Particle Horizons in Cosmology

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Jounl of Moden Physics, 01, 4, 1194-1199 hp://dx.doi.og/10.46/jmp.01.4916 Published Online Sepembe 01 (hp://www.scip.og/jounl/jmp) Picle Hoizons in Cosmology K.

1 Jounl of Moden Physics, 01, 4, hp://dx.doi.og/10.46/jmp Published Online Sepembe 01 (hp:// Picle Hoizons in Cosmology K. Punchnd Ro School of Mhemicl nd Sisicl Sciences, Hwss Univesiy, Hwss, Ehiopi Emil: Received My 9, 01; evised June 15, 01; cceped July 1, 01 Copyigh 01 K. Punchnd Ro. This is n open ccess icle disibued unde he Ceive Commons Aibuion License, which pemis unesiced use, disibuion, nd epoducion in ny medium, povided he oiginl wok is popely cied. ABSTRACT The imponce of invesiging picle hoizons in ode o inepe cosmologicl soluion of Einsein s field equions hs been descibed. We hve pesened he fomul nd sudied he picle hoizons in some of he cosmologicl models pesened in ou elie ppes. I is well known h he Fiedmn-Robeson-Wlke (F-R-W) models, he enegy densiy of he fee gviionl field denoed by ε, equivlenly denoed by McCllum pmee ξ, vnishes bu he picle hoizons exis nd hus he fome hs no being on he le. Howeve, we hve shown in ou models pesened heein h ε is eled wih picle hoizons. Fuhe, i is shown h s ε gows, he segmen of he coesponding picle hoizon deceses nd hus he dius of he coesponding visible univese deceses. Keywods: In Homogeneiy; Anisoopy; Cosmology; Picle Hoizon 1. Inoducion Cosmology dels wih he lge scle sucue of he univese, which by definiion conins eveyhing, viz., boh obsevble nd non obsevble. To undesnd he physicl nue of he univese s whole, emps wee mde duing 19 h cenuy wihin he fme wok of he Newonin heoy of gviion. Bu, hese effos did no fucify since Newonin gviion ssumed insnneous popgion of gviionl inecion fo which hee hd no been ny expeimenl jusificion. The pogess of moden cosmology hs been guided by boh heoeicl nd obsevionl dvnces. The subjec elly ook off in 1917 wih he fis cosmologicl soluion given by Albe Einsein bsed on his genel heoy of eliviy o heoy of gviion. Since hen wide nge of cosmologicl models hve been consuced, sudied nd nlyzed wih vying objecives. The Fiedmn-Robeson-Wlke (F-R-W) cosmologicl models, deived bsed on he win ssumpions of spil isoopy nd homogeneiy povides sisfcoy descipion of he obsevble univese fo consideble p of is hisoy. Howeve, he exisence of inhomogeneiies in he fom of glxies nd cluses s well s he nisoopy in he cosmic bckgound diion could no be explined wih he help of hese models. Cosmologicl models wih inhomogeneous densiy hve been sudied [1-]. I hs been shown [4-7], h he enegy densiy of he fee gviionl field ε is eled o boh nisoopy nd in homogeneiy. A lo of isoopic nd homogeneous cosmologicl models hve ppeed in lieue bu few models wih he chceisics of nisoopy nd in homogeneiy. An emp hs been mde o fill he gp by consucion of wide nge of nisoopic nd inhomogeneous cosmologicl models [8-1]. Howeve, in ode o inepe cosmologicl soluion of Einsein s field equions, one should invesige some specil specs like hoizons [1,14]. I hs been sudied even hoizons in cosmology [15]. In he pesen ppe we will descibe mhemiclly nd povide gphicl epesenions of picle hoizons which exis in some of he models pesened in ou ppes [8-1]. The Eigen vlue of he confoml Weyl enso in Peov s clssificion [16] is denoed by ε nd is known s he enegy densiy of he fee gviionl field s i lwys coupled wih he meil enegy densiy ρ. In Secion, following [17,18], we hve defined picle hoizons nd lis some popeies of he hoizons. In Secion, we hve inoduced he mos genel spheiclly symmeic spce-ime meic filled wih nisoopic fluid. We hve given he soluions of Einsein field equions, nd expession fo he enegy densiy of he fee gviionl field epesened by ε. In Secion 4, we pesened he equions of kinemicl quniies, McCllum quniy ξ, nd Rychudhui s equion fo he spheiclly symmeic models. In Secion 5, we hve given he fomul fo picle Copyigh 01 SciRes.

