Università degli Studi di Napoli Federico II, Largo S. Marcellino, Napoli, Italy

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1 O o athatcal cocto btw th Cyclc Uv Iflatoay Uv -ac Iflato -ac cooloy a vaou cto of Nub hoy Mchl Nall Datto Scz lla a Uvtà l Stu Naol Fco II Lao S. Macllo Naol Italy Datto Matatca Alcazo R. Caccool Uvtà l Stu Naol Fco II Polo ll Scz ll colo Mot S. Alo Va Cta (Fuootta 6 Naol Italy Abtact h a a vw a th of o tt ult that ha b obta vaou ach coc th ba collo t a M-thoy (Cyclc Uv -ac flato a -ac cooloy. I th Scto w hav cb o uato coc coc voluto a Cyclc Uv. I th Scto w hav cb o uato coc th coolocal tubato a B Cuch/B Ba ac-t th M-thoy ol of a B Cuch/B Ba tato a o uato coc th oluto of a bawol B Cuch/B Ba Cooloy. I th Scto w hav cb o uato coc th at Eyotc cuvatu tubato bfo th B Ba o uato coc th ffctv fv-oal thoy of th toly coul htotc t a a au vo of N fvoal uavty wth fou-oal boua a o uato coc th coll ba a th o of th Hot B Ba. I th Scto w hav cb o uato a th ull y coto volato coc th flatoay ol a o uato coc th voluto to a ooth uv a yotc cotact ha wth w >. I th Scto w hav cb o uato coc th aoxat flatoay oluto oll away fo th utabl axu of -ac t thoy. I th Scto 6 w hav cb vaou uato coc th -ac uac ol zta t zta olocal cala fl a -ac a alc uatu cooloy. I th Scto 7 w hav how vaou a tt athatcal cocto btw o uato coc th -ac Iflato th -ac uatu cooloy th zta t a th ba collo t a M-thoy. Futho ach cto w hav how th athatcal cocto wth vaou cto of Nub hoy cally th Raauja oula uato th Aua Rato a th Fboacc ub. Dcat to th oy of Pofo Aatol Alxvch Kaatuba (97- athatcal u who oal ach a fo alway ouc of w at ato

2 . O o uato coc coc voluto a Cyclc Uv. [] [6] h acto fo a cala fl coul to avty a a t of flu a N x Uv wth l lt ρ a hooou flat ( aβ Σ ρ a V S x N a' a ' N. (. W u to t cofoal t a to t vatv wth ct to. N th la fucto. h bacou oluto fo th cala fl ot ( a V th cala ottal. h uato of oto fo avty th att a cala fl a tahtfowaly v by vay (. wth ct to a N a aft whch N ay b t ual to uty. Ex t of o t t th Et uato a H a&& a π G & V β ρr β ρ (. π G & V β ρr β ρ (. M M wh a ot a o t vatv. a -la th Fa cotat a Wth a th tajctoy th a a ' 6 ( aβ ρ ( a a V ( a'. (.b Now w olv th uato of oto atly bfo a aft th bouc. Bfo th bouc th a lttl aato t c t ha b xotally lut th V bco c utc-oat acclat ha. Futho th ottal a' lbl a u off to u fty. h Fa cotat a ' a th a 6 cala fl uato ( a ' ' wh ot cofoal t vatv. h al oluto / 6 l( AH a A A AH A AH ( A AH ( a ( a (.

3 / 6 wh. W choo to b th t wh a vah o that < bfo collo. A a tato cotat whch coul b t to uty by cal ac-t cooat but t covt ot to o o. h Hubbl cotat a f t of th ba cal facto a a ' / a a a ' / a whch at ta th valu H ( a H ( ctvly. R-x th cala fl a a fucto of o t t a w obta < h tato cotat l H ( t. (. H ha a atual hycal ttato a a au of th cotacto at of th xta-o. W b that wh th ba aato all o ca u th uual foula fo Kaluza-Kl thoy y (.b wh th fou-oal l lt y th ffth atal cooat whch u fo zo to L a L a aat wth th o of lth. hc w hav that: H L & (.6 Lt t wh L L th o lth of th xta o L a aat wth o of lth a t th o t th fv-oal tc t a 6 6 t (.7 wth t b FRW o t. Notc that a hft ca alway b coat fo by a cal of L. A th xta o h to zo H t to a cotat H (. Iatly aft th bouc cala tc y oat a H a aly cotat. h 6 tc y of th cala fl cal a a a aato cal a a o th fo oat at all a. It covt to -cal a o that t uty at cala tc y-aato ualty t a ot th coo Hubbl cotat H. h Fa cotat. (.b th a ( a' H ( a (. a th oluto

4 H l H ( out H a H H. (.9 h ba cal facto a / 6 / 6 H 6 a a A H H out / 6 / 6 H 6 a a A H H out. (. ( H th cotat A H / H ( out 6 ha b f o that w atch a a a to th co oluto v (.. A fo th co oluto w ca cout th Hubbl cotat o th two ba aft collo. hy a ± H out H H o th otv a atv to ba ctvly. Fo H < H th ca of latvly lttl aato oucto atly aft collo a xa but a cotact. Wha fo H H > both ba cal facto xa aft collo. If o cala ottal t th cala fl woul cotu to oby th oluto (.9 cov to ( out H C l. (. H V w h valu actually la tha fo H < H th ca of wa oucto of aato. Howv th c of th ottal V alt th xo (. fo th fal t valu of th cala fl. A co th ottal wll tavll th otv cto H uc to a oalz valu Hˆ ( out < H( out o that th fal t valu of th cala fl ca b all tha. If th th ca th a v co zo ta v to xao hotly aft aato oac. If aato oac occu wll aft ha co th ottal wll. (. ov a aoabl tat fo th fal t valu f w u th coct valu Hˆ ( out. h c of (. ly utoo: whl th Uv tc y oat a ow at t a ca loathcally wth t. Howv wh th Uv bco aato oat a a t Hubbl a ca a cov to th ft lt abov. Wth a th. (.6-. w ot th follow cocto wth ub thoy:

5 / / / / 6 / / 7 7 Φ ( Φ 6 / 7; / 7 7 Φ ( Φ / 6 ; / 7 7 Φ ( Φ / 6 ; / 7 7 Φ ( Φ / ; / 7 7 Φ ( Φ 7 / 6 Not that wh a a th o that coo to th hycal vbato of a ut a th hycal vbato of th booc t. H w hav u th follow xo: ( Φ / 7 wth Φ that th Aua ato a atual ub a 7 a th coactf o of th M- hoy.. U th follow ottal V c V ( F (. w co th oto of bac a foth aco th ottal wll. V ay b accuatly aoxat by c V. Fo th u xotal ottal th a l cal oluto ( t a t V V c ( t (. c whch a xa o cotact Uv oluto acco to whth t otv o atv. A th of th xa ha of th cyclc cao th a o of acclat xao whch a th Uv ty hooou a flat follow by oll ow th ottal V to th wll. Aft ha oll uffctly a th cal facto ha bu to cotact th Uv accuatly follow th abov cal oluto ow th wll utl cout th ottal u. Lt u co th bhavou of u all hft th cotact ha. I th bacou cala fl uato a th Fa uato w t B δ a H H B δh wh B a H B a th bacou uatt v fo (.. o la o δ o obta δ & δ & δ (. t t wth two laly t oluto δ t a t wh <<. I th cotact ha th fo oluto ow a t t to zo. Howv th oluto ly a ftal hft th t to th B Cuch: δ &. W xt th co a outo collo vlocty whch w hav aatz a H ( a H ( out. Wth th cal oluto (. w ca calculat th valu of co vlocty by tat th facto of th ottal F. (. a a Hav fucto whch uty fo > a zo fo < wh th valu of at th u of th ottal. W cout th vlocty of th fl a t aoach a u y covato at th ju V to f th vlocty aft co. I th cal oluto th total y a

6 aoach fo th ht & V t valuat fo jut to th lft of u a acco to. (.6 a th ut ual th total y & & 6 / t 6 V / at th. Hc w f that H V c 6c (. (. Not that fo th. (. w obta: [ H c V ( 6c ( ] wh th ub coct wth th o that coo to th hycal vbato of a ut by th follow Raauja fucto: coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w'. At th bouc th oluto atch to a xa oluto wth ( out ( H ( > H χ (.6 wh χ a all aat whch a bcau of th latcty of th collo. W hall ly au a all otv χ v a follow th voluto fowa t. Sc χ all th outo oluto vy aly th t v of th co oluto a tat bac aco th ottal wll aft th bouc: th cal oluto v (. but wth t otv. W ca tat χ a a tubato a u th oluto. (. cu abov δ t a t. O ca tahtfowaly cout th tubato δ H th ow o by atch at a bfo. O f valu at th u. Byo th ot δ H ow a bacou cal oluto H cay wth a fl ha atta th valu B δ H χh / c wh 6 / t c / B H th bacou c fo la c wha th. h atu occu wh th cala 6

