`Derivation of Weinberg s Relation in a Inflationary Universe

Transcription

1 1 `Derivain f Weinberg s Relain in a Inflainary Universe Iannis Iraklis aranas Yrk Universiy Deparen f Physis and Asrny 1A Perie Building Nrh Yrk, Onari MJ-1P CANADA e ail: iannis@yrku.a Absra We prpse a derivain f he epirial Weinberg relain fr he ass f an eleenary parile and in an inflainary ype f universe. Our derivain prdues he sandard well knwn Weinberg relain fr he ass f an eleenary parile, alng wih an exra er whih depends n he inflainary penial, as well as ubble s nsan. The derivain is based n Zeldvih s resul fr he slgial nsan Λ, in he nex f quanu field hery. The exra er an be undersd as a sall rrein he ass f he eleenary parile due inflain. This er als enables us alulae, he iniial value f he field φ O fr w kinds f penials hsen, whih akes Weinberg s relain pssible. Clsed and fla and pen universes give he ass f he pariles lse he ass f a pin, 10 MeV/ r as he ne als predied by Weinberg s relain. Keywrds: inflainary slgy, quanu field hery, eleenary parile, ass f he pin, slgial nsan. 1.Inrduin I is a well knwn resul ha he ass f an eleenary parile an be bained as a binain f he fundaenal nsans f physis naely, G, 0 and h. [1] naely: 1 / = h (1) G This is knwn as he Weinberg s frula and i is purely epirial. In his paper we firs ffer a siple derivain fr i using Einsein s field equains, and in a inflainary del universe wih he help f Zeldvih s definiin f he slgial nsan Λ. The assupin fr using Zeldvih s definiin is ha here igh be a pssible relain beween he sae nsans and he definiin f he slgial nsan, in he nex f quanu field hery, saisfying als he general hery f relaiviy, and prbably iplying a relain beween irs and ars..thereial Bakgrund In an inflainary universe, he law f expansin f is radius resebles ha f he De- Sier universe. The Friedann equain in he vauu dinaed ase has as is firs sluin he equain given by he relain belw: []

2 [ ] R( = R exp () where R( is he radius f he universe a any ie, and R is se iniial radius, and finally is n he ubble s paraeer a arbirary ie unless k = 0. Fr he purpse f ur alulain, was aken be nsan. In he nep f he an inflainary slgial senari is a funin f he inflainary field φ, whih iself is a funin f ie. Therefre we have ha :[] [ φ() ] 8V [ φ() ] = () P where V(φ) is he inflainary penial funin f he inflainary field, P is he Plank ass, when naural unis are used. Fr he abve field, he dified equains in he slgial sense ake he fr given belw when he energy densiy ε and he pressure p f he si fluid an be replaed ε wih ε +ε and p wih p +p and p and ε an be defined as fllws: [] φ φ φ ε = + V ( φ), p = V ( φ), φ = () wih ε and p are respeively he energy densiy and pressure due inflainary field. Using () he field equains an be nw wrien as fllws: 8G = ( ε ε) + + Λ (5) R( 8G + = Λ ( p + p ) () () R R. (6) I is nvenien nw wrie he equains abve in ers f ubble s paraeer. Therefre we aan als have: G ( = 8 ( ε ε) + + Λ (7) G q ( ( + ( = Λ 8 ( p + p ). (8). Analysis Befre ninuing ur analysis, i wuld be gd elabrae n ur assupins in he nex f general relaiviy hery used in wriing dwn hese equains. Firs, i is assued ha we are dealing wih a regin whih is wihin he hrizn disane a he ie under nsiderain. This regin hen underges rapid expansin, being re r less independen fr he res f he universe. The eri used fr his regin is a Rbersn- Walker ype under hgeneus and isrpi spae. We als ignre he spaial variain f he field φ, whih bees unifr in value all ver his regin. The value

