lc ols No n Enstn-Guss-onnt-ln Systm I Inducn II Modl (cn sc quns oundy conons) III lc ols (ozon dus ft sngulty) V Summy Ts To (Os Insttut of Tcnology JAPAN) Nouyos Ot (Kn Unvsty JAPAN) T wo ws suppotd y JSPS KAKENI (79)
Inducn No G quntum gvty? sngulty? Stng oy? Ely Unvs? Effctv oy of supstng oy (tc stng Mtsv Tsytln 87) S κ d x [ g ( ) F { ( G µν gµν) µ ν ( ) ( ) ( µνσρ µν σρ µν µν ) ( µ ν µν ) ] ( ) µνλ ν ρ ρσλ µ σ 8 µν µν ( ) } - Supstng oy fundmntl oy ow lc ols ly unvs dscd n suc oy? - Cn unsolvd polm suc s sngulty d ngy solvd n suc oy? - Wll stng oy vfd y osvn /o xpmnts?
Inducn No W wll study - lc ol solun n ffctv stng oy wt Guss-onnt tm dln g od tm - comp m wt pvously nown soluns suc s oulw-s solun solun wt modl wtout g od tm of dln Guo Ot To 8 W fnd tt - T dln fld ffcts stuctu of lc ol soluns muc - T low lmt fo mss ( ozon dus) of solun n - T ft sngulty pps t non-zo dus n - T g od tm of dln fld dos not ffct soluns so muc xcpt fo -dm cs - Fo cgd solun no xtm solun wn dln fld ncludd
Inducn No Pvous wos out stngy Stngy lc ols wt U() Gug Fld A) Spclly Symmtc lnc /o Axonc lc ol K Md G W Gons NP98 7 (988) Gfnl G T oowtz A Smng P (99) A Sp S Tvd F Wlcz Mod PLA 77 (99) S Gddngs A Smng P 7 (99) M Cvtc Youm PL7 (99) A Cmoo K Vd??? ) Mssv ln Mssv Axon o Cosmologcl Constnt T J Alln M J owc A Ll PL7 7 (99) Ggoy J A vy P7 (99) S J Poltt L Wlts P 7 (99) L Wlts g-qc/98 (99) S J Poltt J Twmly L Wlts P 7 (99) C) ottng lnc /o Axonc lc ol A Cmpll t l PL no (99) A Sn PL 9 no7 (99) S Mgnm N Stwt PL98 99 (99) A Gc' Gltsov O Kcn PL 7 7 (99) ) Supdnc fom lnc lc ol K S??? (99) J Kog K Md WU-AP//9 (99) Stngy lc ols wt Non-Aln Gug Fld A) SU() lnc lc ol G Lvlsvl Mon PL9 7 (99) E E onts V Gl'tsov PL (99) G Lvlsvl Mon NP 7 (99) T To K Md P8 no (99) Y M Co t l PL8 no (99) ) SU() lnc lc ol E E onts V Gl'tsov PL 9 (99) Klus J Kunz A Sood PL7 no (99) C) Mssv ln Axon C M O'Nll P no 8 (99) Stngy lc ols wt g Cuvtu Tm A) G oulw S s PL (98) pologcl lc ols G C Pys v () 8 G C Y S Myung Y Z Zng P () 89 G C PL 8 () 7 T To Md P7 () T To Md P7 () 7 ) G oulw S s PL7 no 9 (98) V G Clln C Mys M J Py NP 7 S Mgnm N Stwt P7 no 9 (99) A Cmpll N Klop K A Olv PL8 99 (99) A Cmpll t l NP99 7 (99) S Mgnm P no 9 (99) M Cmpnll C O Lous J Ausc g-qc/9 (99) C) E E onts V Gt'tsov PL (99) P Knt t l P no8 9 (99) S O Alxyv M V Pomznov p-t/9 (99) T To Ym K Md P no 79 (997) P Knt K Tmv PL9 (997) C M Cn V Gl'tsov G Olov P7 8 (7) C M Cn V Gl'tsov G Olov P78 (8) Z K Guo N Ot T To PTP 8 (8) Z K Guo N Ot T To PTP (9) N Ot T ToPTP (9) 99 N Ot T To PTP (9) 77 N Ot T To PTP (9) 77 N Ot T To PTP () 7 K Md N Ot Y Ssgw P8 (9) -Loop Effct P Knt K Tmv IOA-/9 (99)
