Rick s Cosmology Tutorial: Chapter 6

Transcription

1 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs Chaptr 6 Th Shiftig utro:proto Ratio I Th Early Uivrs: Th First Fiv Miuts With Som Rmarks About utrio Douplig Last Updat: Ju 006. Itrodutio As xplaid i Chpatr, th asy story of th arly uivrs starts at aroud ~0.00 to ~0.0 sod. Th physis of th uivrs is rlativly asy aftr this tim baus thr ar fw partil spis prst. This is baus th tmpratur is low ough to sur that th oly partils that a b ratd out of th availabl thrmal rgy ar ltros/positros ad utrios (ad, of ours, photos). W say othig of dark mattr purly out of igora about what it is. At slightly arlir tims th xt lightst partils, muos, would hav b abudat (mass ~05 MV). At still arlir tims, mor ad mor of th havir partils would hav xistd i larg umbrs. Prod bakwards i tim. At tims arly ough for th strogly itratig partils to b prst i larg umbrs, startig with pios of mass ~5-40 MV, thigs bom xtrmly ompliatd. This is baus hug umbrs of diffrt hadros a th b formd by strog itratios. For this raso, w pik up th story at aroud sods. At this tim, th typial thrmal rgy (~kt) is muh largr tha th rst mass of th ltro., th ltros, positros ad all th utrio spis ar prst i umbrs omparabl with th photos (i.. as dtrmid by th rlativisti Frmio blak-body sptrum, ad h th sam as th photo dsity apart from th umrology fators s th Gral Physis part of th wb sit for a drivatio). owvr, thr ar also utros ad protos prst. Ths ar th survivors from th arlir, diffiult, priod, bfor 0.00 sods. Thy ar far fwr i umbr, about ulo for vry x 0 9 photos. This baryo:photo ratio aot b drivd from osidratio of th asy priod, aftr 0.00 sods. To driv it thortially would rquir aalysis of th diffiult arlir priod i whih th strog ular itratios wr domiat. Th ulos ow xtat ar th fw survivors from th vry arly aihilatios i whih ulos outumbrd atiulos oly vry margially. This misbala i ulo ad atiulo umbrs implis a brak dow of tim rvrsal symmtry. Thus, a thortial prditio of th photo:baryo ratio rquirs a thory i whih tim-rvrsal symmtry is ot rsptd, or is brok. W will ot stray ito ths dp ad murky watrs. Istad w shall b ott that, at prst, our kowldg of th baryo:photo ratio oms from obsrvatioal masurmts rathr tha thory. W shall simply assum that this ratio is o of th God giv uivrsal paramtrs. At suffiitly arly tims th typial thrmal rgy, ~kt, would hav b grat ough to rat ulo/atiulo pairs. At ths tims, th ulos, ad may othr hadros, would hav b roughly as abudat as photos, utrios, t. Cosqutly, oly -i-x0 9 ulos hav survivd from this priod. It is ot surprisig that this is a vry small fratio. If harg ojugatio symmtry (or, quivaltly, PT symmtry) wr xat, th umbr of survivors would b prisly zro, baus th umbr of ulos ad atiulos would math xatly. Thus, th tir ivtory of ormal mattr i th uivrs today is solly th rsult of th dviatio of th laws of physis from xat harg ojugatio symmtry. By Pag of 4