2 K. P. RAO 1195 hoizons of he mos genel spheiclly symmeic meic. We hve invesiged he exisence of picle hoizons in some of he models discussed in ou elie ppes nd deived fomule fo picle hoizons in hese models. The digmmic epesenions of he picle hoizons sudied hee hve been povided in Figues 1-4. In Secion 6, we pesened diec compison beween he Even nd Picle hoizons nd gphicl epesenion is povided in Figue 5. In ll he figues he dil coodine is ken long he hoizonl xis while he ime coodine is ken long he veicl xis. The ppe ends wih concluding emks in Secion 7.. Picle Hoizons 1) Even hoizon Conside n obseve nd phoon on is wy o he obseve long null geodesic. I cn hppen h he Figue. Digm showing he picle hoizons descibed by (8) of he spce-ime meic descibed by (6) coesponding o 18,15 nd 1. Figue 1. Digmmic epesenion of he picle hoizon descibed by (1) of he spce-ime meic descibed by (9). Figue. Digmmic epesenion of he picle hoizon descibed by () of he spce-ime meic descibed by () * * * * * * Figue 4. Digm showing h he picle hoizon of he spce-ime meic descibed by (6) is mde up of cuved segmens coesponding o vious vlues. spce-ime is expnding such e h he phoon neve ges o he obseve. As Eddingon hs pu i, ligh is hen like unne on n expnding ck, wih he winning pos (obseve) eseeding fom him foeve [18]. In such cses hee will be wo clsses of phoons on evey null geodesic hough he obseve: hose which ech he obseve finie ime nd hose who do no. They e seped by he ggege of phoons (ligh fon) h ech excly. The ligh fon is clled obseve s even hoizon. The exisence nd moion of n even hoizon depend on he fom of expnsion pmee. The even hoizons in cosmology hve been sudied by [15]. * 0. * Copyigh 01 SciRes.