fm In ognl dvn of ffctv cn t ws ft dvd n fomgvtnl w κ Enstn -dmnsonl constnt µνσρ µν µ ν w tt fv-dmnsonl lc v nctv ( ) n qut µν fm numcl popts dlnc ntpt nfnd n stng oy n tnsfomd stng Also t convnnt In( ) 9 wols nvstgt modynmc of lc wt slop p σρ [] µν ols µν coffcnt gvn n tms of gg Noffctv ' ognl Enstn soluns fm SoT w t tnsfom t Enstn fm duc dmnsons us mnsonl cpcty ofn non-dlnc lc cocn In dvn of G tm gn w fnd tt G fv-dmnsonl lc ols v qut nctv µν µ νν gρσλ µ ( ) ] notn n mguty δg sgn [ up n tnsfom µν µν [ ]( S-mtx clculn stng oy ( ) postv µνλ fo σ nsoluns µν ) cpcty () ρ g mss ut cngs smll mss fvµν popts fom o-dmnsonl T t of non-dlnc lc 8 tms ou sults n Enstn fm So w tnsfom t owv t lwys ngtv s mss vd f dln ol ngtv fo lg mss ut cngs sgndfnn postv fo smll ' mguty fld δgµν mss nµν fv µ κ -dmnsonl gvtnl constnt dln fld F gug fld -fom /8 nsons w o tn fv t cpcty lwys ngtv fo ot g dvtv tms dmnsonl soluns owv t lwys ngtv s mss vd f dln λ γ γ µνρσ µν numcl g gvn ( ) ntms off Gpmt µ( )Λ µνρσ () d x coffcnt gg slop µν G κ on-dlnc lc ols Scn dvotd conclusons fld ddd In dmnsons o tn fv t cpcty lwys ngtv fo ot Acn In ognl dvn of ffctv cn t ws ft quns G cocn dvd n fm fom Enstn [] ( ) d tx g ndlnc non-dlnc lc ols Scn S dvotd conclusons S-mtx clculn stng oy n tnsfomd n stngfm Also convnnt κ w lnc v st- cus w loo fo soluns n wc t fld zo T constnt µ fld dfnn oy ntpt ou Enstn-G sults n Enstn fm So w tnsfom t n Enstn fm duc dmnsons dffnc of soluns wt wtout g od tm of dln fld W v us fld dfnn mguty δgµν [ gµν {oy µν µenstn-g ( ) }] δ lnc µ v n w νγ / w stt cus w loo fld cosmologcl tm wt study symptclly flt soluns dln couplng λ toug w gug [ ( ) ] [ ] otn up g od tms ffctv cn fo tc owng low-ngy nducd s dffnc of soluns wt の論文の式 () の Gµν のところは Gµν µ ν stng: ' wt dln coup g od tms W consdt lso ncludd cosmologcl tm followng low-ngy ffctv cn fo tc stng: pocss doppd llowd cus ffctv low-ngy cn cn γ γ Λ Λ d x g ( ) pp () computd の S µ t d x fom g scttng ( ) ( ) F γ G stng µ( ) Gµν のところは ()Gµν d off G 笹川君の論文の式 pκ fld dfnn wn mpltuds n os κ g ov pocss γ In doppd ffctv cn ws ognlly computd ns fm All t mns tt no tms Enstn d x g ( ) od () µ G κ wc systm µνρσ clm mnd up fld dfnn wn t d off µνon uvtu dln fld d fom of cn f y ltd t wy cnnot tt W µνρσ µν G w γ / w v st cus w loo fo soluns n wc t fld compu ( )zo Not tt Also not tt ffctv µνρσ cn ws µν ognlly m8πg () w dln G µnducd of -dmnsonl gvtnl constnt cuvtu /8 (> ) Tfld of fld quns so tscl pocdu lgtmt constnt s fdffnc µνρσ µν solun solutly pfd fom of cn y ltd mtc fld stngt od soluns wt wtout g tm of dln fld W v lso ncludd cosmologcl tm G tm () gg slop pmt G comnn gvtnl constnt /8 (> ) t gvn n tms of gg slop pmtκ study systm 8πG -dmnsonl M pp Λ γ/(γλ) wt dln couplng flt soluns n t δ λ toug w study symptclly Lt gg us consd mtc g numcl coffcnt gvn tms slop pmt fld stngt () pocss fld od n F f llowd ng constnt of ds T dmguty n coc of of dln In ov