2 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs ormal mattr w ma baryoi mattr, plus ltros, i.. xludig dark mattr ad dark rgy., it should rally b o surpris to fid that ormal mattr aouts for oly about % to 4% of th avrag dsity of th uivrs. At th tim of writig, dark mattr is thought to aout for 5-7%, ad dark rgy for th rst (roughly 7%)., a ubiasd prsptiv might b that th ulos formd a quit uimportat miority ostitut of th arly uivrs, wh photos ad utrios wr domiat, ad that th sam rmais tru today, si dark mattr ad dark rgy ar ow domiat. owvr, w hav a partiular itrst i th baryoi miority ostitut. This is what w ad our world ar mad of. Morovr, it is what vrythig w a s is mad of, v th most distat of astroomial objts whih ar atually visibl. owvr, thr is a importat ot of autio hr. W urrtly hav vry littl ida at all of th atur of 96%-98% of th uivrs. It is ot lar whthr th dark mattr ad dark rgy that sm to b prst i th uivrs a aris out of th stadard big bag modl. Util this is rsolvd, th stadard modl must b rgardd as ttativ. vrthlss, th stadard modl has som vry imprssiv susss. Th prditio of th xist, ad approximat tmpratur, of th mirowav bakgroud is th most obvious. Prhaps mor imprssiv, though, is th orrt prditio of th primordial hlium abuda. This drivatio is th subjt of this Chaptr.. Evolutio of th utro:proto Ratio i Outli W brifly prviw i this Stio what will b ovrd i mor dtail i th subsqut stios. Wh w start our story, at ~0.00 sods, th utros ad protos ar arly qual i umbr. This applis wh kt is larg ompard with th mass diffr M M p. At ~0.5 sods kt boms omparabl with M M p. Cosqutly, it is ow sigifiatly mor diffiult to mak utros out of protos tha vi-vrsa, ad th utro:proto ratio has fall sigifiatly blow 50:50. I th first sod or so, utros ad protos a itrovrt via th followig ratios (mdiatd by th wak ular for):- p, p Ths ar th ratios whih provid th mhaism for maitaiig th utro:proto ratio i thrmal quilibrium. If ths ratios ar prvtd from ourrig at som tim, as idd thy ar at aroud s (s blow), th utro:proto ratio will frz at its valu at that tim xpt for th atural bta day of th utros. Baus th tmpratur at whih th abov ratios frz out dtrmis th d of th fast phas of utro dstrutio, it is th ky fatur whih dtrmis th osmi hlium abuda. If idd this is distit from ordiary mattr. Fr utros ar ustabl with a half-lif of about 5 miuts. Thy ar stabl oly wh ombid with protos withi a ulus. Pag of 4

3 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs Asid: th utrio ratios with th ulos, togthr with th purly lptoi itratios lik ad, ar th mhaism by whih th utrios maitai thir thrmal quilibrium with th rst of th uivrs (th utrios do ot itrat dirtly with photos). If th purly lptoi itratios as at th sam tim as th utrio-uloi itratios, th th utrios will doupl from th rst of th uivrs at this tim. W shall show that this is tru blow. Virtually o of th osmi bakgroud utrios will vr itrat with aythig agai i th lif of th uivrs. Aftr th utrio-ulo ratios hav asd, th utro:proto ratio otius to fall, but ow far mor slowly du oly to th bta day of th fr utros. If othig itrvd to sav thm, utros would thrfor b dstid to disappar from th uivrs. If othig itrvs withi a priod of th ordr of svral of thir half-livs, it will alrady b too lat to prvt th umbr of utros big far smallr tha th umbr of protos for th rst of th lif of th uivrs. owvr, somthig dos itrv ad withi th rquird timsal (amly withi 5 miuts, th half-lif of a fr utro big about 5 miuts). This somthig is th formatio of stabl uli. utros bom stabl wh boud with a proto to form a dutrium ulus, providd that th tmpratur is ot so high as to aus photodisitgratio. Thus, th formatio of stabl dutros ostituts a sod ruial frz out i th utro abuda. W shall show blow that dutros bom stabl at ~-5 miuts. This is th ruial tim whih dtrmis th uivrsal utro abuda vr aftr. Si virtually all osmi utros ar i th form of hlium uli (othr lmts big a tiy proportio) this also dtrmis th osmi hlium abuda i ompariso with that of hydrog. th tim of this sod frz out is th sod dtrmiig fator i th prditio of th osmi hlium abuda. Th ourr of both frz outs is ruial to th hlium (utro) abuda. If ithr wr substatially dlayd, th osmi hlium abuda ould b drastially rdud, pottially to virtually othig. owvr, this would probably hav o grat impliatios for th subsqut volutio of th uivrs (s Cosmi Coiids Part ). O th othr had, if th first frz out wr muh arlir, th abuda of utros ad protos would b virtually qual. Th primordial uivrs would th b virtually pur hlium with vry littl hydrog. This would hav profoud ffts o th hmistry of th uivrs for vr aftr. Cotaiig o hydrog, th uivrs would otai o watr, o hydroarbos ad o protis as w kow thm. ot that th first frz-out ivolvs th wak ular for, whras th sod ivolvs th strog ular for.. Dtails: Phas Equilibrium Ratio I this first phas, th ulo-utrio ratios p, p ar ativ. Thy provid th mhaism by whih thrmal quilibrium btw th dsity of utros ad th dsity of protos a b ahivd. ot that ths ratios our i both dirtios. O mas of fidig th utro:proto ratio (i priipl) is to auratly alulat th rat of ah of ths four ratios. Ths will Pag of 4