7 c χ l V V c χ c D. (.7 A a byo D th tc y ovwhl th atv ottal a th fl a oto th latau V wth H aly cotat a ual to Hˆ c c χ (. χ ( out H ( utl th aato att a vacuu y bco fcat a H th a away to zo. Not that w ca wt th. (. a follow: 6 Hˆ ( out c χ χ 6 c c V 6c. (.b Alo th uato lat wth th ub.. wth th o that coo to th hycal vbato of a ut by th follow Raauja fucto: coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w' a wth th ub ( / that lat to th hycal vbato of th booc t by th follow Raauja fucto: coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w'. h t t to th lft of th ottal wll < tally tcal th co a outo ta fo χ << aly c t. (.9 V 7

8 Fo th outo oluto wh ha lft th cal oluto but bfo aato oato th fto. (.6 ay b tat to v th t c th B Ba at ach valu of t & Hˆ Alo th uato ca b wtt a follow: Hˆ ( out. (. t & Hˆ / c χ χ 6 c c V 6c. (.b h t. (. a cohycal cal. h coo foula fo th t bfo th B Cuch vy fft. I th cal oluto (. o ha fo la c t ( / c( c / 6 t. (. V c c h la xotal facto a th t to th B Cuch fa lo tha th t fo th B Ba fo ach valu of. h ffct u to th ca H aft th bouc whch tu u to th otv valu of χ. A th cala fl a byo th ottal wll t u oto th otv latau V. h valu of H ( out aly cacll th aa aco th ottal wll a uc to Ĥ v. (.. Oc aato oato b th fl ucly cov to th la t (Hubbl-a lt of. (.9 aly C l H ˆ out / H (. wh H th Hubbl au at tc-aato ualty. Alo th. (. ca b wtt a follow C l c χ χ 6 c V c 6c / H. (.b h c obvou: th aytotc valu of o th ato of H ( out Ica ( out ˆ to H. H ˆ uh futh lw low H lay aato oato allow th loathc owth of th tc y oat ha to cotu fo lo. h oluto of th cala fl uato aft xa. (.9 fo la covt to o t t a a atch & H t ta V (. a a ( t

9 wh a abov w f a ( t to b uty at tc-aato ual ty. W hav that ay ach t axal valu ax a tu aou u th aato att o utc oat och. Fo xal ax ach th aato a f fo. (. t ax t V < C t t V wh t th t of att oato. Fo tu aou th att a w u 6 t V C t V ( ( (.. (. cc Fally f th fl u to vy la C o that V / V ( C c xotally all th oly tu aou th utc-oat a. Fo ou cao to b vabl w u th to b a ubtatal och of vacuu y oato (flato bfo th xt B Cuch. h ub of -fol N of flato v by uual low-oll foula N cc V (.6 V c fo ou ol ottal. Fo xal f w a that th ub of bayo Hubbl au b lut to blow uty bfo th xt cotacto whch ctaly ov-ll uaat N that th cyclc oluto a attacto w t o N 6. h aly fulfll f C of o uty Plac ut. Hc th. (.6 ca b wtt a follow: N c V V c C 6. (.6b Wth a th. (.-..6b w hav th follow athatcal cocto wth th Aua ato: 9 / 7 7 Φ ( Φ / ; / 7 7 Φ ( Φ / ; 7 ( Φ / 967 ; ac ( π / 7 acco 76 9 π 7 7 Φ 9 / Φ / ; / 7 acco π ( Φ 76 ; 66 ; ( Φ 967 ; ; ;

10 π 7 ( Φ / 76 ; ac ( Fo th foula v abov w ca alo calculat th axal valu C th cyclc oluto: fo la c a fo t >> χ t t t l C χ (.7 t wh w u H t th b of th aato-oat och. Fo. (.7 w obta t t c N c V V. (. χ h uato ov a low bou o t. h xt ca to ta V. h u V c N 6 w f t co. I th ca th axu tatu of th Uv GV. h ot vy fft to what o f l flatoay ol. W hav how that a cyclc uv oluto xt ov w a allow to a thouh th Et-fa ulaty acco to th atch coto. (. a (.6. Scfcally w au that H( out ( χ H( wh χ a o-atv cotat coo to ba who latv aft collo at tha o ual to th latv bfo collo. Ou aut how that fo ach χ th a uu valu of H ( out that fctly cyclc. Now w how that a ca vlocty fctly coatbl wth y a otu covato a collo btw a otv a atv to ba ov a at ty of aato at o th atv to ba. W hall au that all oth xta o a oul a fx a th bul ac-t btw th ba ttl ow to a tatc tat aft th collo. W hall ta th t of aato o th ba aft collo a b v. By o Ial atch both tal a fal tat a wll a covato of total y a otu w hall b abl to coltly fx th tat of th outo ba a atcula th xao at of th xta o H ( out t of H (. h tal tat of ty ba wth to a a wth coo vloct v < a v > oby v v ; E tot ; v v v v P tot. (.9 v v h ft uato follow fo Ial atch o th two ba a th aoach a uat th th ba cal facto. h co a th uato a th fto of th total

11 y a otu. h th uato (.9 ly that th co ty tat ha v v E a that th total otu tot ( LH P tot < (. ( LH( wh w tfy v (. w ca wt th. (. alo a follow: v wth th cotacto of th ffth o ( LH. Fo th. P tot L c V 6c L V c 6c <. (.b h coo uato fo th outo tat a aly obta by lac wth ρ fo th otv to ba a wth ρ fo th atv to ba au th t of aato ouc at th collo o ach ba ρ a ρ ctvly a v fo a cohycal calculato a a both otv. Wt v ± ( out tah( θ ± wh θ ± a th aocat at o obta two oluto h ( θ Ptot Ptot ; h ± Ptot Ptot ( θ (. wh ρ ρ wth ρ a ρ th t of aato o th otv a atv to ba ctvly aft collo. Both ρ a th ft oluto wth ( th vloct of th otv a atv to ba a th a aft th collo a thy w bfo t. I th co wth ( th otv to ba cotu th atv y cto but th atv to ba alo ov th atv y cto. h coo valu fo v ± ( out a V a ρ a au to b otv. I v ( out P tot tot Ptot ( ( P ( P tot v ( out ± P tot tot Ptot ( ( P ( P tot V P tot Ptot ( Ptot ( ( /( P ( tot o V P tot P ( Ptot ( P ( tot tot (. wh th ft oluto fo V hol fo th ( ca a th co fo th ( ca. W a tt th latv of th ba th outo tat c that v th xao at of th xta o v ( out v( out LH( out coa to th latv LH th co tat. W f th ( oluto v

12 H H ( out ( v ( out v ( out v P tot Ptot (. ( P ( tot a th ( oluto H H ( out ( ( Ptot P tot tot ( P ( P tot. (. wth P tot v by (. both ca. W ot that w ca wt th lato abov to.. v ( out v( out LH( out alo a follow: ( out v( out LH ( out L ( χ v c V 6c. (.b At th ot w to co how th t of aato ρ a ρ o th latv of aoach of th ba. At vy low LH ( << o xct th out ba collo to b aly aabatc a a xotally all aout of aato to b ouc. h ( oluto ha th of both ba aly ual bfo a aft collo: w au that t th oluto ath tha th ( oluto whch al th low vlocty lt. A LH ( ca w xct ρ a ρ to ow. Now f w co ρ a ρ to b both << P tot << th th co t th oato oat. If o aato ouc o th atv to ba ρ > ρ th H H ( out ( ( ( ρ χ (.c ρ a o χ all a otv. h th coto cay to obta cyclc bhavou. Cocvably th ba to ca cha fo to ' t at collo. h w obta ( χ ( ρ ρ t. (. Fo th ( oluto w ca tahtfowaly t a u lt fo H ( out / H( ( χ. Co fo xal th ca th th ba to ucha at collo t. h xo (. v H ( out / H( a a fucto of a P tot. It att at fx a P tot wh P tot a ual to P tot ( t allt valu. Fo wh ualty hol. Fo P tot < t axz fo P tot t axz wh t allt valu a P tot wh t ual to 7. h o tha