tms doppd T cus ffctv low-ngy cn cn - T stng tstng nvstgtd Md Ot Ssgw 9 n δos of γ fm wn constnt of y dln fldmpltuds T mguty coc mnd upmodl fldofn dfnn d off fom scttng computd n stng mdmnsonl ducn systm ncouplng fm ds d δ ds d Also tt ffctv cn ognlly Enstn All t mns tt no fm t constnt w ft m dmnsonl ducn offm systm n stng /( computd ) T n ng not Enstn fm w would gtws γif systm tt W solutly pfd fom of cn f y ltd t wy on cnnot clm wc Enstn fm w would gt γ /( ) T s:γotn t pysndcp study n f fo () dmnsons owv f wn go Enstn ft cng 8) systm Gmg () ˆ )Gm() ) δ S d x g ( f w ( : t gotcp S G coc fo xmpl γ n f 8) fo owv ft go Enstn Elcnc consd mtc fld κ stngt s wt γlt us n m dmnsonl ducn w gt ddss: otn t pysndcp Elcnc ddss: t gotcp n tn dmnsons γ dmnsonl ducn w gt nsons ot cocs v fm own ut convnnt δt wt n m () ds gt d F f cocs v own gt convnnt n ny dmnsons ot w γ λt() ou nlys of lc ols vlu n nyγ dmnsons So n t pp Λ ut t followng sm vlu foofou of Gllon lc olsmodfd n ny dmnsons nt coos γ -T mnly n s ou study nlys lc ol modl sm popts of gvty So n t pp w t scond vwpont coos γ mnly n ou followng of lc ol (ffyt Espos-Fs Vlmn 9) ow Elcnc sults dpnd on t coc owv w lso xmn study ( ( ) ) ddss: otn t pysndcp Elcnc ddss: t gotcp od s ow fo ot vlus of γ soluns n In fou dmnsons n sults T dpnd on t coc owv w lso xmn lc fowll ot tt vlus of γ n fou dmnsons n T constncy of ou sults wt f ol 8) soluns In fct w fnd M µ µ () lso svs cc constncy of ou sults wt f 8) In fct w wll fnd tt t sml vos ltoug som dffnc n ls In Modl
EOM oundy conons No sc Equns [ ] - G µν [ µ ν ] g µν( ) [ γ µν (γ ρ σ γ ρ σ )P µρνσ ] µ µ ν ( ) 8( ) ( ( F µ Fν F βf ) β - γγ F [γ γ G µγ( µ ν µ µ γ( ) )] - µ ( g γ F µν) G µν µν g µν [ ] µν µν µρ ρ ν ρσ µρνσ µ ρσλ νρσλ g µνg P µνρσ µνρσ g µ[σ ρ]ν g ν[ρ σ]µ g µ[ρ g σ]ν oundy conon - T xtnc of gul lc ol ozon - gulty of domn of out communcns - Asymptc fltnss t sptl nfnty ( )
γ tm γwt dlnλ symptclly ncludd F cosmologcl couplng λ toug w study flt sol Λ lso ( ) γ µ( ) () ( ) ( ) ( ) δ γ G ν w 笹川君の論文の式 G ypsufc wt const lmnt of µν( )-dmnsonl psnts pp () の ln Gµν のところは µ N o 7 )( ) volum Σ fo ± In ov pocss g tms doppd T llowd cus ffctv low-n od fld zo T constnt µ γ cust w fld loo fo soluns n wc t ( )γ fom γ d offfom Φ 8 scttng δmpltuds γ δcomputd followng Eq () mndquns up fld dfnn wn t n st uns wt wtout g od tm of dln fld W v - All t mns t Also not tt ffctv cn ws computd nn Enstn fm ognllyflt dln couplng λ toug w study symptclly soluns t ( γ ) on ltd f y wy γ γ solutly pfd fom of cn t cnnot clm wc systm ( ( δγ )( µ ν 9) ( ) ) γ ( ( ) のところは Gµν ( ) Φ( δ ) f δ study systm () Sttc spclly symmtc spctm: ms doppd T llowd cus ffctv low-ngy cn cn Lt us consd mtc fld stngt ( )[( ) ( )] n t d off fom scttng mpltuds computd n stng os δγ γ f µ ( ) γγ γ λ ( ) gnlly computd n Enstn fm t mns µ Allδ Λ tt no ds d F f W y ltd t wy on cnnot clm wc systm tt γ γ δ ( ) ( ) γ ( )( γ ) µ mtc funcns Φ δ ngt ( ) ) δ γδ 8( ) M ( ( ) γ( ) ( ) ( δ ) ( ) ± ( ) δ Elcnc ()constnt T fld qun fo () dwstudy otn t pysndcp F f flt ddss: Elctc cg: In tpp symptclly soluns cosmologcl ( ) wtout 8( ) cg Elcnc ddss: t gotcp gv xwll fld (7) sly ntgtd γ δ δ EOM oundy conons ( δ )( ) γ f µ ) ( γδ ' δ f ( δγ µ δ γ cp w constnt cospondng cg So ou ts ducd sttng oundy conons fo δγ ust l ov st of quns pvous pps [ ] lds δ dln ntgt n ou (f ) Sclng 8( ) M ( ) ( 9) ( m ± ) () ( ) 8( ) w w v dfnd dmnsonlss vls: / Λ Λ pms n fl T fld vls cn scld y oundy conons symptc vos dvtvs wt spct Nmly w msu ou lngtn unt of In wt fol lso () 8( ) M ( ) on vls fo smplcty W v dfnd ) ± non-dlnc γ ( oulw-s solun: At ozon w sould v ( ) 8( )( ( m)n ( m)( m )( m ) ( n) 8( ) ) ( M γ δ ) ± Λ γ/(γλ) ( ( )γ M ( ) ( ) () ( ) 8( ) w γ ( ) ( ) γ ) δ δ ( ds d 8(( ) )M ( )
EOM oundy conons No 8 - T soluns otnd y ntgtng EOM fom ozon nfnty numclly - At ozon () qun of dln fld coms sngul ( ) sngul t ozons To cov gulty w mpos ( conon: ) ( ( ) c F () [ Cγ { ( ) { C( ) [ C( ) ( ) c C ( ) ( )γ ( ) } C ( )γ [C( ) ] ( ) }] C( ) C( )γ [C( )( ) ] C( ) ( ) {C ( )( ) γ [( ) C C ] [C( ) ( )][C( ) ] } {C( )γ [C ( ) ( 9)C ] [C( ) ] } C( ) 9 γ ] { } C( )[C( )( ) ] ( ) ( ) γ 8 ( ) γ[( )C C ] C ( )γ () X : vlutd t ozon
T dcmnnt - > : no stcn fo pmts llowd gon cf Md-Ot-Ssgw 9 ngtv dcmnnt ^8 < γgm*pê / T ozontl x dcmnnt of Eq () postv n T ozontl x γ / - : T gon w dcmnnt coms ngtv no lc ol solun No 9-8 -7 - - - - - - Coctd: p Out[] EOM oundy conons 7 ( C ) [ C 8 C γ C(C )γ C (C ) (C ) Cγ γ 9C (C C ) (C FI FIG : T gon w dcmnnt of Eq γ vtcl x < ^8-Lgm*pê
( )γ ( ) γ N o A Acn sc quns ( )γ ( ) γ ng low-ngy ffctv cn fo )tc stng n on scm [ ]: U () ( ) ( ) δ ( fld: of dln µν µν Confgun g ( ) F G* g µ ν ( ) ( ) γ ( ) ( ) ( ) δ µ ν ( µν ) ( ) µνσρ µν σρ ( )*- µν ( µν () '()* *- ' γ ( )γ γ Φ 8 δ γ µν ) () () µνλ ν ρ ρσλ σ µ µ - ( µ ( ) 8 µν (lu) µ µ ( γ ( ) δ Φ( δ ) - γ/(γλ) ( ) () M Λ ( ) sonl gvtnl constnt dln fld F gug fld -fom /8 - µνρσ µν µν γ/(γλ) n n tms of gg slop pmt µνρσ G M Λ λ ( δ γ ds d ( ) Λ ft µ µ dvd () fom gnl dvn of ffctv cn tµ ws n Enstn fm -8 (d) ng oy n tnsfomd n stng fm [] Also t convnnt ntpt Gm() Gm () g δ γ/(γλ) tnsfom t n () n fm So w Enstn fm duc dmnsons us ) - δ M Λ ( Φ δ ds ( d guty δgµν µν µ ν gµν [ _tu ( ) ] [ ] otn up p -dmnsons (nutl) 7) symptclly (wtout In t pp wstudy flt soluns cosmologcl constnt T dsλ γ λ δ d () Gmg () ' Mxwll fld (7) Gm() sly ntgtd gv δ -dln場の値は小さく 傾きも緩やかになっている Gm() Gmg () λ γ δ (8) Λ µ( ) d x g ( ) γ γδ F () G f () ( cospondng M (ou 9) ts ducd w constnt cg So sttng Λ γ λ (7) Λ γ λ ( v st cus w loo fo soluns n wc t fld zo T constnt µ o flds δ dln ntgt ov st of quns ust l n ou pvous nc of soluns wt wtout g od tm of dln fld W v