4 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs dpd o th partil dsitis ad th ross stios for ah ratio. Th ross stios will dpd upo th wak itratio ouplig strgth (.g. Frmi s ostat), th partil rgis, ad th form fators for th ulos. Th lattr rquirs spialist iput to b alulatd auratly. Th rat of hag of th utro dsity, whih is qual ad opposit to th rat of hag of th proto dsity, is th foud by subtratig th rat of utro loss i th forward ratios from th rat of utro produtio i th bakward ratios. It is lar that, as th tmpratur falls, th utro dsity will rdu, ad th proto dsity will iras. This is baus i p, p th forward ratios a our v at vaishigly small utrio or positro kiti rgis. I otrast, th rvrs ratios a our oly if th ombid kiti rgis of th partils o th RS xds th mass diffr, M (M p + M ) [or, for th sod ratio, M M p + M ]. If th ratios ar i quilibrium at a giv tmpratur, th to ompsat for this kimati disadvatag, th rvrs ratios must hav a advatag i trms of gratr partil dsitis of th ratig partils, p + ad p + rsptivly. Si th lpto dsitis ar fixd by th Frmio blak body sptrum, ad th umbr of positros is sstially qual to th umbr of ltros at ths tims, it follows that th proto dsity must xd th utro dsity i ordr that th forward ad bakward ratios bala. [This impliitly assums that, apart from th kimati fators disussd, th matrix lmt part of th ross stios for th forward ad rvrs ratios ar th sam. This follows from tim rvrsal symmtry,.g. by xprssig th matrix lmts i trms of Fyma diagrams i whih th forward ad bakward ratio matrix lmts ar simply ojugats]. Whilst th abov skth of a possibl alulatioal rout is orrt i priipl, it has two drawbaks. Th first is that it is ompliatd ad rquirs spialist form fators for th ulos to arry out th alulatios of th ratio rats. Th sod is that it ivolvs th dliat allatio of larg quatitis, i.. th rat of hag of th utro dsity is th diffr i rats of th bakward ad forward ratios. Si it is likly that th ratio rats a oly b stimatd approximatly, it is likly th diffr i th (arly mathd) bakward ad forward ratio rat stimats will b grossly i rror. Fortuatly thr is a solutio to this problm. Istad of tryig to alulat th small rat diffr, w assum it is zro! I othr words, w assum that a quilibrium stat prvails, ad th forward ad bakward ratios ar i bala. A diffrt quilibrium ofiguratio will xist at diffrt tims, i.. at diffrt tmpraturs. W ar thus assumig that quilibrium a b stablishd suffiitly quikly (i.. i xtrmly small fratios of a sod) so that th uivrs passs from o quilibrium ofiguratio to aothr otiuously. [B: Th total umbr of ulos is fixd. th atur of this thrmal quilibrium is vry diffrt from that whih prvails at tmpraturs ~ tims highr, wh ulos a b ratd out of th thrmal rgy. Th total umbr of ulos plus atiulos is th vastly gratr, ad a b ddud from th tmpratur. Th baryo umbr is th sam though, but ot dduibl simply from th tmpratur (at last, ot without iput from som Grad Uifid Thory or strig thory)]. Pag 4 of 4