13 ouh fo u to obta th all valu of χ to a th cyclc cao wo. A ucto ba to at collo t > futh ca th axal valu of th ato. o obta cyclc bhavou w χ to b cotat fo bouc to bouc. hat coa to th to bfo collo th factoal cha to a th factoal oucto of aato ut b cotat. W ot that fo 7 w hav th follow athatcal cocto wth th Aua ato: 7 / 7 / 7 Φ Φ ; 7 7 / 7 9 / 7 Φ Φ O o uato coc coolocal tubato a B Cuch/B Ba ac-t a M-hoy ol of a B Cuch/B Ba tato. [] [] [] [] [6] W co a otv o atv to ba wth coolocal yty but whch ov thouh th fv-oal bul. h oto thouh th wa bul uc xao o cotacto of th cal facto o th ba. h cal facto o th ba oby a of Fa uato ρ± K C H ± ± ρ ± (. 6 M L 6M b b wh ρ ± th ty (ot clu th to of att o aato cof to th ba b ± th ba cal facto a H ± th uc Hubbl cotat o th otv (atv to ba. Choo cofoal t o ach ba a lct th ρ t uato (. bco b ' b Kb C M L ρ b ' C b Kb M L ρ. (. wh ot cofoal t vatv. h coo acclato uato fo b '' a b '' fo whch C aa a v by fftat uato (. a u ( b b ( ρ Pb ρ wth P b th u of att o aato o th b. W ow how that th two uato ca b v fo a l acto ov w uat th cofoal t o ach ba. Co th acto S tn x [ M L( N b' Kb ρ b L( N b' Kb ρ b ] ± ± (. wh N a la fucto touc to a th acto t aatzato vaat. Vay wth ct to b ± a th tt N v th coct acclato uato fo b '' a b '' follow fo (.. h uato a uvalt to (. u to two tato cotat.

14 W wt th acto (. t of a fou-oal ffctv cal facto a a a cala fl f by acoh b b ah. 6 6 Claly a a tafo a a cal facto a a a cala fl u cal of th atal cooat x. o tt o hycally ot that fo tatc ba th bul ac-t Y / L fct At- Stt ac wth l lt Y ( t x. h aato btw th ba v by a coth Ll Ll a 6 o t fo zo to fty a t fo u fty to zo. I t of a a th acto (. bco S t x M L( a& Ka a & S (. whch coz a th acto fo Et avty wth l lt a (( t t γ x x j γ j b th caocal tc o H S o j E wth cuvatu K a a ally coul cala fl. h att acto S covtoal xct that th cal facto aa ot th Et-fa cal facto but ta b acoh( / 6 a b ah( / 6 o th otv a atv to ba ctvly. Now w wh to a u of two vy owful cl. h ft th ato that v th abc of yty th low y o of th fv-oal thoy houl b cbabl wth a fou-oal ffctv acto. h co that c th oal thoy wa cooat vaat th fou oal ffctv acto ut b cooat vaat too. Sc th fvoal thoy local a caual t aoabl to xct th ot th fouoal thoy. If futho th lato btw th fou-oal uc tc o th ba a th fou-oal fl local th covaac lu at wth th abov ult foc th lato to b ( / 6 ( ( / 6 coh µν µν µν h µν. (. Wh w coul att to th ba tc th xo houl t th acto fo att cof to th otv a atv to ba ctvly. Lw w ca fo (. a covaac atly f th ffctv acto fo th fou-oal thoy: M S x R µ (.6 ( S [ ] S [ ] wh w hav f th ffctv fou-oal Plac a M ( G M L π. h two ba ot a t acco to th foula (. a th bacou oluto lvat ot-collo au to cot of two flat aalll ba wth aato

15 t ρ ±. h coo fou-oal ffctv thoy ha aato ty ρ a a al cala fl wth tc y ty ρ. h fou-oal Fa uato cofoal t th a A a' ( ρ a ρa A (.7 a wh w hav f th cotat A a a u th fact that th al cala tc 6 y ρ a. h oluto to (.7 a th al cala fl uato ( a ' ' : ( a A l A (. (. Fo th oluto w cotuct th cal facto o th ba acco to (. obta: b ± ± A (.9 o w that wth th choc of oalzato fo th cal facto a a (.7 th ba cal facto a uty at collo. W ay ow ctly coa th cto (.9 wth th xact fv-oal oluto uat th t la to obta A ( / L ( L tah ( / y ( ( y / L (. tah wh V tah y a ± th valu of th aato ty ρ ± o ach ba at collo. hc th. (.9 ca b wtt alo: y th aty aocat wth th latv vlocty of th ba at collo b ± ( L ( / L tah( y / L ± tah( y /. (.b Futho w f th factoal ty atch o th two ba a f (. o that w hav f tah( y /. (. L Now w cb th tubato of th ba-wol yt t of th fou-oal ffctv thoy. W hall ow cb th cala tubato lotual au wth a atally flat bacou wh th cal facto a th cala fl a v by (.. h tub l lt a Φ Ψ x. (. [ ]

16 Sc th a o aotoc t th laz thoy w hav Φ Ψ. A colt t of tubato uato cot of th aato flu uato th cala fl uato of oto a th Et otu cotat: ( v Φ' δ ' v δ Φ δ '' H δ ' δ ' Φ' H Φ a ρ v ' ( δ (. ' Φ wh ot vatv δ th factoal tubato th aato ty v th cala ottal fo t vlocty.. v v δ th tubato th cala fl a fo (. w hav th bacou uatt a' H a ( [ ( ] a '. [ ( ] W a tt olv th uato th lo wavlth lt <<. Solv all th abov uato fo δ l a o f δ w δ ( ( w... N (. fo aabatc tubato. h coot of th bacou y ty th fouoal ffctv thoy a cala tc y wth w a aato wth w. It follow that fo aabatc tubato at lo wavlth w ut hav δ δ. (.6 I lotual au th factoal y ty tubato a th vlocty ottal tubato th cala fl (co a a flu wth w a v by ( δ ' δ Φ ' δ v. (.7 ' Fo th uato (. abov (a u ' a δ ' t follow that δ δ v. (. ' Mata th aabatcty coto (.6 u to o ( th u that th factoal vlocty tubato fo th cala fl a th aato houl b ual: th aato vlocty t of δ th otu cotat th yl v δ /'. Ex 6

17 δ ( Φ ' HΦ ρ (.9 ρ ' wh ρ ' a. h abov uato ay b u to t th la t a xao of all th uatt of tt about th ulaty. W hall choo to aatz th xo t of th aat cb th coov y ty tubato ε H Φ whch ha th follow xao about : ( ε E( ε ε D (. wh ε a ε a abtay cotat a D ( l... (... Fo aabatc tubato w obta E. (. 9 δ ε l ε O( l (.a v ε ( O( l δ δ O( l v v O( l Φ ε l ε O( l (.b ( δ ε ( l ε O( l 6 (.c ζ M ε ε ( 6 l O( l (. wh ζ M th cuvatu tubato o coov lc touc by Muhaov. Wth a th (.c w ot that obl th follow athatcal cocto wth th Aua ato: 6 9 / 7 7 ( Φ / Futho th. (.b-.c th th ub that lat to th o that coo to th hycal vbato of a ut by th follow Raauja fucto 7

18 coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w' a that a a Fboacc ub. Now w co th oaato of tc tubato thouh a collo of tol ba C wh th bacou ac-t cly M / Z R. h fo that w ta fo th fvoal coolocal bacou tc j ( t y( t t y b ( t y δ x x (. j a w wt th ot al cala tc tubato about th a ( y x t t βyx ( t y ( Φ t Wty t ( Γ ( t y( Ψ δ χ x x j b j j. (. Fo tubato o M C R t tahtfowa to f a au whch th tc ta th fo j χ ( t t y χ δ j j χ x x (. a χ atf a al cala uato of oto o C M R. o b c th au β Γ Φ Ψ χ Φ χ Ψ χ W. (.6 Notc that th o-zo vaabl ca all b lat to χ acco to Γ. (.7 ( Φ Ψ χ W hall hcfoth f to th a th Ml ato coto. Futho o th Z yty w obta Nua bouay coto o χ y ± χ ' (. wh y ± ± y / a th locato of th two Z fx ot. I th ol ac-t th lowt y o fo χ y -t a ha th aytotc fo ( t y Q Pl t χ (.9

