EVENT HORIZONS IN COSMOLOGY

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1 Mahemaics Today Vol7(Dec-)54-6 ISSN EVENT HORIZONS IN COSMOLOGY K Punachanda Rao Depamen of Mahemaics Chiala Engineeing College Chiala Andha Padesh, INDIA dkpaocecc@yahoocoin ABSTRACT The impoance of invesigaing hoizons in ode o inepe a cosmological soluion of Einsein s field equaions has been descibed We have pesened he fomulae and sudied he even hoizons of some of he models pesened in ou ealie papes Diagammaic epesenaion of he even hoizons have been included I is well known ha in F-R-W models he enegy densiy of he fee gaviaional field ε, equivalenly ξ, vanishes bu he even hoizons exis and hus he fome has no beaing on he lae Howeve, we have shown in ou models pesened heein ha ε is elaed wih hoizons Fuhe i is shown ha as ε gows he segmen of he coesponding even hoizon deceases and hus he adius of he coesponding visible univese deceases The sudy of paicle hoizons will be pesened in ou subsequen papes Key wods: In homogeneiy, Anisoopy, Cosmology, Hoizons AMS subjec classificaion: 74E5, 74E, 83F5 Inoducion Cosmology deals wih he lage scale sucue of he univese, which by definiion, conains eveyhing, viz obsevable and non-obsevable To undesand he physical naue of he univese, as a whole aemps wee made duing 9 h cenuy wihin he famewok of he Newonian heoy of gaviaion Bu, hese effos did no fucify since Newonian gaviaion assumes insananeous popagaion of gaviaional ineacion fo which hee is no expeimenal jusificaion The pogess of moden cosmology has been guided by boh heoeical and obsevaional advances The subjec eally ook off in 97 wih he fis cosmological soluion given by Albe Einsein based on his geneal heoy of elaiviy (o heoy of gaviaion) Since hen a vide ange of cosmological models have been consuced wih vaying objecives The Fiedman Robeson Walke (F-R-W) cosmological models, deived based on he win assumpions of spaial isoopy and homogeneiy povide a saisfacoy descipion of he obsevable univese fo consideable pa of is hisoy Howeve, he exisence of in-

2 K Punachanda Rao - Even Hoizons In Cosmology 55 homogeneiies in he fom of galaxies and cluses as well as he anisoopy in he cosmic backgound adiaion could no be explained wih he help of hese models Cosmological models wih inhomogeneous densiy have been sudied by Tolman (934), Oma (949), Bondi (947) and ohes J Kishna Rao (97, 97, 973, 99) has shown ha he enegy densiy of he fee gaviaional field, epesened by ε, is elaed o boh anisoopy and in-homogeneiy A lo many isoopic and homogeneous cosmological models have been appeaed in lieaue bu a few models wih he chaaceisics of anisoopy and in-homogeneiy An aemp has been made o fill he gap by consucing a wide ange of anisoopic and inhomogeneous cosmological models by K Punachanda Rao ( 997, 998, 5, 8, 9 ) Howeve, in ode o inepe a cosmological soluion of Einsein s field equaions, one should invesigae some special aspecs like hoizons (MacCallum, 98; 988) Thus, in his pape we will descibe and give gaphical epesenaion of hoizons of some of he models pesened in ou ealie papes ( K Punachanda Rao 998; 999; 5; 8; 9 ) The eigen value of he confomal Weyl enso in Peov s classificaion (Kishna Rao, 966) is denoed by ε and is known as he enegy densiy of he fee gaviaional field as i always coupled wih he maeial enegy densiy ρ The anisoopy in he 4-dimensional space-ime is descibed by he quaniy denoed by ξ and defined by MacCallum (98) as given below: [( / ) ( R/ R)] ξ = = [( / ) (R/ R)] The quaniies ε and ξ ae equivalen In secion, following Rindle ( 956, 977 ), we defined boh even and paicle hoizons and lised some popeies of he hoizons In Secion 3, we have given he fomulae fo even and paicle hoizons of he mos geneal spheically symmeic meic and deduced he coesponding fomulae fo F-R-W models In Secion 4, we have invesigaed he exisence of even hoizons in some of he models discussed in ou ealie papes ( K Punachanda Rao, 998; 999; 5; 8; 9) and deived fomulae fo even hoizons of hese models The diagammaic epesenaions of even hoizons sudied in Secion 4 have been povided in Appendices o 3 The pape ends wih concluding emaks in Secion 5 Hoizons and hei popeies (i) Even hoizon Conside an obseve and a phoon on is way o he obseve along a null geodesic I can happen ha he space-ime is expanding a such a ae ha he phoon neve ges o he obseve As Eddingon has pu i, ligh is hen like a unne on an expanding ack, wih he winning pos (obseve) eceding fom him foeve (Rindle, 977) In such a case hee will be wo classes of phoons on evey null geodesic hough he obseve: hose which each he obseve a a finie ime and hose who do no They ae sepaaed by he aggegae of phoons (ligh fon) ha each exacly a = This ligh fon is called obseve s even hoizon The exisence and moion of an even hoizon depend on he fom of expansion paamee