5 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs For thrmal quilibrium w a appal to th Boltzma distributio to giv immdiatly th ratio of th utro ad proto dsitis, i.., p M k BT whr, M M M P.9MV () From Chaptr 4, th tim-tmpratur rlatioship for th radiatio ra of th uivrs aftr aihilatio of th ltro/positro pairs is, T( 0 K).0 x 0 t(ss) 0 (a) But aftr aihilatio of th ltro/positro pairs th tmpratur bam a fator of.40 tims gratr tha bfor. So our bst stimat of th tmpratur bfor th aihilatio of th ltro/positro pairs is, T( 0 K) 0.7 x 0 t(ss) 0 (b), usig () ad () w a ddu th utro:proto ratio at ay tim as log as quilibrium prvails. Thus w fid, t (s) T ( o K) :P (%) P (%) 0.00.E E E E E E E E E E E x E E E E E E B: Th figurs i italis ar hypothtial si frz out ours at > s. 4. Dtails: Phas Th Ed of th Equilibrium Th first of th two frz outs ours wh th ratios p, p as. Why should this happ? Th raso is that ths ratios ar ourrig i a xpadig uivrs. Providig that th uivrs dos ot xpad vry muh i th tim btw ulo/lpto itratios th xpasio maks littl Pag 5 of 4

6 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs diffr. owvr, suppos that, for a giv partil, th typial tim btw itratios is T I ad that a partil typially travls a dista L I i this tim. What if th uivrs xpads suffiitly i this tim itrval that th dista L I strths to mor tha L I? I othr words, what if th fratioal xpasio, L/L of th uivrs i tim T I is gratr tha o? This would ma that, at th d of th priod T I, th partil has a gratr dista still to travl bfor it itrats tha it had at th start! Th situatio is illustratd as follows, whr partil is to b imagid as travllig towards partil with whih it would pottially itrat, partil L I partil Tim 0: x x partil partil Tim T I : x x dista has strthd to >L I W olud that if th strai of th uivrs xds uity i th tim priod T I btw itratios th th ratios will as. ow th strai rat of th uivrs is just th ubbl paramtr baus, dr dt R implis, our rquirmt for swithig off th itratios is, dr R d () dt dt T I (4) But si T I is th ma tim btw itratios for a giv partil, its riproal is just th ratio rat pr partil., th itratios swith off if, > ratio rat pr partil (5) W ow d to fid th ratio rat. It is uavoidabl that th atual strgth of th wak ular for should tr our alulatio at this poit. owvr, w will ot attmpt a aurat alulatio ivolvig ulo form fators, t. Istad w shall b guidd by th ross-stio for so-alld psudo-lasti muo-utrio sattrig. I th tr of mass fram th total ross stio of this purly lptoi ratio is, 4 G FE (6) 5.0 x 0 whr G F is Frmi s ostat for th wak ular for, ad E is th M P rgy of th iomig ltro-utrio. M P is th proto mass. I Equ.(6) ad i th valu for Frmi s ostat w ar takig th usual partil physiists librtis with uits, i.. G F a b thought of as havig dimsios /mass or /rgy or lgth aordig to tast. Rsults i physial uits ar obtaid by fatorig usig ad as Pag 6 of 4