19 wth Q a P b abtay cotat. W hav th follow lato: P Q out Q γ l out P P. (. h lato a uffct to t th tc fluctuato aft th bouc. W a oly tt th lo-wavlth at of th ctu a fo th ca of tt P u by coa to Q. A a ult w obta th aoxat atch ul Q P out P. (. out Q h y coto (.6 thouh (. a atf cly fo all t a coactf Ml o Z bacou. Now w wh to u th fou-oal ffctv (oul thoy to f th bouay ata fo th fv oal bul tubato. I ay fou-oal au th fou-oal tc tubato h µν a cala fl tubato δ t th uc tc tubato o th ba va th foula (.: ( lω µν ± h hµν ± µν δ (. wh Ω coh( / 6 a h( / 6 Ω a th tc tubato a factoal.. ± ± δµν a hµν δµν b± hµν. h foula atculaly ay to u fv-oal lotual au. h au ay alway b cho a t coltly au fx. I th au th fv-oal tc ta th fo j ( y b ( t y ( Ψ δ x x ( t y ( Φ t W ty t ( Γ L L L L j. (. I th abc of aotoc t th ba tajcto a utub th au. A at couc that th fou-oal lotual au cala tubato vaabl Φ ± a Ψ ± cb tubato of th uc oty o ach ba ( x ( Φ ( ± b± ± ± ± Ψ± (. a cly th bouay valu of th fv-oal lotual au tubato Φ y Ψ y. U (. a (. w f fo th uc tubato ± Φ L a L ± ± Ψ ± Φ Φ tah δ Ψ Φ tah δ Φ Φ coth δ Ψ Φ coth δ. ( h ba cofoal t ay b x t of t by tat t t ± y (.6 ( t ± 9

20 wh b /. So fo xal th bouay valu of th bul tc tubato Φ L o th otv to ba v xlctly by Φ L ( t y Φ ( t y ( t tah( ( ( t y t / 6 ( ( t y t δ (.7 6 wh y th locato of th otv to ba. Alo th uato (.-.7 w hav th follow cocto wth th Aua ato: 6 9 / 7 7 ( Φ / Now u (. a th follow uato ( y / [ ( y h( y L ( ( y h( y ] O( L t t t ρ ( y / ε tah Ψ( t ζ M ( t l ± L coh (. to f Q a P bfo a aft th bouc fo all coot of th tc tubato a atch acco to th ul v uato (. ult ζ M ht two aat cal-vaat lo wavlth cotbuto th ot-ulaty tat. h ft occu a a ct couc of th cha (. a t of th aout of aato at at th ulaty. h co ootoal to th ffc th t of th aato o th two ba. At la o vloct w hav ζ ε 6 L ( ε Vout V V O( V V L ρ L M out ± ± (.9 wh V a V out a th latv vloct of th ba bfo a aft collo. Not that c P ε atch P fact uvalt to atch ε aco th collo. I t of fouoal aat clu v (. f th abuac of th aato a th factoal ty atch f f (. w f aa at la o vloct ( V V ε ε fv out ζ M out. (. 6 L L h th fal ult lvat to tac tubato aco th ulaty th yotc a cyclc ol. h ult fo th lo wavlth cuvatu tubato altu th fou-oal ffctv thoy oaat to th hot B Ba aft th ba collo : θ / ε Vcoll θ hθ θ / 6 L 9ε tah ζ M (. 6 L coh

21 wh θ th aty coo to th latv V coll of th ba at collo a th ε co foula au V coll all. L th bul cuvatu cal a ha a cal vaat 6 ow ctu. Wth a th. (. a (. w hav th follow athatcal cocto wth th Aua ato a th Fboacc ub: 6 6 ; 6 that a Fboacc ub; ; 6 ( ( wth a that a Fboacc ub; 7 ( Φ / 6. Futho th. (. a a Fboacc ub a coct wth th o that coo to th hycal vbato of a ut by th follow Raauja fucto: coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w'. h -oal ac-t w co a ct ouct of -oal Eucla ac R a a two-oal t-t ac-t ow a coactf Ml ac-t o M. h l lt fo M R thu C C t t θ x θ θ < t < (. wh x a Eucla cooat o R θ aatz th coact o a t th t. h coact o ay th b a ccl whch ca w tfy θ wth θ θ o a Z obfol whch ca w tfy θ wth θ θ a futh tfy θ wth θ θ. h fx ot θ a θ θ a th tt a tol Z -ba aoach at aty θ coll at t to - wth th a latv aty. h obfol ucto th ca of tt th yotc/cyclc ol oally otvat by th cotucto of htotc M thoy fo lv oal uavty. I th ol th bouay ba o ozo to. Howv th to a uboat ffct a t a th ba collo locally wll-oll by M C R. Now co a t loo of au R M thoy fa. It a M πr t th ffctv t to µ L wh L th z of th xta o. h ffctv Et-fa avtatoal coul v by κ κ / L. h avtatoal ottal ouc by uch a loo act o :

22 M Φ κ (. ( A R D wh A D th aa of th ut D -h A D π / Γ( ( D /. Scalz to th ca of tt aly -ba lv-oal M thoy th to µ lat to th lv oal avtatoal coul by a uatzato coto lat to th fou-fo flux a µ π κ (. / wth a t. Euato (. a (. th ly that th tycal avtatoal ottal aou a t loo 6 Φ ( µ R θ / (. 6 6πµ R u to ucal facto. Wth a th (. w ot that a obl th follow athatcal cocto wth th Aua ato a th Fboacc ub: 9 6 ; ; 6 6 ( ( ; a a 9 a Fboacc ub. Futho w hav that / 7 / ; ( Φ ( Φ hc th avtatoal ottal o th cal of th loo of o θ a thfo cottly all fo all collo aty. Sc th a aato of th loo wh thy a ouc of o th z R th ottal Φ th tycal avtatoal ottal thouhout ac. Multly th tt coot of th bacou tc (. by Φ a f t w coclu that th outo tc ha a xao aty of o θ( Cθ wth C a cotat of o uty. W coclu that fo all θ th avtatoal bac-acto u to t loo oucto all. / 7 / 7. O o uato coc th oluto of a bawol B Cuch/B Ba Cooloy. W hall loy a cooat yt whch th fv-oal l lt fo th bacou ta th fo ( t y( t t y b ( t y x (.6 wh y aatz th ffth o a x th th o-coact o. Coolocal otoy xclu tx o yx t a hooty u a b a t of x. h t y at of th bacou tc ay th b ta to b cofoally flat.

23 W f t lt to wo cooat whch th ba locato a fx but th bul volv. h bul tc thfo v by (.6 wth th ba locato fx at y ± y fo all t t. h fv-oal oluto th ha to atfy both th Et uato a th ν ν Ial atch coto o th ba. h bul Et uato a G Λ wh th bul coolocal cotat ( / G G w f G µ δ µ Λ 6/ L. Evaluat th la cobato G a G β wh ν ν β yy β β y (.7 ν ν yy ( β y β (. t / L lb β a ν l( t / L. h Ial atch coto o th ba a b y y t b L (.9 wh all uatt a to b valuat at th ba locato y ± y. Now w x th tc a a of Dchlt o Nua olyoal y a y bou at o by a cotat t y uch that th atf th Ial atch coto xactly at vy o y. o lt th w ft cha vaabl fo b a to tho oby Nua bouay coto. Fo (.9 b / Nua. Lw f w f N ( t y by t (. N t y th o ca aly chc that ( t y y N alo Nua o th ba. Sc N a b / oby Nua bouay coto o th ba w ca xa both a ow N N ( t N ( t P ( y / ( t ( t P ( y wh P ( y a olyoal P ( y y y y b (.... (. atfy Nua bouay coto a ach bou by P ( y < y /( fo th lvat a of y. Not that th t-t coffct th aatz ay alo b xa a a ow y. By cotucto ou aatz atf th Ial atch coto xactly at ach o th xao. Subttut th aatz (. to th bacou Et uato (.7 a (. w ay t th oluto o by o th aty y. At ach o y o cally obta a ub of laly t albac uato a at ot o oay fftal uato t. h ft fw t of th oluto a

24 ( 9t... N ty t y (. N t y... (. t 6 7 a 7 t y t t y... (. t y... (.6. Wth a th. (.-. w ot that a obl th follow athatcal cocto wth th Aua ato a th Fboacc ub: / 7 7 ( Φ 9 / 697 / 7 7 ( Φ / 7 Futho 7 ( a a coct wth th o that coo to th hycal vbato of th booc t a to th hycal vbato of a ut by th follow Raauja fucto:. ; coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w' coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w'. o calculat th aff tac btw th ba alo a acl oc w ut olv th oc uato th bul. Lt u ft co th tuato Bhoff-fa cooat fo whch th bul tc tatc a th ba a ov. h Bhoff-fa tc ta th fo ( Y A ( Y x Y N (.7 wh fo Schwazchl-AS wth a hozo at Y