3 1196 K. P. RAO Figue 5. Digm showing he diec compison beween Even nd Picle hoizons of he spce-ime meic (6) s descibed in he Secion 6. Hee we hve seleced 15. ) Picle hoizon Suppose h vey fis phoons (ligh fon) emied by he obseve big-bng even e sill ound. As his ligh fon sweeps ouwds, owds moe nd moe glxies, he obseve he big-bng nd hese glxies see ech ohe fo he vey fis ime (cosmic insn). Hence, ny cosmic insn his ligh fon, clled he obseve s picle hoizon, divides ll glxies ino wo clsses elive o he obseve: hose ledy in obseve s view nd ll ohes. ) Some popeies of hoizons ) Evey glxy, wihin A s even hoizon, excep A, evenully possess ou of i. Fo if B is such glxy, hen A s hoizon phoon in he diecion of AB is wihin B s even hoizon, nd will heefoe ech B finie cosmic ime. Th is, when B psses ou of A s even hoizon. b) Evey glxy B wihin A s even hoizon emins visible foeve A. Fo, he even hoizon iself bings ls view of B. As B ppoches A s even hoizon in models wih infinie expnsion, is hisoy, s seen A, ges infiniely diled, nd is ligh infiniely ed-shifed. In collpsing models B s ligh ges infiniely blue-shifed s B ppoches he even hoizon. c) As glxies e oveken by A s picle hoizon, hey come ino view A wih infinie ed-shif in bigbng models, nd infinie bleu-shif in models wih unlimied ps expnsion. d) If model possesses no even hoizon, evey even evey glxy is seen on evey glxy. Fo, n invisible even implies he exisence of even hoizon. e) If model possesses no picle hoizon, evey obseve - if necessy by veling fom his oiginl glxy - cn be pesen ny even ny glxy. Fo, in pinciple, his only vel esicion is his fowd ligh cone ceion; bu h would be picle hoizon if ll glxies we no lwys wihin i. f) If n even hoizon exiss, wo biy evens e in genel no boh knowble o one obseve, even if he vels. Fo, conside wo dimeiclly opposie evens ouside n even hoizon. Thei fowd ligh cones cn inesec. Bu o now eihe even mens being in is fowd ligh cone. g) The even nd picle hoizons, if exis, mus coss ech ohe wihin he life ime of he model. Fo, he picle hoizon ws nd he even hoizon will be, he fundmenl picles ssocied wih hem. h) When model, in which boh even nd picle hoizons exis, is un bckwd in ime (i.e. ime evesed), he even hoizon becomes he picle hoizon nd viceves.. Spheiclly Symmeic Spce-Time Meic The mos genel spheiclly spce-ime line elemen cn be consideed s ds e d R d sin d d e (1) whee, R nd e funcions of nd only. We ssume h he meil souce of he gviionl field, filled in he meic, is n nisoopic fluid wih he enegy momenum enso whee b T p u u p g p p () b b b u 0,0,0,exp, u u 1 () exp,0,0,0, 1 (4) nd p nd p epesen especively he dil nd nsvese componens of fluid pessue whee s denoes he pope densiy of he fluid. The Einsein field equions wih he cosmologicl consn included is given by b b b 8T R 1 Rg g b (5) Afe using (1) o () we ge he following: 1 8T1 8p e R R R R e RR R R R 1 R 8T 8T 8p (6) 8 e RR R R R R e RR R R R R 1 R (7) Copyigh 01 SciRes.

4 K. P. RAO T4 8 e R R R R R e R R R R 1 R R R R R 8 e 4 R R R R 4 RR RR e 4 (8) R R R R 4 1 R (9) is he Eigen vlue of he confoml Weyl enso in Peov s clssificion [16]. Hee nd wh follows pime nd ove hed do denoe especively diffeeniion wih espec o nd. As we shll see le since he Eigen vlue is lwys coupled wih he meil enegy densiy, he fome ws inepeed [4-6], s he enegy densiy of he fee gviionl field nd is pesence is eled wih boh nisoopy nd inhomogeneiy. The necessy nd sufficien condiion fo confoml flness of he spce-ime (1) is 0. Using he diecionl deivives exp nd D exp D long he dil nd nsvese diecions especively, Γ nd U e defined s Γ DR R exp (10) exp U u R x D R R (11) We now diecly wie down he equions which goven he evoluion of he sysem s below [19]: D U R (1) DΓ UD (1) D nr nr U R (14) 1Γ 4 4 DU U R p p R 4 8 D p p 8 4 p D R 4 8 D p p 8 D 4 p D R R R (15) (16) (17) D p 8 p p D ln R exp (18) whee n denoes he byon numbe densiy nd we hve 1 wien R R. We my menion hee h he enegy densiy of he fee gviionl field which is coupled lwys o he meil enegy densiy s well s he cosmologicl consn plys significn ole in descibing uly elivisic siuions. We hve ken, fo simpliciy, he coupling consn s uniy. Howeve, in ode o ise he conibuion of subsnilly elive o, we could hve chosen he coupling consn o desied levels. 4. Kinemics of Spheiclly Symmeic Models Spheiclly symmeic soluions cn be clssified ccoding o hei kinemicl popeies. The ssumpion of spheicl symmey implies h oion Wb 0 nd hence he fluid velociy field mus be hype sufce ohogonl. Fom (1) nd () we ge he expessions fo he emining kinemicl quniies, viz. cceleion u, b componens of he she enso, she invin b defined by b nd expnsion of he ime-like conguence u, s below: u exp (19) exp RR (0) exp RR (1) RR exp () In view of hese expession, he nisoopy in he 4-dimensionl spce-ime, denoed by, nd defined by [1] s kes he fom 1 b () RR RR nd similly he Rychudhui equion [0] 1 b (4) u u w w R u u (5) b b b, ; b b b using () cn be ewien s 1 4 p p (6) The quniy h descibed he nisoopy of he 4-dimensionl spce-ime s defined by [1] cn lso be wien s [19] 1 1 RR (7) Copyigh 01 SciRes.