3 56 Mahemaics Today Vol7(Dec-)54-6 (ii) Paicle hoizon Suppose he vey fis phoons (ligh fon) emied by he obseve a a big-bang even ae sill aound As his ligh fon sweeps ouwad, owads moe and moe galaxies, he obseve a he big-bang and hese galaxies see each ohe fo he vey fis ime (cosmic insan) Hence, a any cosmic insan his ligh fon, called he obseve s paicle hoizon, divides all galaxies ino wo classes elaive o he obseve: hose aleady in obseve s view and all ohes (iii) Some popeies of hoizons (a) Evey galaxy, wihin A s even hoizon, excep A, evenually possess ou of i Fo if B is such a galaxy, hen A s hoizon phoon in he diecion of AB is wihin B s even hoizon, and will heefoe each B a a finie cosmic ime Tha is, when B passes ou of A s even hoizon (b) Evey galaxy B wihin A s even hoizon emains visible foeve a A Fo, he even hoizon iself bings a las view of B As B appoaches A s even hoizon in models wih infinie expansion, is hisoy, as seen a A, ges infiniely dilaed, and is ligh infiniely ed-shifed In collapsing models B s ligh ges infiniely blue-shifed as B appoaches he even hoizon (c) As galaxies ae oveaken by A s paicle hoizon, hey come ino view a A wih infinie ed-shif in big-bang models, and infinie blue-shif in models wih unlimied pas expansion (d) If a model possesses no even hoizon, evey even a evey galaxy is seen on evey galaxy Fo, an invisible even implies he exisence of even hoizon (e) If a model possesses no paicle hoizon, evey obseve if necessay by aveling fom his oiginal galaxy can be pesen a any even a any galaxy Fo, in pinciple, his only avel esicion is his fowad ligh cone a ceaion; bu ha would be a paicle hoizon if all galaxies wae no always wihin i (f) If an even hoizon exiss, wo abiay evens ae in geneal no boh knowable o one obseve, even if he avels Fo, conside wo diameically opposie evens ouside an even hoizon Thei fowad ligh cones can no inesec Bu o know eihe even means being in is fowad ligh cone (g) The even and paicle hoizons, if exis, mus coss each ohe wihin he life ime of he model Fo, he paicle hoizon was and he even hoizon will be, a he fundamenal paicles associaed wih hem (h) When a model, in which boh even and paicle hoizons exis, is un backwad in ime (ie ime evesed), he even hoizon becomes he paicle hoizon and vicevesa 3 Fomulae fo hoizons In his secion, we deive he fomulae fo boh he even and paicle hoizons Fo his pupose le us conside he mos geneal spheically symmeic line-elemen ds e d R ( d sin d ) e d () whee, R and ae funcions of and only Wih he coodinaes and suppessed in he space-ime descibed by he meic (), he equaion of moion ( - elaion ) of a phoon emied a (, ) owads he oigin galaxy a (, ) is given by / / e d e d ()