25 A coh ( Y / L Y ( Y coh( Y / L ( Y / L ( Y / L ( Y / L ( Y / L coh h N coh h. (. At th Y -cooat of th ba t by th aat Y ; th ubut tajcto Y ± ( ca th b t by tat th Ial atch coto whch a ( Y / L ± V wh V ( Y / / N a th o of th otv- a tah ± Y ± ± ± ± atv-to ba ctvly. Fo th t futh follow that Y lat to th aty y of th collo by tah y ch( Y / L. Fo th uo of au th tac btw th ba a atual choc to u acl oc that a othooal to th fou talatoal Kll vcto of th tatc bul coo to hft x a. a th x a cooat to b fx alo th oc th w f that Y cotat fo a aff aat alo th oc. o a th cocto to ou oal ba-tatc cooat yt call that th tc fucto b t y A Y a thu Y ( bb t bb y wh w hav touc th cotat aat w hav t y ( t t y (.9 b θ θ tah y V / c. Aot y ow a th aff ( b b ( b θ t b b b t ( b b t ( t b θ (.6 y t y wh t to b a ow a a fucto of y. W ca olv th uato o by o y u th aatz y y y t c y (.6 wh th cotat c a thlv y. U th oluto fo th bacou oty v th. (.-(.6 a o th bouay coto that t ( y t w obta t y 7 ( t 6t y t 7t 6 t ( t 79t y t t y t y O( y c t t y t 9t t t 6 c t y t y t y y O y 6 (.6 t t( 6t y t ( t t y 6 c t y t t y O( y (.6 t y t ( t y c O( y (.6 t t 7t t y c y O( y (.66 6 (.6

26 ( y 6t y c O ( t c 6 O( y (.6 7 c 7 O( y. (.69 Subttut t x / y a y ωy x ω x / y O y.. to lowt o y th oc a tajcto of cotat t ly olly alo th ω cto. Hc th lt th aff a tc aato of th ba f wth th follow uato w f x x ( y Ll O( y L ω O ωx x (.7 ut caly a. o chc th th aff tac btw th ba v by a L y 6 ( t t y ( t t 9t y ( t t y t t' y t y t y y ( t t t 79t y O( y (.7 whch to lowt o y uc to 7 x x x a x O x 6 O y (.7 L 6 at wth th xao of (.7. W obta alo th follow uato: δ L x ξ b ( x x x ξ ( x ~ x ω ( x ~ ~ ~ ~ ( A J ( x B Y ( x ( A J ( x B Y ( x ( x. (.7b o valuat th tubato δ a th aff tac btw th ba co ν µ ν µ ν µ ν κ µ ν µ ν x& δ δ µν δ ( δ δ δ ν x x& x& µν x& x& µν x& x& µν κ x x& x& µν x& x& ρ σ ρ σ ρ σ ρσ x& x& ρσ x& x& ρσ x& x& (.7 wh ot cat fftato wth ct to th aff aat a o to th co l w hav tat by at a a u of th bacou oc uato µ ν µ & x σ µν σ x& x& a th cotat ν µν x& x&. If th ot of th oc o th ba a utub th xo futh lf by th vah of th ufac t. Covt to cooat wh t x / y a y ωy to lowt o y th utub

27 oc l uly th ω cto a o th tub aff tac tcal to th follow tub tc tac δ y x xξ tγly y ω. (.7 L b Exlctly w f δ a L ( B At ( ( y B t A t 9t y ( B At y t t ( 69t ( ( ( t At t 7t At t B 7t 6 y y B t 6 B t 6 6 ( t 966t 7t At ( t 97t 6t 7 y O( y whch ubttut x / y t a o t of O y uc to ~ ~ δ B B 9 ~ 7 ~ a 7 Ax x Ax Bx Ax Bx Ax O x L x 96 (.7 (.76 ~ wh B By. Alo th xo accoac wth th xao of (.7. Howv th tub aff a tc tac o ot a at O. Wth a th. (.7 a (.76 w hav th follow athatcal cocto wth Aua ato a wth Fboacc ub: / 7 / 7 / 7 7 Φ ( Φ / 6677 / 7 7 Φ ( Φ / y 6 / 7 / 7 / 7 6 / 7 7 Φ Φ ( Φ / 7 7 / 7 7 Φ ( Φ / ; / 7 / 7 / 7 7 / 7 / 7 ; / 7 7 / 7 ; / 7 / 7 / 7 7 / 7. / 7 7 Φ ( Φ 7 / / 7 7 Φ ( Φ / ( Φ / ( Φ 7 / 7 ; 7

28 Futho w hav that: ; 6; wh a 6 a Fboacc ub; 9 ; a a Fboacc ub. ; 7 ; ; wh. O o uato coc th at yotc cuvatu tubato bfo th B Ba. [6] [7] [] [6] Wth a th yotc tubato clu avty w co th acto fo N coul fl tact oly thouh avty: N N x R ( V ( (. wh w hav cho ut whch π G M Pl. I a flat Fa-Robto-Wal bacou wth l lt t a ( t x th cala fl a Fa uato a v by & H & V (. a wh H a& / a H & V ( (. a ( / V V wth o uato l. Aoth uful lato H & &. (. c If all th fl hav atv xotal ottal Et-cala uato at th cal oluto a ( t l( At V V th a wll-ow th A V. (. c c c hu f c >> fo all w hav a vy lowly cotact uv wth <<. W focu o th toy tubato c th a local au-vaat uatty a o th ca of oly two cala fl. h toy tubato uato

29 9 V V V V V δ δ & & & & & & & & & & & & (.6 flat act lac by Φ θ δ δ δ & & & & & & & & & & & & & & && a V V V V V a H. (.7 Fo lcty w wll focu attto o taht l tajcto cala fl ac. Sc θ & th toy tubato ot ouc by th Nwtoa ottal Φ a w ca olv th uato ath ly. W hall au that th bacou oluto oby cal yty o that γ & &. Dot vatv wth a touc th -cal toy fl a S δ δ (.. (.7 bco '' ' ' S V a a a S δ δ. (.9 h cucal t ov th ctu of th tubato th ' ' a V a a. (. Wh th uatty aoxatly w wll aa t aly cal-vaat tubato. It cutoay to f th uatty H H w γ ε & & &. (. I th bacou cal oluto γ ε c. (. W oc by valuat th uatty (. a xao v ow of ε a t vatv wth ct to N wh a a N / l wh a th valu of a at th of th yotc ha. Not that N ca a th fl oll owhll a th cotact yotc ha oc. W obta th ft t (. by fftat (. obta ε ' ' H a a a. (. h co t (. fou by fftat (. twc wth ct to t a u th bacou uato a th fto of N. W obta

30 Fally to x ( a '/ a ah. (. a V a H ε 6ε ε N O ε H t of th cofoal t. Fo (. w obta H ' H ε (. whch tat to ( ε H. (.6 Now t ( / u th tal a u tato by at w ca -wt th a H ε ( ε ε '. (.7 ε U th a ocu oc o th tal th xo ca b wtt a ( ε ε ' ( ε ( ε ' ε '. (. ε Now u th fact that ε ' Hε N a that to la o / ε H ca b lac by t valu th cal oluto (wth cotat ε H ε w ca -wt th co t o th htha a ε N ( ε ( ε ' ( ε (.9 ε ε whch how that th t of o / ε a ca thu b lct. Altoth w obta N ε N H ( ε ε. (. ε ε U (. a (. wth (. w ca calculat th cucal t t th toy tubato uato a'' a V ε N a. (. ε ε h vato fo cal-vaac th ctal x of th toy tubato th v by ε N. (. ε ε h ft t o th ht-ha th avtatoal cotbuto whch b otv t to a th ctu blu. h co t th o-avtatoal cotbuto whch t to a th ctu.

31 Now f R to b th cuvatu tubato o coov atal lc fo N cala fl wth al Kahl tc j o cala fl ac th laz Et-cala fl uato la to H D j R & j Ψ (. H& Dt a wh th N toy tubato j & & l & & j (. δ l δ a jut th coot of δ othooal to th bacou tajctoy a th oato D / Dt jut th oc oato o cala fl ac. h lfy bcau th cala fl ac flat o th tc j δj a D / Dt uc to a oay t vatv. Co oly two cala fl w hav & δ / & & & & δ /. (. & Fo a taht l tajctoy fl ac th ht-ha of (. vah v f th toy tubato ozo. W au that th cala fl bouc occu aft th yotc ottal a tu off o that th uv tc-oat fo th ot-of-vw. h cala fl tajctoy & ~ γ & fo t < tb a & ~ γ & fo t > tb wth & cotat a atv th vcty of th bouc. h bouc la to a lta fucto o th ht-ha of (. D δ & b Dt (.6 ( t t b ( t wh t b th t of th bouc of th atv-to ba. A ca b aly fo (. f th toy tubato alay hav acu a cal-vaat ctu by th t t b th th bouc la to th tataou covo to cuvatu tubato wth cly th a lo wavlth ctu. W ca tat th altu of th ult cuvatu tubato by tat uato (. u (.6. Sc w hav au th uv H / t. Sc th toy tubato tc-oat at th t (& δ & δ & & δ (.7 / caocally oalz t ctu v by δ h (. π t u to o-cal vaat cocto. h xo oly hol a lo a th yotc bhavou tll uway: th yotc ha at a t t aoxatly v by V / ( c t. Aft t th toy tubato oby & δ t & δ whch ha th oluto δ A Bl ( t.

