cndin Journl of Physics Pg of Einsin s hory of grviion rmd s modifid grviy pproch. Hnc diffrn modificions of Einsin s hory hv bn proposd o xplin h cosmic cclrion of h univrs. Mos significn mong hm r fr grviy [8], fr,t grviy [9] whr R is h Ricci sclr nd T is h rc of h nrgy momnum nsor. Som ohr lrniv horis of grviion r sclr-nsor horis of grviion formuld by Brns nd Dic [] nd Sz nd Bllsr []. Hr, w focus our nion on Sz Bllsr [] sclr nsor hory of grviion. In Brns-Dic hory sclr fild φ, which hs h dimnsion of h invrs of grviionl nd which inrcs qully wih ll forms of mr, hs bn inroducd in ddiion o h usul mric nsor fild g ij. Subsqunly Sz nd Bllsr proposd nw sclr nsor hory of grviion. In his hory h mric is compild wih dimnsionlss sclr fild in simpl mnnr. This coupling givs sisfcory dscripion of h w filds nd n nigrviy rgim pprs in his hory. This hory lso suggss possibl wy o solv h missing mr problm in non fl FRW cosmologis. Hologrphic dr nrgy is nohr lrniv o h soluion of dr nrgy problm. This is bsd on h hologrphic principl. According o his principl h nropy of sysm scls no wih is volum, bu is surfc r []-[]. A cosmologicl vrsion of his principl ws proposd by Fischr nd Sussind [4] nd Cohn l. [5]. Rcnly, Grnd nd Olivros [] suggsd nw hologrphic Ricci dr nrgy modl wih dnsiyρ M pl ηh ς H. Lr, Chn nd Jing [7] modifid his modl by ssuming h dnsiy of dr nrgy conins h Hubbl prmr H, h firs ordr nd h scond ordr drivivs. Th xprssion of h nrgy dnsiy of modifid hologrphic Ricci dr nrgy is givn by ρ H H& HH &&. Thus if w h whol univrs ino ccoun, hn h vcuum nrgy rld o h hologrphic principl my b viwd s dr nrgy, usully clld hologrphic dr nrgy. Hnc svrl uhors, in rcn yrs, lv invsigd hologrphic dr nrgy modls in nisoropic spc-ims. This is bcus of h fc h obsrvionl d suggss h h nomlis found in h cosmic microwv bc ground CMB simuld incrsing inrs in nisoropic modls. Also, hs modls will crinly hlp for br undrsnding of h rly sgs of voluion of h univrs. Kirn l. [8] hv discussd minimlly inrcing hologrphic Binchi yp-v dr nrgy modls in Sz-Bllsr sclr hps://mc.mnuscripcnrl.com/cjp-pubs

4 Pg of Cndin Journl of Physics nsor hory of grviion. Vry rcnly, Rddy l. [9] hv discussd Binchi yp-ix minimlly inrcing hologrphic dr nrgy modl in Sz-Bllsr sclr-nsor hory of grviion. Rddy l. [] hv discussd Knowsi- Schs minimlly inrcing hologrphic dr nrgy modl wih linrly vrying dclrion prmr in Sz-Bllsr hory. Rmsh nd Umdvi [] hv obind minimlly inrcing hologrphic dr nrgy modl in Sz- Bllsr hory using linrly vrying dclrion prmr proposd by Arsu nd Drli []. Amir nd Ali [] hv discussd Ricci dr nrgy of mndd FRW univrs in Chrn-Simon modifid grviy. Ali nd Amir [4] hv sudid hologrphic dr nrgy modls in Chrn-Simon modifid grviy. Ds nd Suln [5]-[] hv sudid Binchi yp-ii nd VI nisoropic modifid hologrphic Ricci dr nrgy cosmologicl modls. Snhi l. [7]-[8] hv discussd som Binchi yp modifid hologrphic Ricci dr nrgy modls in gnrl rliviy nd Brns-Dic sclr-nsor hory of grviion. Ro nd Prsnhi [9] hv sudid Binchi yp-i nd III modifid hologrphic Ricci dr nrgy modls in Sz-Bllsr hory. Th bov discussion nd h invsigion hv moivd us o considr h modifid hologrphic Ricci dr nrgy modl in Knowsi-Schs spc im in h frm wor of Sz Bllsr sclr nsor hory of grviion.. Mric nd h Fild Equions W considr h Knowsi Schs spc-im givn by ds d A dr B dθ sin θ dφ whr A nd B r h funcions of h cosmic im only. Th fild quions givn by Sz nd Bllsr [] for h combind sclr nd nsor filds r n, Rij Rgij wϕ ϕϕ, i, j gijϕ, ϕ Tij Tij nd h sclr fild ϕ sisfis h quions n, i n, ϕ ϕ nϕ ϕ ϕ ; i, Also w hv h consrvion quion hps://mc.mnuscripcnrl.com/cjp-pubs