5 1198 K. P. RAO whee he uppe nd lowe signs especively coespond o expnding nd concing models. I hs been suggesed h physiclly meningful cosmologicl model should evolve in such wy h ss fom 0 big bng nd ends up wih 1 in he blck hole, h is big cunch. 5. Fomul fo Hoizons Wih he coodines nd suppessed in he spceime meic descibed by (1), he equion of moion, elion, of phoon emied, owds he oigin glxy, is given by e d e d. In he spce-ime model, descibed by [1] 1 ds d f () d sin d 4 d The equion of moion (8) kes he fom (8) (9) 1 1 d 1 d (0) In (9), picle hoizon exiss, since he condiion fo which 1 d 0 is sisfied. Thus, he equion of he hoizon is given by d 1 d 1 0exp1 (1) 0 0 nd hus he picle P1, 1 comes ou of picle hoizon log If he obseve O 0, nd he picle P whee, see ech ohe hen 1, 1 1. We hve given he digmmic epesenion of his picle hoizon in Figue 1. We now conside he spce-ime model descibed by [1] sin ds k d L d d d () whee k 0, L 0 nd 0 L. The equion of picle hoizon of () is given by 1 1 k () whee 1 is he ime when he picle comes ou of he picle hoizon. If he obseve O 0, nd he picle P, 1 whee 1, see ech ohe hen 1. We hve given he gphicl epesenion of his picle hoizon in Figue. We now conside he spce-ime model descibed by [1] 1 d ds d G d sin d (4) in which picle hoizon exiss nd is equion is given by e (5) whee 0 0 is consn nd he picle comes ou of he picle hoizon 1. If he obseve O, 0 nd he picle P, 1 whee 1, see ech ohe hen 1. We now conside moe genel meic descibed by [1] n ds d d sin d d (6) Wih he consn sisfying 1 n 1, in which picle hoizon exiss nd is equion is given by n 1n 1 n (7) If he obseve Oo, nd he picle P, 1 1 n 1 n 1 1 whee 1, see ech ohe, hen n. I is cle h in he spce-ime meic (6), n is eled o by n 1 1. Thus, he equion of picle hoizon (7), in ems of, kes he fom 1 1 (8) I ppes, fom (8), h he evluion peiod of he picle hoizon is divided ino hee ps especively: iniil, inemedie nd finl epochs. As inceses fom he ode of zeo o he ode of 1, he following popeies of picle hoizons my be obseved: 1) In he iniil epochs, he picle hoizons wih incesing vlues of s nee disnces nd un wy fom he obseve wih fse es. ) In he inemedie epochs, he fhe hoizons e ove ken by hei peceding ones in sysemic mnne. Th is, in he pocess he nees hoizon ove kes ll ohes nd becomes he fhes o he obseve, while he fhes hoizon llows ll ohes o ke ove i nd becomes he nees nd so on. ) In he finl epochs, wih he evesed lengh scles, i.e. lge he disnce of hoizon fom he obseve highe he coesponding vlue of, he picle hoizons expnded wih slowe es. We hve given digmmic epesenion of he picle hoizon descibed by (8) fo 18, 15 nd 1 in Figue, in which he bove dwn conclusions e clely illused. 4) Smll segmens of he hoizons of s e nged o fom coninuous cuve nd his cuve will emin s he ue picle hoizon of he obseve. These segmens wih gowing fom is minimum o mximum e nged in sequence fom he fhes o he nees. We hve given digm in Figue 4 which demonses he popey 5). Copyigh 01 SciRes.