32 Match th oluto to th ow o oluto t th yotc ha o f that by t b th toy ow by a atoal facto of l( t / t b. Eloy th Fa uato to lat & ~ γ & to H utt vyth toth a to th Plac a w f fo th vaac of th atal cuvatu tubato th cal-vaat ca R c V ~ γ h ( l( t / tb R (.9 π M Pl ( ~ γ fo th fctly cal-vaat ca. Notc that th ult oly loathcally o t b : th a c o th u valu of th ffctv ottal a th aat c... Io th loath Obvato o th cut Hubbl hozo cat 9 (.9 th u cv M Pl o aoxatly th GU cal. h of cou tly cott wth th htotc M-thoy. Wth a th. (.9 w hav th follow athatcal cocto wth th Aua ato: 6 / 7 77 π hav that π 7 ( Φ 6 / 9 / M Pl 7 ( Φ / 7966 R. If w ta M th w Pl. Futho w obta ac ( 7966 acco( π π wth a Fboacc ub. If th toc tubato a uly covt to cuvatu tubato th cuvatu tubato ht th ctal tlt v (.. W ow b by -x. (. t of N th ub of -fol bfo th of th yotc ha (wh N ( ε N a ε >> : lε. (. ε N h xo tcal to th ca of th Nwtoa ottal tubato xct that th ft t ha th oot. I th xo ε (N au th uato of tat u th yotc ha whch ut ca fo a valu uch at tha uty to a valu of o uty th lat N -fol. If w tat ε N th th ctal tlt N. (. N H w that th of th tlt tv to. Fo aly xotal ottal ( th ctal tlt / N. lhtly blu bcau th ft t oat. Howv th a wll-otvat xal whch th uato of tat o ot ca laly wth N. W hav touc to aatz th ca. If >. th ctal tlt. Fo xal

33 .97 fo. h xal t th a that ca b achv fo th tocallyuc cuvatu tubato th lt ol ouhly.97 <.. Fo coao f w u th a tat ocu fo th Nwtoa ottal fluctuato th cyclc ol (au thy covt to cuvatu fluctuato bfo th bouc thouh ffct w obta.9 < <. 97. h a a wth th tat obta by a t aaly ba o tuy flato ottal ctly. Wth a th valu of a. w hav th follow athatcal cocto wth th Aua ato: 67 / 7 < 67 / ( Φ ; / 7 / ( Φ ; / 7 / ( Φ ; A co way of aalyz th ctal tlt to au a fo fo th cala fl ottal. Co th ca wh th two fl hav t ottal that ca b oll a ( V c a & γ &. h. (. bco V ( γ c Pl c (. c M wh w hav u th fact that c ha th o of v a a to th facto of Plac a. h c of M Pl claly cat that th ft t o th ht a avtatoal t. It alo th c that a a blu cotbuto to th ctal tlt. h co t th o-avtatoal t a a cly wth th follow flat ac-t ult c (. c althouh th at ot at all obvou at tat t of th calculato. Fo a u xotal ottal whch ha c th o-avtatoal cotbuto zo a th ctu lhtly blu. Fo laubl valu of c a γ / ay th avtatoal c about o ct a th ctal tlt. alo cott wth ou al tat. W ot that c lat wth th Aua ato by th follow athatcal foula: / 7 / [ ] / / Φ Φ

34 I th cyclc ol th t of th ottal ut ca a th fl oll owhll o that th yotc ha co to a whch coo to c >. If c cha fo o tal valu c >> to o valu of o uty at th of th yotc ha aft cha by a aout th c c /. Wh c la th o-avtatoal t. (. tycally oat a th ctal tlt a fw ct towa th. β Fo xal uo c a c ; th th ctal tlt β. (. β whch coo to.97 < < fo otv < β < at wth ou al tat. Wth a th valu w hav th follow athatcal cocto wth th Fboacc ub:. W ot that atv ottal of th ty wth vy la valu of c hav b au to a atually t thoy. Ou xo fo th ctal tlt of th tocally uc cuvatu ctu ca alo b x t of th cutoay fat-oll aat Not that ε /( ( γ ε ε V V c. h th ctal tlt η V V ( γ ε η. (.. (.6 M Pl h ult ca b coa wth th ctal x of th t-lay (Nwtoa ottal tubato wh th coo foula ε η. (.7 M Pl H th ft t aa avtatoal but t ha th oot of th avtatoal cotbuto to th tocally uc fluctuato ctu. So th tlt tycally a fw ct. Fally fo flato th ctal tlt 6ε η (. wh th ult x t of th low-oll aat ( V ε / V a η M Pl V / V. H w hav val th facto of M Pl to llutat that both flatoay cotbuto a avtatoal o. h v th a a fo a th Nwtoa ottal tubato th cyclc ol. M Pl /

35 . O o uato coc th ffctv fv-oal thoy of th toly coul htotc t a a au vo of N fv-oal uavty wth fou-oal boua. W wll ow bfly vw th ffctv cto of toly coul htotc t thoy a -oal uavty wth boua v by Hoava a Wtt. h booc at of th acto of th fo S S SG S YM (.9 wh S SG th fala -oal uavty S SG IJKL I... I R G IJKLG ε CII I G I... I G 7 I... I κ M 7 (. a S YM a th two E Ya-Mll tho o th obfol la xlctly v by / / ( κ t F tr t F tr κ SYM. πκ π M πκ π M (. ( H F a th two E IJ au fl tth a C IJK th -fo wth fl tth G C. I o fo th abov thoy to b uytc a wll a aoaly f th IJKL [ I JKL] Bach tty fo G houl cv a cocto uch that / κ ( ( G IJKL { J δ ( x J δ ( x πρ } IJKL π π (. wh th ouc a v by J tf F tr R. (. ( ( ( Wth a th. (. a (. w hav th follow athatcal cocto wth th aua ato a th Raauja oula uato: / / Φ 7 ; 7 ; 6Hz 77 ; ; 67 / / Φ 9969 W ot that a a π Fboacc ub whl lat to th hycal vbato of th booc t by th follow Raauja fucto:

36 coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w'. Whl th taa b of th cocto to th au cocto ( tf F tr R (.b la to vah ouc t th waly coul htotc t Bach tty th t ca o lft wth o-zo ouc ± tr R o th two hyla. A a ult th atytc to fl G a hc th co t th avto uyty vaato δψ I JKLM D Iη ( ΓIJKLM IJΓKLM G η... (. o ot vah. Wth a th athatcal cocto wth th Aua ato a Fboacc ub w ot that: / / Φ ; a a Fboacc ub. Futho a lat to th hycal vbato of th booc t by th follow Raauja fucto: coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo Now lt u tat wth th zoth o tc t w'. A B ( x V / Ω x x µ ν η x x R (. µν wh Ω AB a Calab-Yau tc wth Kahl fo ω Ω. (H a a b a holoohc ab ab a at-holoohc c. o tac of th cal ot of th oluto w hav touc oul V a R fo th Calab-Yau volu a th obfol au ctvly. o / o κ th tc ca b wtt th fo A B ( bˆ µ ν η x x R ( γ ( x V / ( Ω h x x ˆ µν AB AB (.6 AB 6

37 wh th fucto bˆ γˆ a h AB o x a th Calab-Yau cooat. Futho / G ABCD a G ABC cv a cotbuto of o κ fo th Bach tty ouc t. h al xlct fo of th cocto a wth ˆ ( / / / b RV x πρ (.7a ˆ γ V ( x πρ / / EF AB R V ( x / Ω AB (.7c GABCD ε ABCD ωefε ( x h πρ G ABC (.7 H ( x κ πv π / ω tr ε th t fucto whch X ( Ω ( Ω R R (.7b (.7 6 v Ω. (. X fo x otv (atv. Wth a th. (.7 a (. w hav th follow athatcal cocto wth th Aua ato: 6 7 / 7 7 ( Φ / ; 7 97 ; 666 π 7 / 7 ; / 7 7 / 7 / 7 ( Φ ( Φ / 7 7 ( Φ 6 / ( Φ / 6 ; / 7 I th fv-oal ac M of th uc thoy th obfol fx la cottut fouoal hyufac whch w ot by M (. h wll b a E 6 au fl A µ ( accoa by auo a au att fl o th obfol la M. W wll t th au att fl to zo th follow. h fl cott of th obfol la M cot of a E au fl A µ a th coo auo. I ato th aoth otat bouay ffct whch ult fo th o-zo tal au fl a avty cuvatu. Mo cly ot that. π 6 κ ( ( AB AB ΩtFABF ΩtRABR πv X X / F. (.9 AB 7