5 Cndin Journl of Physics Pg 4 of ij ij T T ; whr j R ij is h Ricci nsor, R is h Ricci sclr, ω nd n r rbirry dimnsionlss consns nd 8π G c in h rlivisic unis. Also h nrgy momnum nsors for mr Tij nd for h nisoropic hologrphic dr nrgy Tij r dfind s 4 j T dig[,,,] ρ i m j T dig[ ρ, p, p, p ] dig[, ω, ω, ω ] ρ 5 i r θ φ r θ φ whr ρm, ρ r h nrgy dnsiis of mr nd h hologrphic dr nrgy nd p r, p nd p θ φ r h prssurs of h hologrphic dr nrgy long r, θ nd φ xs p rspcivly. Hr ω is h quion of s EoS prmr of h fluid nd ρ ω, ω, ω r h EoS prmrs in h dircions of r θ φ r, θ nd φ xs rspcivly. Th nrgy momnum nsor of hologrphic dr nrgy cn b prmrizd s j Ti dig[, ω, ω δ, ω γ ] ρ For h s of simpliciy w choos dviions from ω long θ nd φ rspcivly. Now h fild quions for h mric h form ω ω nd swnss prmrs δ nd γ r h r && & B B B B B w n ϕ & ϕ A&& B&& AB & & w ϕ & ϕ A B AB ωρ n A&& B&& AB & & w A B AB n ϕ ϕ & ω δ ρ ω γ ρ AB & & B& w & AB B B n ϕ ϕ ρ ρm hps://mc.mnuscripcnrl.com/cjp-pubs

6 Pg 5 of Cndin Journl of Physics A& B& n & ϕ && ϕ & ϕ A B ϕ A& B& A& B& B& & ρm ρm & ρ ω ρ δ γ ρ A B A B B whr ovrhd do dnos diffrniion wih rspc o im. Th following r h physicl nd gomricl prmrs o b usd in solving h Sz- Bllsr fild quions for h spc-im givn by quion. Th vrg scl fcor of h Knowsi-Schs spc im is dfind s AB Spil volum is givn by V AB whr is h vrg scl fcor of h univrs. 4 Th vrg Hubbl s prmr H is givn by A& B& H H H H A B whr H, H, H r dircionl Hubbl s prmrs in x, y nd z dircions. 5 Th mn nisoropy prmr is dfind s i Hi H H whr H i i,, rprsn h dircionl Hubbl prmr. Th sclr xpnsion θ nd shr sclr r givn by A& B& θ A B A& B& AB & & σ A B AB hps://mc.mnuscripcnrl.com/cjp-pubs

7 Cndin Journl of Physics Pg of. Soluions of fild quions nd h cosmologicl modl From quions 8 nd 9, w obin δ γ 9 Equions 7-, in viw of Eq. 9, rduc o four indpndn quions in svn unnowns A,B,φ,δ,ρ m,ρ h sysm: nd ω. Hnc w us h following ddiionl condiions o solv i Shr sclr is proporionl o h sclr xpnsion so h [] A B ii Th hybrid xpnsion lw givn by [] AB / nd iii modifid hologrphic Ricci dr nrgy dnsiy givn by [7] ρ H H& HH && Now from Eqns. nd, w hv A B Also, from Eqs. nd h sclr fild ϕ is givn by ϕ&ϕ ϕ n / which on ingrion cn b pu in h form ϕ n n ϕ d ψ whr ϕ nd ψ r consns of ingrion. 4 5 Now h mric cn b wrin s hps://mc.mnuscripcnrl.com/cjp-pubs

8 Pg 7 of Cndin Journl of Physics ds d dr dθ sin θdφ 4. Physicl discussion of h modl Equion dscribs Knowsi-Schs modifid hologrphic Ricci dr nrgy cosmologicl modl in Sz- Bllsr sclr nsor hory of grviion wih hybrid xpnsion lw proposd by Arsu l. []. Th following r h xprssions for physicl nd inmicl prmrs of h modl: Spil volum is V 7 Th vrg Hubbl s prmr is H & Th sclr xpnsion is θ Th shr sclr is H σ Th vrg nisoropy prmr is Th dclrion prmr is q 7 hps://mc.mnuscripcnrl.com/cjp-pubs

9 8 Fig. : Plo of dclrion prmr vrsus cosmic im for 7. nd.. From Eqs. nd 8, w hv h nrgy dnsiy of modifid hologrphic Ricci dr nrgy s ρ From Eqs. 7, nd h EoS prmr of modifid hologrphic Ricci dr nrgy cn b found s 7 w ϕ ω 4 From Eqs. 8, nd, w g h swnss prmr s 9 δ 5 Pg 8 of hps://mc.mnuscripcnrl.com/cjp-pubs Cndin Journl of Physics