3 f urvaure k = 0 has been used in he line eleen whih f urse iplies a fla spaial geery. Fr sipliiy, we le ε = p = 0. Using () we an nw subra (6), (5) and he new values f ε and p we bain equain: R R () () = G φ Using () we subsiue in (9) fr () R, R we finally bain: G φ = 0 (10) whih iplies ha: φ = ns = φ (11) (). Nex fr (8) upn subsiuin f he derivaives R, R we bain: G 8G [ φ() ] = Λ φ () + V [ φ() ] (1) Slving f he slgial nsan Λ we have: 8GV Λ = φ (1) Using Zeldvih s relain As a nex sep, we will ake use f Zeldvih s resul [5], where he bains an expressin fr he slgial nsan Λ fr he energy ensr f a plarized vauu in he quanized hery f fields. Zeldvih gives he fllwing expressin: 6 G Λ =. (1) h Subsiuing in equain (1) in (1) and slving fr he ass f he eleenary parile we have: 1/ GV φ h 8 =. (15) G Observing (1) we see ha he firs par f he RS equain is he well knwn Weinberg s relain dified during inflain by an exra er whih invlves he inflainary penial V(φ ) and ubble s paraeer (φ ), bh f he alulaed a he iniial value φ f he salar field φ. Sebdy an rerieve Weinberg s relain if he send parenhesis in he RS bees ne. This is pssible when he salar field φ akes iniial values φ whih an be alulaed fr he hie f differen penials V(φ). There wul als be a value f he ubble paraeer given by (16) fr he value = GV φ (16) whih he riginal Weinberg relain an be rerieved. (9)

4 5 Exaine Weinberg s relain fr w differen inflainary penials Nex we will exaine Weinberg s relain fr w differen kinds f inflainary penials, naely a assive salar field and als a self-ineraing salar field given by:[6] V ( φ) = φ = φ h (17) λ λh V ( φ) = φ = φ where λ is diensinless nsan and has he diensins f ass. Fr equain (10) we an nw use he fa ha φ( = φ se iniial value f he field. S using(15) and (10) alng wih (1) we have afer subsiuin in (1) ha: h 8G 1 = φ G h h 8 Gλhφ = (19) G We an nw see ha he ass f he eleenary parile depends n he iniial value f he inflainary penial φ and he ubble paraeer (φ ). If he send square brake n he RS bees equal ne hen sebdy exaly erieves he undified r riginal definiin by Weinberg f an eleenary parile s ass. This urs when: φ = h G 1/ 1/ φ = λgh fr he firs and send penial respeivelly. 6. The ase f k = 1 r lsed universe The k = 1 ase rrespnds a lsed Friedan vauu dinaed universe whih evlves arding he law. Therefre he field equains an be wrien as fllws: 8G + = ε + Λ (1) R 1 8G + Λ p + = () R ( Subsiuing in (1) and () as befre he expressins fr ε and p and subraing (1) fr () we bain he fllwing equain: (18) (0)

5 5 R ( R( 1 G + φ ( R( + = R(. () R ( In he k = 1 ase he radius f he universe evlves arding he law:[7] [8] 1 R ( = s(. () If we nw subsiue () in () we again bain ha: φ ( = 0 r φ( = ns = φ (5) Using () and subsiuing in () we bain afer siplifying ha: 8G Λ = [ [ ] + sh φ V φ (6) Bu in any slgial del = 1, and herefre (6) finally bees: G Λ =.0 8 V φ. (7) Using nex Zeldvih s definiin f he slgial nsan we bain: 1/ GV φ.0 h 8 = G whih iners f he w penials bee: h G =.0 φ G h h φ G =.0 λhφ G φ Again as befre we an rerieve Weinberg s riginal relain if fr exaple: 1/ h φ = / (8) (9) (0) (1) φ = 1.08 () λgh 7. The ase f k = -1 r pen universe This ase rrespnds an pen Friedan vauu dinaed universe whih evlves arding he law: [9] 1 R ( = sinh[ ], ()f fllwing he sae seps as befre we bain he fllwing equain fr he ass f he eleenary parile: = and again: G 1/ GV φ.76 h 8, ()