38 I vw of th bouay acto (. t follow that w wll ta coolocal ty t wth oot o th two boua. Not that th z of tho t t by th a cotat v by. (. whch t th atu of th o-zo o. h bouay coolocal t a aoth otat t ouc th -oal bacou a a oluto of th fv-oal thoy. W ca fo th Kaluza-Kl ucto o th tc A B V / β x x V / β Ω ABx x. (. h colt cofuato fo th atytc to fl that w u th ucto v by C βγ Gβγδ [ Cβγδ ] C AB Α ωab GβAB F β ωab F β Αβ β Α 6 CABC 6 ξωabc GABC ξωabc (. a th o-zo o G ABCD ( x EF ε ABCD ωefε (. 6 wh wa f. (.. W ca ow cout th fv-oal ffctv acto of Hoava-Wtt thoy. U th fl cofuato (.9 (. w f fo th acto (.9 (. that S S S S (. av hy bou wh β βγδε Sav R FβF ε ΑFβγ Fδε κ (.a M Shy V V V V ξ ξ V GβγδG κ M ε βγδε ( ( ξ ξ ξ ξ Α V G βγδ ε ε ε (.b ( S { V } V bou M M 6 VtF M µν κ πgu I th xo w hav ow o hh-vatv t. h -fo fl tth ubjct to th Bach tty βγδ. (.c G βγδ κ ( ( G µνρσ { J δ ( x J δ ( x πρ } µνρσ π GU (. whch follow ctly fo th -oal Bach tty (.. h cut J hav b f. (.. h fv-oal Nwto cotat κ a th Ya-Mll coul GU a x t of -oal uatt a (

39 κ κ v / κ π GU. (.6 v κ Sc w hav coactf o a Calab-Yau ac w xct th bul at of th abov acto to hav ht v ucha a thfo to coo to al N uavty fv o. Accoly lt u coa th ult (. to th ow N uavtyatt tho fv o. I th tho th cala fl th uval hyultlt aatz a uatoc afol wth cot tuctu M Q SU ( / SU U. Hc to coa ou acto to th w houl ualz th th-fo C βγ to a cala fl σ by tt ( th bul ε ε ε ε ( σ ( ξ ξ ξ ξ Α G βγδ V εβγδε. (.7 h th hyultlt at of th acto (.b ca b wtt a wh ( V σ ξ ξ v u v Shy h uv V κ (. M u u u u. h covaat vatv f a Α wth u (. h a ol tc huv u vkq ca b cout fo th Kahl ottal ( S S CC K Q l S V ξξ σ C ξ. (.9 u Coutly th hyultlt cala aatz a Kahl afol wth tc h uv. It ca u b otat that a Kll vcto o th afol. o aalyz th uyty ot of th oluto hotly to b cu w th uyty vaato of th fo aocat wth th thoy (.. hy ca b obta th by a ucto of th -oal avto vaato (. o by alz th ow fv-oal tafoato by atch oto au fou-oal N tho. It uffct to th booc t oly. Both aoach la to βγ β γ / j ( γ δ γ F ε V ( ξ ( ξ ( ε δψ D ε βγδε Vε Gβγδε 96 βγδε δζ Vε Gβγδ ε j β j V γ β Vε V ε ( x ( jε (.6 βγ j j j ( jε V ε ( x γ ( jε / j γ ε V γ ( ξ ( ξ ( ε wh a th Paul atc. hc w that th lvat fv-oal ffctv thoy fo th ucto of Hoava-Wtt thoy a au N uavty thoy wth bul a bouay ottal. j j 9

40 h thoy (. ha all of th ut cay fo uch a th-ba oluto to xt. Gally o to hav a ( D -ba a D -oal thoy o to hav a ( D -fo fl o uvaltly a coolocal cotat. h coolocal t ov by th bul ottal t th acto (.. Fo th vwot of th bul thoy w coul hav ult th-ba oluto wth a abtay ub of aalll ba locat at vaou lac th x cto. A wll ow howv ltay ba oluto hav ulat at th locato of th ba to b uot by ouc t. h atual caat fo tho ouc t a th bouay acto. Gv th aoaly-cacllato ut th tct th obl oluto to tho t a a of aalll th-ba coo to th obfol la. It cla that o to f a th-ba oluto w houl tat wth th Aatz µ ν a( y x x η b( y y (.6 µν V V ( y wh a a b a fucto of y x a all oth fl vah. h al oluto fo th Aatz atfy th uato of oto v fo th acto (. v by / a a H b b H V b H a H y c (.6 wh a b a c a cotat. W ot that th bouay ouc t hav fx th fo of th haoc fucto H th abov oluto. Wthout cfc foato about th ouc th fucto H woul cally b lu toth fo a abtay ub of la c wth lo ±. h of ach c woul th cat th locato of th ouc t. h cty of atch th bouay ouc at y a πρ howv ha foc u to co oly two uch la c aly y [ πρ] a y [ πρ]. h c a lu toth at y a πρ. hfo w hav yh ( δ ( y δ ( y πρ (.6 whch how that th oluto t two aalll th-ba locat at th obfol la. W t that th oluto olv th fv-oal thoy (. xactly wha th oal / fo Calab-Yau oluto wa oly a aoxato to o κ. It tahtfowa to how that th laz vo of (.6 that th xao to ft o O( κ / coc wth Wtt oluto (.6 (.7 uo aoat atch of th tato cotat. Hc w hav fou a xact alzato of th laz -oal oluto. W tll hav to chc that ou oluto v half of th uyt. Wh β a V a th oly o-zo fl th uyty tafoato (.6 lfy to δψ j β j D ε ε ( y V γ ( ε δζ V γ Vε ε ( y V ( ε j β j. (.6 Wth a th. (. (. (.7 (.6 a (.6-.6 w hav th follow athatcal cocto wth Aua ato a wth th Raauja oula uato:

41 / 7 7 ( Φ / 79 7 ( Φ 767 / 966 / 7 ; 767 / / Φ 99 ; π / 7 7 ( Φ / 7 7 ( Φ / 6 / 7 ; / 7 7 ( Φ / 79 / 7 7 ( Φ / 7 / 7 7 ( Φ / 97 / 7 7 ( Φ / 9 / ( Φ / 69 Futho th ub a 96 a coct wth th o that coo to th hycal vbato of a ut a to hycal vbato of th booc t by th follow Raauja fucto: coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo ;. ; ; ; t w' ; ;

42 coπtxw' πx w' x atlo cohπx πt w' w' tw' 7 lo t w'. W ot alo that a Fboacc ub. h Kll o uato δψ δζ a atf fo th oluto (.6 f w u that th o ε v by / j ε ε γ ε ( (.6 H ε j wh ε a cotat ylctc Majoaa o. h how that w hav fou a BPS oluto v fou of th ht bul ucha.. O o uato coc th coll Ba a th O of th Hot B Ba W hav v th fv-oal ffctv acto of htotc M-thoy th ct ubcto (.. Now w hall u a lf acto cb avty γδ th uval bath oulu of th Calab-Yau th-fol a fou-fo au fl Α γδεζ wth fl tth F Α a a l bul M-ba. It v by M F S x R M! µνκ ε ξ h Αγδεζ! ( γ δ ε ζ M ( ( µ X ( ν X ( κ X ( ( M X (.66 wh γ δ ε ζ... µ ν h ac-t a fv-oal afol M wth γ ( cooat x. h fou-oal afol M a th vbl h a bul µ ba ctvly a hav tal cooat ξ ( a to M. Not that ha o of a. If w ot β a β th th vbl ba ha to M th h ba ( β M a th bul ba β M. It tahtfowa to how that th to of th bul ba β M ut alway b otv. Futho o ca aly uc that th to o th vbl ba M ca b th otv o atv. W wll ta > o that th to o th vbl ba atv. Futho w wll choo β uch that ( β > that th to of th h ba otv. h to h µν th uc tc ( γ µ ( µ o M. h fucto X ( ( ξ( a th cooat M of a ot o M wth cooat ξ(. γ µ I oth wo X ( ( ξ( cb th b of th ba to M. h BPS oluto of Lua Ovut a Wala th v by