10 9 From Eqs., nd, w hv h nrgy dnsiy of mr s 4 9 w m ϕ ρ Th ovrll dnsiy prmr Ω is givn by Ω ΩΩ 4 m m w H H ϕ ρ ρ 7 Fig. : Plo of EoS prmr of MHRDE vrsus im for.7 nd.,.8,.,.8, w. Pg 9 of hps://mc.mnuscripcnrl.com/cjp-pubs Cndin Journl of Physics

11 Cndin Journl of Physics Pg of Fig.: Plo of ovrll dnsiy prmr vrsus im for.8,.,.8, w,. nd.7. Fig. rprsns h bhvior of dclrion prmr wih im. I is obsrvd h hr is smooh rnsiion of h univrs from rly dclrd phs o l im cclrion. Fig. dpics bhvior of EoS prmr of MHRDE vrsus im. I cn b sn h h modl nrs quinssnc rgion from dus modl sinc ω iniilly nd will nvr cross h phnom divid lin i.., ω. Th vriion of vrg dnsiy prmr Ω is givn in Fig.. This shows h Ω iniilly incrss rpidly wih im nd ulimly bcoms qul o on. This shows h h univrs bcoms fl l ims. All h bov rsuls r in good grmn wih h obsrvions of modrn cosmology. 5. Conclusions In his ppr, w hv discussd h modifid hologrphic Ricci dr nrgy modl in Knowsi-Schs univrs in h frm wor of sclr-nsor hory of grviion proposd by Sz nd Bllsr []. W hv usd h hybrid xpnsion lw nd rlion bwn mric ponils o obin drmin soluion of h fild quions. This soluion rprsns h modifid hologrphic Ricci dr nrgy univrs in sclr-nsor cosmology. I is obsrvd h h modl is xpnding spilly. I cn b sn h Hubbl s prmr, xpnsion sclr, shr sclr, sclr fild, nrgy dnsiis of mr nd dr nrgy divrs iniilly i.., nd hy ll bcom consn in infini im. I is wll nown h sclr filds ply significn rol in dr nrgy cosmology. Th rsuls obind r in ccordnc wih h rcn obsrvions of hps://mc.mnuscripcnrl.com/cjp-pubs

12 Pg of Cndin Journl of Physics cosmology nd w hop h hs rsuls will b usful o hrow br ligh on our undrsnding of sclr nsor cosmology. Acnowldgmn: Th uhors r grful o h rfr for consruciv commns which hv improvd h prsnion of his wor. Rfrncs []. A. Riss, l.: Asron. J., []. A. Riss, l.: Asrophys.J.7, 5 4. []. S. Prlmur, l.: Asrophys. J. 57, [4]. E.J. Coplnd, l.: In. J. Mod. Phys. D, 5, 75. [5]. B. Rr, P.J.E. Pbls: Phys. Rv. D, 7, 998. []. R. Cldwll: Phys. L. B 545,. [7]. T. Chib, l.: Phy. Rv. D, 5. [8]. S.M. Croll, l.: Phys. Rv. D 7, [9]. T. Hro, l.: Phys. Rv. D 84, 4. []. C.H. Brns, R.H. Dic: Phys. Rv. 4, []. D. Sz, V.J. Bllsr: Phys. L. A, []. L. Sussind: J. Mh. Phys., []. R. Bousso: JHEP, 7, [4]. W. Fischlr, L. Sussind: hp-h/ [5]. A. Cohn, l.: Phys. Rv. L. 8, []. L.N. Grnd, A. Olivros: Phys. L. B, 9, [7]. S. Chn, J. Jing: Phys. L. B, 79, [8]. M. Kirn, l.: Asrophys. Spc Sci. 54, [9]. D.R.K. Rddy, l.: Prspc im J. 7,. []. D.R.K. Rddy, l.: Prspc im J. 7,. []. G. Rmsh, S. Umdvi: Asrophys. Spc Sci., 5. []. O. Arsu, T. Drli: In. J. Thor. Phys. 5,. []. M.J. Amir, S. Ali: In. J Thor. Phys. 54, 5. hps://mc.mnuscripcnrl.com/cjp-pubs

13 Cndin Journl of Physics Pg of [4]. S. Ali, M.J. Amir: In. J Thor. Phys. 55, 595. [5]. K. Ds, T. Suln: Asrophys. Spc Sci.,, 4 5. []. K. Ds, T. Suln: Asrophys. Spc Sci.,, 5. [7]. M.V. Snhi, V.U.M. Ro, Y. Adiy: Cn. J. Phys. 95, 8 7. [8]. M.V. Snhi, V.U.M. Ro, Y. Adiy: Cn. J. Phys. 95, [9]. V.U.M. Ro, U.Y.D. Prsnhi: Eur. Phys. J. Plus, 4 7. []. C.B.Collins, l.: Gn. Rliv. Grvi., []. O. Arsu, l.: JCAP,, 4. hps://mc.mnuscripcnrl.com/cjp-pubs