7 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs rquird. Thus, to gt a ross stio i m - from Equ.(6), ad for a utrio rgy xprssd i J, w would us 5 - G F.0x0 / M p.44x0 m/j. I applyig Equ.(6) to a ulo targt, w shall assum that th thr quarks of whih th ulo is omposd ah bhav lik th massiv lpto, ad ah bhav idpdtly. ow a utro is omposd of th quarks udd, ad a proto of uud. Thus, th ratios p, p ivolv th ovrsio of a d quark ito a u quark., just two of th thr quarks i th utro will do as a suitabl targt for th ratios, that is th d quarks. Thus, w shall assum th ulo ross-stio is just twi Equ.(6), ad w assum th ross stio is th sam for both possibl ratios, i.., p p G FE (6b) 8 [Asid: Equs.6,6b apply i th tr-of-mass fram, E big th iomig utrio rgy i that fram. I th fram i whih th targt partil of mass M is statioary, ad th utrio rgy is E L, th ross stio is, for xampl, E L / M G FM. (6) E / M This is th form whih is usful for laboratory data i whih th targt is statioary. For utrio rgis whih ar vry larg ompard with M (.g. for ular targts ad utrio rgis of may GV), this boms, L G ME (6d) F L Thus, i th laboratory fram th ross-stio dpds oly liarly upo th utrio rgy, rathr tha quadratially as i Equ.6. For th as of prst itrst, i whih both partils hav thrmal distributios of rgy, ithr th tr of mass or th laboratory systms apply. Th radr may b ord that th sigifiat diffr btw th ross stio xprssios i th lab ad CoM systms implis that th atual thrmal distributio may aus ithr to b sriously i rror. This is ot th as baus our appliatio, for tims >0.00 s, ivolvs typial thrmal rgis of <60MV at most. This is small ompard with th ulo mass (arly GV). I this limit w s that Equ.(6) rdus to Equ.(6), baus th ulo is virtually statioary v i th tr of mass fram. ot also that th sigifiat fatur of Equ.(6) is that th ross stio for utrio itratios irass aordig to th utrio rgy squard. This is partly why th osmi utrio bakgroud is so hard to dtt th rgy of th utrios today is so vry low, v lowr tha that of th mirowav bakgroud photos (as w hav s i Chaptr 5). This is, of ours, i additio to th fat that th vry small siz of Frmi s ostat, G F, maks utrios itrat vry wakly.] Pag 7 of 4

8 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs To us Equ.(6b) w shall tak th utrio rgy to b th avrag of thir (Frmio) sptrum, i.., E.5k B T (7) Th objtiv is to stimat th typial tim, T I, btw ulo/utrio itratios for a giv ulo. ow th rlativ vloity btw th ulo ad th utrio with whih w ar supposig it will itrat a b tak to b th vloity of light, (to a vry good approximatio, i.. utrio masss ar rtaily vry small ompard with th rgis i qustio, whih ar of ordr k B T ad h gratr tha aroud MV ovr th priod of itrst)., for a fftiv ross-stio, our hos ulo swps out a volum pr uit tim. If th umbr dsity of utrios is th umbr of utrios with whih our ulo will itrat pr uit tim is thus, i.. this is th ratio rat pr ulo, ratio rat pr ulo = = (8) T I ow th umbr dsity of ltro utrios is giv by th Frmio blak body distributio, k BT 0.09 (9) (s th Gral Physis stio for a drivatio). Thus, Equs.6b, 8 ad 9 will giv th ratio rat for p pr utro. But thr is also th ratio p to osidr. Th argumt is xatly th sam, as is th ross-stio, i our simpl approximatio. Th dsity of positros pr spi stat is also th sam as that of th ltro utrios (otig that th positros ar rlativisti at ths tims), but aoutig for th two spi stats givs a xtra fator of two. Ovrall, thrfor, th positro ratio auss th ratio rat to b thr tims fastr. (ot that w shall fid that th ratios frz out wll bfor th ltro/positro aihilatios tak pla at ~4 sods. This is importat si othrwis th dsity of positros would ot b giv by a xprssio lik (9). Rathr, thr would b virtually o positros lft ad th sod ratio, p, would otribut gligibly to th ratio rat pr utro.). Puttig Equs.(6b), (7), (8) ad (9) togthr, iludig th fator of thr for th positro ratio, ad ovrtig th ross stio to physial uits, givs, T I 8.0x0 5 ( ) (M ) P 4.5 (kt) 0.09 kt 7.4x0 0 kt M P 5 4 (0) avig foud th ratio rat pr partil, (Equ.0), w ow d th ubbl paramtr. (B: It is th valu of th ubbl paramtr at th tim of itrst that is dd, ot its urrt valu!). This is giv i trms of th dsity of th uivrs as Pag 8 of 4