43 ( y( N A x B D ( y y D Y ; BD ( y A NB D ( y y < ( A NB D ( y F fo Y F β fo y > Y (.67 Y ; wh ( y y C ( y ( β y C βy D fo y < Y D fo y > Y (.6 a A B C N a Y a cotat. Not that A B C N a ol a Y ha th o of lth. h vbl a h bouay ba a locat at y a y R ctvly a th bul ba locat at y Y Y R. W au that C > o that th cuvatu ulaty at D o ot fall btw th bouay ba. Not that y l th o of all volu whl y R l th o of la volu. Not that t th oluto of th fou-fo uato of oto to. (.66 yl cly th bul acto wth cha th tval y Y a cha β th tval Y y R. h foulato of th acto. (.66 u th fou-fo Α atculaly uful wh th thoy cota bul ba a th ca yotc thoy. h follow uato H& c δ ach c ach (.69 c x th ty tubato t of th t lay at th t of collo co th xotal ottal V Y v th th. (.69 yl. If w δ v. (.7 C Now w cout th ctu of uatu fluctuato of th ba δ Y a u th ult to cout th t lay. Fo th calculato of uatu fluctuato t uffct to wo at th lowt o β /. Wthout lo of alty w ca thfo t A N. I that ca th bul ba Laaa v by µν L β βm B D( Y η µ Y νy V ( Y. (.7 Not that th a wth L β v th follow uato βm A B L D ( Y Y& β N V ( Y N (.7 wh w t A N a atal at of Y to zo. Lt u ft co th atally hooou oto of th ba whch wll b cb by Y (. It ov by th follow uato of oto

44 D ( Y Y& V ( Y E (.7 wh E a cotat. E. (.7 ly th tatt that th y E of th bul ba cov to th o β /. Sc w hav cho th vbl ba to l at y a th h uv to l at y R w focu o th bach Y & < whch ca th bul ba ov towa th vbl ba. h oluto to. (.7 th v by ( Y D( Y ' Y ' ( E V ( Y (.7 ' wth a wth th collo occu at. Lt u ow co fluctuato aou th bacou oluto Y (. Naly f Y Y ( δy ( x wth δy ( x << Y ( w ca xa th acto to uaatc o δ Y L fluc D V V [ δy& ( ( δy ] D V E D δy Y Y (.7 wh w hav u. (.7 a wh w hav touc D D a V Y V Y fo lcty. h y lato th fluctuato uato a v fo th acto (.7 x x f a& a t t x f ; x ( (.76 wh f D δy a wh a t f by a&& a t t V V D D Y Y. (.77 h fluctuato. (.76 ca b coa wth th coo uato fo th tubato of a cala fl wth o ottal a ally coul to a FRW bacou wth cal facto a ( a& δ & δ& δ. (.7 a Df f a δ. (.7 bco x x f a & x f. (.79 a Lt u ow cu th Hubbl hozo fo th tubato. Rcall that uual cooloy (. (.79 w hav x ( a ( hya ( hyh (. a

45 wh H a / a& th Hubbl au a v fo th cal-facto a. By fto a o a to b out th Hubbl hozo wh t wavlth la tha th Hubbl au. Fo. (. w that th occu wh x <. hfo a o wth altu f co out th hozo wh x O(. Slaly ou cao w ca wt wh x / D ( H (. hy hy t / hy / D. h ol of th Hubbl au lac by ( Y D( Y Y ( E V ( Y / / H t D D (. whch to b thouht of a a ffctv Hubbl au fo th tubato. So th lth cal at whch altu fz o a B (ath tha a t but th altu tlf a v fo. (.76 o a t. h fatu of two fft cal facto ovl act of yotc cao. Wth a th coao to flatoay cooloy w hav that flato th wavlth a ttch ulually whl th hozo aly cotat. I th yotc cao th wavlth a aly cotat whl th hozo h. W ca obta a ctu whch cal-vaat. Wt th uato fo th tubato th fo of. (.76 uful c o ca a off fo t th ctal lo of th ow ctu. It t by th valu of ( a& & / t a t. I atcula o obta a cal-vaat ctu f ( a& & / t a t wh th o obv o th CMB co out th hozo. Cob. (.7 a (.77 w f a&& a t t Y D D Y Y V V D( Y ' Y ' ( E V ( Y '. (. h ctu wll b cal-vaat f th ht ha of. (. ual wh th o of tt co out th hozo. hc w hav: a&& a t t Y D D Y Y V V D( Y ' Y ' ( E V ( Y '. (.b Wth a th. (.b w hav th follow athatcal cocto wth th Aua ato: / 7 / 7 / 7 7 Φ ( Φ / / 7 ; ; ( Φ / 7 ( Φ 7 / 7 / 7 ( Φ ; 6. W ot alo that ub a Fboacc ub. 7 / 7 / 7

46 hc w ca wt th. (.b alo a follow: [ a&& a t t V V Y D D Y Y / 7 7 ] D( Y ' Y ' ( E V ( Y ' / 7 Φ ( Φ / (6 966 / 7 ; W ca a a ottal ( Y V of th fo that ht ult fo th xcha of wa M- ba. W woul l to th of V a th ottal v fo th uottal W fo th oulu Y th low y thoy. ycally uottal fo uch oul a of xotal fo fo xal W cy (. wh c a otv aat wth o of a. h coo ottal cotuct fo W a th Kahl ottal K acco to th uual cto V K / M j l K DWD jw WW. (. M l j wh D / K / M l th Kahl covaat vatv K K / Kj K / a a u ov ach ufl lct. E. (. a (. ly that V cay xotally wth Y. H t wll uffc to fo th calculato u a l xotal ottal aly V Y v (.6 Y wh v a a otv ol cotat. Not that th ca wh th ottal at by th xcha of wa M-ba th aat of th fo c ν / wh c a cotat th to of th M-ba a ν th volu of th cuv o whch t wa. h tubato o of tt a tho whch a wth th cut Hubbl hozo. A th wavlth coo to tho o a out th ffctv Hubbl hozo o th ov bul-ba th altu bca fx. Scal vaac wll u D >> u th o. W ow that f th ottal V lbl coa to E th ctu of fluctuato ot cal-vaat. Hc w co th lt wh E << V. h coto a fo th uato of oto fo Y. (.7 atf f Y & tally o uvaltly f th bul ba b aly at t. Fo th ba to b aly at t o ut hav E V tally. A th ba tav th ffth o V ca xotally wha E cotat. Hc th coto E << V autoatcally atf. h bul ba b aly at t cly what w xct fo a aly BPS tal tat. Aly th coto E <<. (.7 uc to V 6

47 Y D D Y (.7 v Y '/ ' Y ' Y v D wh w hav lct th ot cotbuto at Y. O th oth ha. (.77 v a&& a t t v Y D D. (. Cob th abov two xo w obta: D v Y a& t / D a&& ; / ; a t t D at D a&& t. (.9 at D D h ht ha of. (.9 aoxatly ual to th lt of la D. hc w hav a&& t. (.9b at D D Alo h wth a th ucal ult of. (.9b w hav th follow athatcal cocto wth Aua ato: / 7 / 7 / 7 7 Φ ( Φ / / ; 6. / 7 ( Φ / 7 ( Φ 7 / 7 / 7 ( Φ 7 / 7 ; ; Hc th xotal ottal of. (.6 ult a aly cal-vaat ctu of tubato ov that E << V a D >> a atf wh o a out th ffctv Hubbl hozo. W xt cout th tubato altu by u. (.76 to calculat Y. h coto E << V a D >> ut b atf wh wavlth a out th hozo. h coto ca b lax oc th o wll out th hozo. I th lt that D >> wh th lvat o co out th hozo. (.76 uc to x x f [ x ] f (.9 7

48 wth oluto ( C J ( x C J ( x f x / / / (.9 wh ± / o wll-wth th hozo (.. x >> J a Bl fucto. h coffct C a f C a fx by u that b Mowa vacuu fluctuato that fo >> 6βM B x. (.9 U th tal coto w f th follow altu fo o out th hozo (wth x << / f f. (.9 / ( π ( ( π / 6βM B Subttut. (.7 a u f D δy w f Y v Y. (.9 ( π Fally w f th t-lay / βm D B by Y Y& / 6π βm B D (.9 wh w hav u th uato of oto fo Y. (.7. Not that th t-c of l a cay coto fo th valty of th t-lay foal. h facto of D D to b valuat at t wh a v o co out th hozo Y ( u th oto of th bul ba. Lt D ot th valu of D at hozo co fo o. Sc hozo co occu wh x o uvaltly wh (. (.7 v D lo v C. (.96 Subttut. (.9 a (.96 to. (.7 w f v δ. (.97 / π βm B( C D h xo fo δ ca aually wth ca coo to a ctu tlt lhtly towa th blu. h blu tlt u to th fact that th xal D ca a th ba ov. hat th ctal x