4 K Punachanda Rao - Even Hoizons In Cosmology 57 Fom () we noe ha deceases as inceases In ode o check he condiion fo exisence of hoizons, () mus be inegable and ha may be possible when, R, and ae sepaable in and Hence, wihou loss of genealiy he funcions, R, and can be sepaaed in and as shown below: R (, ) = f ( ) Ř () (3) λ (, ) = α ( ) + S ( ) (4) ν (, ) = β ( ) (5) and hus, () educes o ( )/ e d e S / d (6) If ends o a posiive limi as ends o infiniy fo a fixed,, hen he phoon fom, neve eaches he oigin, and hus, is an even beyond he even hoizon When he condiion e S / d fo an even hoizon o exis is saisfied, hen he coodinae of he even hoizon and is given by he inegal e ( )/ d e S / (7) d (8) If he model has a fuue big-bang a f, whee f > and denoes pesen ime, hen he uppe limis of he ime inegaions in boh he equaions (7) and (8) ae o be eplaced by f Similaly, when he condiion e S / S / d, o e d (9) fo a paicle hoizon o exis is saisfied, hen he coodinae of he paicle hoizon a is given by he inegal ( )/ S / S / e d e d, o e d () The ime inegals in paenhesis of (9) and () ae o be used when he definiion of S / e exends o negaively unbounded values of ime We now deduce he equaions fo boh he even and paicle hoizons in he F-R-W models by subsiuion in (3) o (5), he following: f ( ) = [ + ( k / 4 ) ] () S ( ) = log R ( ) () α ( ) = - log [ + ( k / 4 ) ] (3) β ( ) = (4) Thus, he equaions of even and paicle hoizons a, ae, especively, given by an ( / ), k, k R ( ) d, (5) log[( ) /( )], k

5 58 Mahemaics Today Vol7(Dec-)54-6 R ( ) d, o R ( ) d, (6) povided ha he condiions (7) and (9) ae especively saisfied Hee we menion few cosmological models in which eihe of, boh of o none of he hoizons exis The seady sae model has an even hoizon wheeas he Einsein desie / 3 model, in which he scale faco R ( ) α, has a paicle hoizon All he in-flexional and oscillaing non-empy isoopic models have boh he hoizons wheeas Milne s model has none 4 Even hoizons In his secion, we will pesen fomulae and sudy even hoizons of some of he models discussed in he ealie papes ( K Punachanda Rao, 998; 999; 5; 8; 9) In he space-ime model, descibed by K Punachanda Rao e al (998) ds d ( d d sin d ) (7) he equaion of moion of a phoon emied a (, ) owads he obseve a, is given by d d (8) An even hoizon exiss in (7), since he condiion fo which is saisfied Thus, he equaion of he even hoizon is given by d d d, (9) ( ) () whee we have chosen = as he lowe limi of he inegal, since = is no he cene in T models The paicle P wih = cosses ino he hoizon a ( ) [( )( )] () If a signal fom P(,), whee <, eaches he obseve O a (,) hen he elaion among,, and τ is given by () The diagammaic epesenaion of even hoizon descibed by () has been given in Appendix We now conside a moe geneal meic given by K Punachanda Rao e al (998) n ds d ( d sin d ) d (3) wih n saisfying < n < fo which he equaion of moion of a phoon fom (, ) o, is given by d n d (4)