9 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs disussd i Chaptr, ad th dsity of th uivrs is simply that of blakbody radiatio, i.., G 4 T k B ad. ad Stfa' s ostat, () 60 Th fator aouts for th ltros/positros ad utrios as wll as th photos. With th hlp of th usual umrology (s Chaptr 5), this fator is s to b, 7 8 x xx 4 8 () whr th trms i th umrator rprst th ltros (ad positros), th thr utrio spis, ad th photos rsptivly. (Th domiator of is rquird baus th two photo mods ar iludd i th dfiitio of Stfa s ostat)., usig (0, ) ad quatig th rsultig to /T I, as giv by (0), givs, k G 4 B 4T 0 kt 7.4x0 () TI M P whih rdus to, 0 k BT 0.74 x 0 M p whr, G p 0 0 M 7.76 x (4) i.. 0 o k BT 0.8 x 0 M p h T 0.9 x0 K (5) Cosultig th abov Tabl w s that this tmpratur ours at about 0.65 sods wh th utro:proto ratio is about 5.9% : 84.%. ot that, otrary to th imprssio giv by Equs.(-5), th frz out tmpratur dos ot dpd upo th ulo mass (whih appars i ths quatios simply baus th Frmi ostat has b xprssd as ~0-5 / M ). Th xpliit xprssio for th frz out tmpratur i trms of th ostats o whih it truly dpds is, p fo F 6 6 kt 0.9G G (5b) 5. Dtails: Phas Bta Day of th utros O th ulo/lpto ratios p ad p hav asd, th oly mas by whih th umbr of utros a rdu is through bta day, p. Thus, aftr ~0.65 sods, th utro umbr dsity rdus xpotially aordig to, Pag 9 of 4

10 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs (t 0.65) /0 (t) (0.65) (6) whr t is th tim i sods, ad 0 ss is th half-lif of a fr utro (B: this is from Rowa-Robiso, though othr sours us half-livs of ~886 ss. Th lattr is giv by th 004 ditio of th Partil Data Booklt). Baus th total umbr of ulos is fixd, i Equ.6 may b itrprtd ithr as ithr th absolut umbr of utros, or th as th prtag or fratio of th total ulos whih ar utros. Equ.6 holds from th tim of frz out of th ulo/lpto ratios (~0.65 sods) util th tim wh uli bom stabl. O th utros ar safly withi a ulus thy ar stabl. 6. Dtails: Phas 4 Th Formatio of th First uli Th lightst ompoud ulus is th dutro, p. Th dutro is quit lightly boud, th bidig rgy big oly.mv. Equatig th avrag photo rgy (.7kT) to this bidig rgy, w might xpt that th dutros would b stabl agaist thrmal fissio at tmpraturs blow ~.5 x 0 0 o K. Si th tmpratur is lss tha this aftr ~0. s, it might b thought that th dutros would b stabl from this tim owards. owvr, this pays isuffiit rspt to th hug abuda of photos ompard to utros. Ev at tmpraturs, say, oly o-tth of.5 x 0 0 o K, a small prtag of th photos will hav rgis i xss of th.mv rquird to fissio a dutro ito its p ad ostituts. Si thr ar so may photos, oly a vry small proportio d hav rgis abov.mv i ordr to fissio all th dutros. I fat, w d oly as may photos with rgis >.MV as thr ar dutros. Ad th umbr of dutros is ot gratr tha th umbr of utros. At 0.65 sods, ~6% of th ulos ar utros, ad thr ar.9 x 0 9 photos pr ulo (s Chaptr 4), h th umbr dsity of utros is, ad th umbr dsity of photos is, x 0.84 x 0 (7) 9.9 x0 kt 0.46 (8) Th blakbody formula for th umbr of photos pr uit volum with rgy btw E ad E + de is, d E de. E kt (9) Pag 0 of 4