n ty constnt (T typos n ou pvous pps [ scond tm; y sould v sgnγ) λ T cngs mg Λ usd gnt soluns fo dffnt cosmologcl constnt γ / ()γ ful consd svl sclng symmts of ou fld quns ( ld quns () () v sft symmty:gmg () Gm() d No δ ds (δ ) d ds δ d C Symmty sclng γ/(γλ) ) (M Λ ozon dus sngulty 9 s -dmnsons (nutl) M Λ γ/(γλ) γ/(γλ) M Λ oundy cono -tm n Eq () dos not contut t w stct ou consdn cs Comnng t w qun mnng : ( ) C ( )γ [C( ) ] ( ) ) C( ) C( )γ [C( )( ) ] C( ) C ( )( ) γ [( ) C µ µ C ] [C ) C( )γ [C ( ) ( 9)C ] [C( /Sgm _pys s )( ) ] ( ) ( ) ( )γ C( )[C( ( )γ C (_pys ) C ( ) _pys () s Gm() Gmg () Gm() Gmg () δ δ 9 9 ( 7) ( )ozon dus 8 ( ) 8 8 7 8 () Λ γ λ ( ) 9 M_pys () µ µ µ µ () Λ γ λ ( 8) () γ/(γλ) M Λ γ/(γλ) M Λ 8 8 oulw-s 7 7 9 M_pys 7 () M_pys M_pys ( ) µ ( ) ( 9) ft sngulty µ 7 ( ) M Λ γ/(γλ) µ () 7 7 7 δ 8 8 )C C ] 8 _pys 8 _pys _pys 9 δm_pys M_pys µ (8) λλ > ) γ[( < (ds) () Λ γ λ <(8) (ds) λλ > /Sgm v dfnd - T ft sngulty pps t non-zo dus t t ds ) M d M Λ γ/(γλ) ds( δ δ ( ) ds ds d d ( ) Gm() Gmg () δ δ - T no lc ol solun low ctcl mss (ozon dus) ds d d Gm() Gm () Gm() Gm () g ( ) δ g δ ( ) M ( ) 7 (μ ) M 7 (μ ) Gm() G δ Λ γ λ on T gulty conon t ozon λλ < Λ γ λ (ds) (ds) λλ < Λ γ λ ( 7) 8 8 >( 7) > M
µ () oulw-s δ ds d M Λ γ/(γλ) M_pys M_pys MGm Gm() g () δ δ ds d () () () () µ µ µ µ ( 9) cntl sngulty 8 M_pys M_pys () M 8 8) ( - Fo ctcl solun λλ dvgs t som dus n doc > (ds) < > (ds) Λ γ λ ( 7) () - T ft sngulty pps fo lg solun M ( 9) λλ > < (ds) < (ds) (8) 7 7 p_pys ) γ[( S/Sgm t t v dfnd 7) μ cs) - T lmt fo ozon dus ( (cf ) ( low Λ γ λ Gm() Gmg () δ M 9 8 8 () n ty constnt (T typos n ou pvous pps [ scond tm; y sould v sgnγ) λ T cngs mg Λ usd gnt soluns fo dffnt cosmologcl constnt µ γ / ()γ M Λ () (8) Λ γftλsngulty γ/(γλ) Gmg () δ () p_pys ( ) )C C ] () µ Λ γ λ _pys µ () δ Gm() Gmg () Gm() δ Scwzscld ( 7) ( ) ( ) ful consd svl sclng symmts of ou fld quns ( ld quns () () v sft symmty:gmg () Gm() () δ No 7 δ ds ( ) d ds ds δ d d γ/(γλ) ) (M Λ M Λ γ/(γλ) C Symmty sclng s -dmnsons (nutl) ozon dus sngulty γ/(γλ) M Λ oundy cono -tm n Eq () dos not contut t w stct ou consdn cs Comnng t w qun mnng : ( ) C ( )γ [C( ) ] ( ) ) C( ) C( )γ [C( )( ) ] C( ) C ( )( ) γ [( ) C µ µ C ] [C ) C( )γ [C ( ) ( 9)C ] [C( /Sgm _pys s )( ) ] ( ) ( ) ( )γ C( )[C( ( )γ C (_pys ) C ( ) _pys () s λλ < _pys _pys λλ: > -: dgm FIG FIG - dgm VII LACK OLE
-dmnsons (nutl) No ozon dus sngulty µ µ oulw-s 8 M_pys M - T lc ol solun fo ny mss - T sngulty locts t cnt s non-dlnc soluns (cntl sngulty)
Cgd Soluns (-) No ozon dus -/ -/ -/ '()**( oulw-s µ µ M_pys M nn ozon 8 - T no lc ol solun low ctcl mss (ozon dus) - T ctcl solun not xtm lmt (cf oulw-s solun)