6 K Punachanda Rao - Even Hoizons In Cosmology 59 and he equaion of even hoizon, when n, is given by d n d n n ) The paicle P wih = cosses ino he even hoizon a )] ( (5) [/( n)] [( n )( (6) and if he ligh signal fom P (, ), whee <, eaches obseve a (, τ ) hen n n n (7) I is clea ha, in he space-ime meic (3), n is elaed o he measuing quaniy ξ by n = [ ( + ξ ) / ( ξ ) ] We now wie down he equaions of even hoizons (5) in ems of ξ ( eplacing n ) which may help us in undesanding he behavio of hoizons in ems of anisoopy Thus, (5) akes he fom (8) /( ) [( )/ ] I appeas, fom (8), ha he evoluion peiod of he even hoizon is divided ino hee pas especively iniial, inemediae and final epochs As ξ inceases fom o, he following feaues of even hoizons may be obseved: (i) In he iniial epochs, he even hoizons ( wih inceasing values of ξ ) sa a lage (ii) disances and un owads he obseve wih fase aes In he inemediae epochs, he fahe hoizons ove ake hei peceding ones in a sysemaic manne Tha is, in he pocess he fahes hoizon oveakes all ohes and becomes he neaes o he obseve, he neaes hoizon allows all ohes o oveake and becomes he fahes and so on (iii) In he final epochs, wih he evesed lengh scales, ie shoe he disance of hoizon fom he obseve highe he coesponding value of ξ, he even hoizons conac wih slowe aes and collapse o he obseve ogehe a an infinie ime We have given diagammaic epesenaion of he even hoizons descibed by (8) fo ξ = /4 and 3/8 and / in Appendix, in which he above dawn conclusions ae made moe clea (iv) Small segmens of he hoizons of ξ s ae aanged o fom a coninuous cuve and his cuve will emain as he even hoizon of he obseve These segmens wih ξ gowing fom is minimum o maximum ae aanged in a sequence fom he fahes o he neaes We have given a diagam in Appendix 3 o demonsae he popey (iv) 5 Conclusions We have deived he equaions fo he even hoizons wheeve hey exis, in case of he soluions we have discussed in ou ealie papes Also, we have demonsaed hem gaphically I is well known ha in F-R-W models ε vanishes bu hoizons exis and hus he fome has no beaing on he lae Howeve, we have shown in Appendix ha ε, equivalenly ξ, is elaed wih hoizons As ε gows, he coesponding hoizon segmen deceases and hus he adius of he coesponding visible univese decease in case of even hoizon The sudy of paicle hoizons will be pesened in ou subsequen papes

7 6 Mahemaics Today Vol7(Dec-)54-6 Appendix Diagammaic epesenaion of he even hoizon () of he space ime meic (7)

8 K Punachanda Rao - Even Hoizons In Cosmology 6 Appendix Diagammaic epesenaion of he even hoizons (8) of he space-ime meic (4) coesponding o ξ = /4, 3/8, and / Appendix 3 Diagam showing ha even hoizon of he space-ime meic (4) is made up of cuved segmens coesponding o vaious values of ξ

9 6 Mahemaics Today Vol7(Dec-)54-6 Refeences Bondi H (947) Mon No Roy Ason Soc 7, 4 Kame D, Sephani H, MacCallum MAH and Hel G (98) Exac soluions of Einsein s Field Equaions, (ed) Schmuze E (Univesiy Pess: Cambidge) 3 Kishna Rao J (966) Cu Sci, 35, Kishna Rao J (97) Gen Rel Gav,, Kishna Rao J (97) J Phys (London), A5, Kishna Rao J (973) Gen Rel Gav, 4, 35 7 Kishna Rao J (99) Pamana J Phys 34, 43 8 MacCallum MAH (98) in he Oigin and Evoluion of galaxies (ed) Sabbaa VD, Wold Scienific Pub Co 9 MacCallum MAH (988) in Highlighs in gaviaion and cosmology (eds) Iye BR, Aji Kembhavi, Nalike V and Vishveshwaa CV (Univesiy Pess : Cambidge) Ome G C (949) Asophysics J, 9, 64 Punachanda Rao K e al (998) Mahs Today, XVI, 5 Punachanda Rao K (999) Mahs Today, XVII, 9 3 Punachanda Rao K (5) Mahs Today, XXI, 3 4 Punachanda Rao K (8) Mahs Today, 4, 7 5 Punachanda Rao K (9) Mahs Today, 5, 34 6 Rindle W (956) Mon No Roy Ason Soc 6, 66 7 Rindle W (977) Essenial Relaiviy : Special, Geneal, and Cosmological, Spinge Velag 8 Tolman R C (934) Po Na Acad Sci US, 69

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