11 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs, from (8) ad (9), th fratio of all photos whih hav rgy gratr tha E is, d E de E x kt E kt x x dx (0) whr x = E /kt. Comparig with Equ.(7), w s that th fratio giv by th RS of (0) must b lss tha 0.84 x 0-0 if th dutros ar to b stabl agaist photodisitgratio., th oditio for stabl dutro formatio is,. x x x dx.0 x 0 0 () with x =.MV/kT. Si this larly orrspods to x >>, a vry good approximatio for th itgral is, x dx x x. x dx x x x () x x from whih w fid that x = 9.45, ad h T = 8.75 x 0 8 o K. Rallig from Chaptr 4 that, aftr ~4 ss, th tim-tmpratur rlatio is T t 0.0x0, this tmpratur ours at tim ~6. ss. Fially, substitutig this tim ito Equ.(6) givs th utro fratio wh uli ar first stabl as, 5.6 /0 5.9% x 5.9% x % () ow, i Equs.(7,) w assumd that th utros wr 6% of th ulos. Stritly, w should ow rpat th alulatio rplaig 6% i Equs.(7,) with.9%. owvr, arryig this out w fid that it maks oly ~ s diffr to th tim at whih th dutros bom stabl (spifially, 7.4 s), ad th prditd utro fratio is hagd gligibly (to.88%). A furthr ssitivity study o this rsult is to us a utro half-lif of 886 s i Equ.(). This givs.6% utros lft wh th dutros bom stabl at ~7 ss, ad this is probably th mor aurat stimat. Aothr possibl rfimt to th alulatio is basd o rogisig that quality of th umbr of utros ad high-rgy photos oly allows th formatio of uli to start. To our i volum w prsumably wat to wait a littl logr util th dutros will hav a dgr of umrial advatag. For illustratio, if w st our ritrio to b that thr ar oly half as may photos with rgy >.MV as utros, th w fid th tim at whih this ours is 4. ss, oly ~6 ss latr. Th rsultig utro fratio is thrfor gligibly diffrt (amly.5%). Pag of 4

12 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs Similarly, rquirig that thr is oly suffiitly high rgy photo pr 00 dutros, w fid that th tim boms 8 s. This implis a rmaiig utro fratio of.9%. Mor aurat alulatios a b arrid out takig ito aout th rat of th ratios formig dutros, ad thir subsqut ovrsio to hlium uli. owvr, takig it o trust that ths ratios ar rapid o th dutros ar stabl, it is lar from th abov sopig stimats that about % of th ulos ar utros at this stag. 7. Dtails: Phas 5 Th utro Satuary Th Cosmi lium Abuda Th prditio of th osmi hlium abuda rquirs o xtra work. W d oly ot that o dutros bom stabl, all th utros th prst quikly d up 4 i th form of hlium-4 ( ). Esstially this is baus th uli havir tha th dutro hav largr bidig rgis, with hlium-4 big th largst (ad h th tightst boud) of thos that a b formd durig this arly stag of th uivrs. Th bidig rgis ar; am ulus total bidig rgy, MV bidig rgy pr ulo, MV dutrium D.. tritium hlium- hlium-4 lithium Li Si hlium-4 ivolvs protos as wll as utros, ad giv th rsult that a gligibl proportio of othr uli ar formd, it follows that th osmi hlium abuda by mass should b doubl that prditd for th utros wh th formatio of uli starts, i.. x % = 6%. Th bst thortial stimats, ad idd th masurd valu, ar 4%-5%. This agrmt is o of th gratst triumphs of th Big Bag thory. Iidtally, diffrt sours sm to diffr as rgards what squ of ular ratios is most sigifiat i ladig to th produtio of hlium-4. All bgi with th produtio of dutrium. Som th list, 0 D tritium produtio D D tritium produtio D hlium- produtio D D 0 hlium- produtio 4 hlium-4 produtio 4 0 hlium-4 produtio Pag of 4