6 K. P. RAO Compison beween he Even nd Picle Hoizons [] H. Bondi, Monhly Noices of he Royl Asonomicl Sociey, Vol. 107, 1947, p [4] J. Kishn Ro, Cuen Science, Vol. 5, 1966, p. 89. In he spce-ime meic descibed by (6), boh Even [5] J. Kishn Ro, Genel Reliviy nd Gviion, Vol. nd Picle hoizons exis. The Even hoizon [15] is, 1971, pp n hp://dx.doi.og/ /bf descibed by he equion 0. The Picle 1 n [6] J. Kishn Ro, Jounl of Physics (London), Vol. A5, 197, p n hoizon is descibed by (7) s nd is [7] J. Kishn Ro, Genel Reliviy nd Gviion, Vol. 1 n, 197, p. 51. gphicl epesenion is shown in Figue. Hee doi: /bf [8] J. Kishn Ro, Pmn: Jounl of Physics, Vol. 4, n nd 0 1. We now mke diec com- 1990, p [9] J. Kishn Ro, Jounl of he Insiue of Mhemics, Vol. 51, 1995, p. 57. pison beween he Even nd Picle hoizon. Even hoizon is spheicl ligh fon conveging owds he obseve while Picle hoizon is h diveging wy fom he obseve, s ime pogesses. Howeve, Even nd Picle hoizons e invesely popging (opposiely velling) ligh fons (spheicl) o nd fom n obseve. Gphicl compison, illusing his fc, is shown in Figue Conclusion We hve deived he equions of picle hoizons fo some cosmologicl soluions. Also, we hve demonsed hem gphiclly. I is well known h in F-R-W models ε vnishes bu hoizons exis nd hus he fome hs no being on he le. Howeve, we hve shown in Figue 4 h ε, equivlenly ξ, is eled wih hoizons. As ε gows, he coesponding picle hoizon inceses nd hus he dius of he visible univese inceses. 8. Acknowledgemens My pofound hnks e due o he efeee fo his vluble nd consucive commens nd o my collegue D. Ayele Tye Goshu fo his simuling discussions. REFERENCES [1] R. C. Tolmn, Poceedings of he Nionl Acdemy of Sciences of USA, Vol. 0, 194, p. 69. [] G. C. Ome, The Asophysicl Jounl, Vol. 109, 1949, p [10] K. Punchnd Ro, Unpublished Ph.D. Thesis, Bhvng Univesiy, Bhvng, [11] K. Punchnd Ro, e l., Mhemics Tody, Vol. 16, 1998, p. 5. [1] K. Punchnd Ro, Mhemics Tody, Vol. 17, 1999, p. 9. [1] K. Punchnd Ro, Mhemics Tody, Vol. 1, 005, p. 1. [14] K. Punchnd Ro, Mhemics Tody, Vol. 4, 008, p. 17. [15] K. Punchnd Ro, Mhemics Tody, Vol. 5, 009, p. 4. [16] K. Punchnd Ro, Mhemics Tody, Vol. 7, 011, p. 54. [17] M. A. H. McCllum, In: V. D. Sbb, Ed., The Oigin nd Evoluion of Glxies, Wold Scienific Pub. Co., 198. [18] M. A. H. McCllum, In: B. R. Iye, A. Kembhvi, J. V. Nlike nd C. V. Vishveshw, Eds., Highlighs in Gviion nd Cosmology, Univesiy Pess, Cmbidge, [19] A. K. Rychudhui, Physicl Review, Vol. 98, 1955, pp doi:10.110/physrev [0] W. Rindle, Monhly Noices of he Royl Asonomicl Sociey, Vol. 116, 1956, p. 66. [1] W. Rindle, Essenil Reliviy: Specil, Genel nd Cosmologicl, Spinge-Velg, Copyigh 01 SciRes.

EVENT HORIZONS IN COSMOLOGY

Mahemaics Today Vol7(Dec-)54-6 ISSN 976-38 EVENT HORIZONS IN COSMOLOGY K Punachanda Rao Depamen of Mahemaics Chiala Engineeing College Chiala 53 57 Andha Padesh, INDIA E-mail: dkpaocecc@yahoocoin ABSTRACT