6 6 h G =.76 φ G h h φ G =.76 λhφ G φ (5). (6) As befre Weinberg s relain fr he ass f he eleenary parile an be rerived if: 1/ h φ = 0.18 (7) φ = 0.89 λh 1/ 8 Nuerial alulains fr all ases T ge an esiae fr he ass f an eleenary parile in differen universes and in njunin wih Zeldvih s relain we will use relain (15) fr eah del universe. Fr ha we hse = 1/ = 1/10-5 se -1. Firs assue ha he penial energy densiy V(0) f he field be equal he quanu densiy f aer 5 9 ρ quanu = = g /, and als as a send value G h ρ riial = V(0) = 10-9 g/ - when = 1/ =10-17 se -1. Tha an be based n a njeure ha is reenly prpsed ha he urren expansin f he universe is erely a deayed sae f inflain. Therefre we bain: Case k = 0 8 = = g = plank (9) 5 = 1.01 = g Case k =1 8 = 1.7 = g = plank (0) 5 = 1.7 = g Case k = -1 8 = 1.16 = g = plank (1) = 1.16 = g ere we us ne ha he ass f he pin is 10 MeV/ = g where he ne alulaed using Weinberg s relain and fr = se -1 is 60 Mev/ = g [10]. Fr (9), (0), (1) we an als see ha during he inflainary perid (15) predis a ass f he rder f 10 - Plank and als a ass whih is lse he ass f he pin bserved day and shwn abve. Pariles an be reaed when he field φ sars sillaing near he iniu f V(φ), is energy is ransferred he pariles as a resul f hese sillains. The pariles hen reaed llide wih ne anher, and apprah a sae f herdynai equilibriu. (8)

7 7 Cnlusins In ur paper, an expressin f Weinberg s epirial relain fr he ass an eleenary parile has been derived using Einsein s field equains in an vauu dinaed inflainary universe wih a nnzer slgial nsan. Fr ha, Zeldvih s definiin f he slgial nsan in he hery f quanized vauu was used. The fa ha Weinberg s relain an nw be prven via Einsein s field equains akes he resul n an epirial ne. S Zeldvih s definiin f he slgial nsan uld pin a pssible relain beween irs and ars and in a re fudaenal way. The expressin derived fr he ass f he eleenary parile is nw dified, by an exra er whih furher depends n he inflainary penial V(φ ) alulaed a an iniial value f he inflainary field φ. Tw well knwn inflainary penials were used, naely a asive salar field, and a self ineraing salar field. Expressins fr he asses were alulaed a he iniial value f he field φ. Fr bh fields, expressins fr he values φ were given suh ha he riginal and undified Weinberg relain an be rerieved. All alulains were dne in a fla, k = 0, lsed, k = 1, and pen k= -1, universe. Finally esiaes fr he asses f he eleenary pariles were given fr he hree pssible del universes during he era when inflain akes plae, and als a he presen era where he expansin f he universe an be hugh as a deayed sae f inflain. Referenes [1] S. Weinberg, Graviain and Cslgy, Wiley, 197, p [] J., A., Peak, Cslgial Physis, Cabridge Universiy Press, 1999, p. 6. [] A. D. Linde, Parile Physis and Inflainary Cslgy, New Yrk, arwd, 1990, p.5. [] ] J. N. Isla, An Inrduin Maheaial Cslgy, Cabridge Universiy Press, 199, p. 18. [5] Ya. B. Zel dvih, Svie Physis;, Uspeki, 1968, 11, p.81. [6] ] J. Bernsein, An Inrduin Cslgy, Prenie all, 1995, p. 17. [7] J., A., Peak, p. i., p. 6. [8] A. D. Linde, p. i., p.. [9] A. D. Linde, p. i., p.. [10] S. Weinberg, p. i., p. 60. [] R. Tenreir and M. Quirs, An Inrduin Cslgy and Parile Physis, Wrd-Sienifi, 1998, p.10. [5] T. Fuaase and K. Maeda, Physis Review D, 9, 1989, p.99.