13 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs Othrs stat that, aftr th dutros absorb a proto to yild hlium-, th xt ratio is, 4 hlium-4 produtio Ad Rowa-Robiso, aftr produtio of hlium-, lists th ratio squ as big, 0 tritium produtio 4 0 hlium-4 produtio Prsumably all ths ratios our at various rats. It rally dos ot mattr if w ar itrstd oly i stimatig th hlium-4 abuda. All that mattrs is that a pathway to hlium-4 xists ad is rapid o dutros ar stabl. 8. Summary Th orrt prditio of th osmi abuda of hlium is a grat triumph. Thr is o a priori raso why th amout of hlium should b ~5% rathr tha 99% or 0.000%, so th prditio is imprssiv. Morovr, arryig out th ular bala alulatios for th othr uli (hlium-, dutrium ad lithium-7) all produ rsults i agrmt with obsrvatio. Agai this is imprssiv, si, if th origi of ths lmts i a Big Bag firball wr ot orrt, th abudas of ths lmts would hav o a priori raso to all b osistt with th sam timtmpratur history. Ths miority uli ar prst oly i vry small proportios (th ordr of 0.0% for dutrium ad hlium-, ad part i 0 9 for lithium-7). W shall driv ths abudas vry roughly i Chaptr 6B. Rviwig th abov drivatio of th hlium-4 abuda, w s that th most ssitiv fatur is th tmpratur at whih frz out of th ulo/lpto ratios our. Up util that tim th utro:proto ratio is hagig rapidly du to th xpotial dpd upo th tmpratur ad th utro / proto mass diffr., th most sigifiat paramtr i th alulatio is th strgth of th wak ular itratio, th Frmi ostat (G F ), si this dtrmis th frz out tmpratur. Th utro:proto ratio at this frz-out tmpratur is primarily what lads to th prditd hlium-4 abuda. Th strgth of gravity also plays a part i th alulatio. Th frz out tmpratur is proportioal to G /6 (s Equs.4,5b). Thus, if G wr a fator of t biggr, th osmi hlium abuda would b roughly doubld (i.. los to 50%). Othr paramtrs whih play a part i dtrmiig th hlium (or utro) abuda ar: th utro/proto mass diffr; th dutro bidig rgy; th utro liftim; th photo:baryo ratio; ad fially th tim-tmpratur rlatioship. Th latr is sstially giv i trms of G (s Chaptr ) ad so is ot a idpdt quatity. Similarly, th utro liftim a b xprssd (arguably) i trms of G F ad th utro/proto mass diffr (s Appdix A), ad so is also ot a idpdt quatity. Pag of 4

14 Chaptr 6 Th Shiftig utro-proto Ratio i th Early Uivrs Th dutro bidig rgy dpds upo th strgth of th strog ular for, as dos (prsumably) th utro/proto mass diffr. It is thrfor ot tirly obvious that ths ar idpdt quatitis, though this is implid by thoris suh as SU(5) GUTs. (Whthr this is also tru i strig thoris I do ot kow). Thus, igorig thos ostats whih mrly dfi our dimsios (,, k B ), th idpdt paramtrs whih may b varid i osidrig diffrt uivrss ar:- G, G F, M -M p, B D, W may xpt othr paramtrs to aris i th followig Chaptrs whih dpd upo th strog ular for. W will d to b autious as to whthr ths ar dpdt o, or idpdt of, M -M p ad B D. Pag 4 of 4

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