When Does The Universe Become Transparent? Last updated 24 June 2007

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1 Chaptr 9 Last updatd Jun 007. Introduction In Chaptr 8 w saw that bfor 85,000 yars th majority of th ordinary matrial in th univrs was in th form of fr lctrons and protons. In othr words, th ordinary mattr was mostly ionisd gas, or plasma. Aftr 6,000 yars, howvr, only a fraction of a prcnt was still ionisd. Bfor 85,000 yars th chargd lctrons, protons and nucli will intract with any light, or, mor gnrally, with any lctromagntic radiation. Such intractions would disrupt and scattr any cohrnt bams of light. In othr words, th univrs is opaqu bfor 85,000 yars. But at what tim dos th univrs bcom transparnt? Ar thr sufficintly many fr chargd particls at 6,000 yars to rndr th univrs still opaqu? Of cours, thr ar dgrs of transparncy. Th clarst glass will b opaqu if thick nough. Th standard of transparncy w hav in mind is an xtrm on. Astronomrs wish to b abl to s th whol of th obsrvabl univrs. This is not mrly a tautology. In this contxt, th obsrvabl univrs is th rgion which can in principal b obsrvd without violating th prcpts of rlativity (as discussd in Chaptr 5C). On th othr hand, by s w man th ability to actually form an imag from th light, or othr lctromagntic radiation, from ths distant parts of th univrs. To b abl to form an imag from light originating at such a distanc, th man fr path of photons must b comparabl with, or largr than, th siz of th obsrvabl univrs. This is fasibl only bcaus th man dnsity of mattr in th univrs is so xcdingly small. What intractions btwn th photons and mattr can occur? Onc all mattr is in th form of nutral atoms in thir ground stat, th photons can no longr intract with th lctrons. This is bcaus, to do so, th lctrons would nd to b raisd to a highr nrgy lvl, thr bing no lowr nrgy lvls unoccupid (by dfinition of th ground stat, and thanks to th xclusion principl). But w hav sn in Chaptr 8 that, by this tim, th photon nrgis ar far too small to xcit th atomic lctrons. This, of cours, is xactly why th nutral, ground stat, atoms ar stabl. Hnc, whn all mattr is in th form of nutral atoms, th only possibility for photon intractions is with th atomic nucli. W will s blow that th cross sctions for photon-nuclus intractions ar vry small compard with photon-lctron intractions (that is, intractions with fr lctrons). W hav sn in Chaptr 8 that by 6,000 yars, ssntially all hlium nucli and 99% of hydrogn nucli hav bcom nutral atoms. For this rason, 6,000 yars is oftn takn as th tim at which th univrs bcam transparnt. Howvr, vn aftr this tim thr will b som fr protons and lctrons lft (lss than %). Th fraction of rmaining fr protons (and lctrons) falls to <0 - by 57 o K (56,000 yars). Howvr, vn aftr that tim, thr ar still lithium ions, and hnc a i.. xcluding dark mattr and dark nrgy It is intrsting to not, howvr, that this is tru only bcaus th lctron transition from th ground stat (n = ) to th first xcitd stat (n = ) rquirs a largr photon nrgy than subsqunt transitions would, i.. from n = to n =, tc. If th rvrs wr th cas it would b possibl to absorb photons by intrnal transitions whilst th atom could not b ionisd. Pag of 9

2 matching numbr of fr lctrons, i.. about lctron for vry 0 8 initial fr lctrons, which prsist until around million yars. In viw of th prsistnt nonzro dnsity of ions and fr lctrons, is th claim that th univrs bcam transparnt at ~6,000 yars valid? In othr words, is th absolut dnsity of fr lctrons and ions sufficintly low aftr 6,000 yars to rndr th univrs transparnt? It will radily b sn that th qustion hings on th siz of th primordial photon:baryon ratio, sinc, for a givn rmnant fraction of fr protons and lctrons, this is what dtrmins thir absolut dnsity. To addrss this w nd to stimat th man fr path of photons at diffrnt pochs, comparing it with th corrsponding siz of th obsrvabl univrs. Equivalntly, w nd to stimat th typical tim btwn photon intractions, and find whn this first xcds th ag of th univrs.. Calculation of th Photon Intraction Tim W shall nglct photon intractions with atoms and ions, and concntrat upon th photon intractions with fr lctrons. This assumption will b justifid at th nd of this Sction. Th photon-lctron diffrntial cross-sction (Compton scattring) is givn by th Klin-ishina formula (s Bjorkn & Drll Equ.7.7 or Mandl & Shaw, Equ.8.8). In th cas of unpolarisd particls, and in th non-rlativistic limit, th total crosssction rducs to th Thomson formula, i.., lab 8 mc 8 (mc ) 6.68 x 0 9 m () whr m is th lctron mass (0.5MV). ot that th dpndnc on cancls out of Equ.(). ot that w ar safly in th non-rlativistic rgim hr, sinc th typical photon nrgis ar <V at th tims of intrst. Hnc, w nd not worry about th distinction btwn th laboratory fram and th cntr-of-mass fram. Th photon numbr dnsity at tmpratur T is givn as usual by, k BT 0.6 () c Th lctron dnsity bfor hydrogn rcombination, but aftr hlium rcombination, is givn by, () 9.9 x0 whr.9 x 0 9 is th ratio of photons to baryons (s Chaptr ). Th flux of photons is givn as usual by, Pag of 9

3 J c () Th numbr of photons intracting pr scond with a givn lctron is J. Thus, th numbr of photon-lctron intractions pr scond pr unit volum is, Intractions pr sc pr m = c (5) Altrnativly, th numbr of intractions which ach photon undrgos pr scond is, Intractions pr sc pr photon = c (6) Taking th rciprocal, th avrag tim btwn intractions for any givn photon is, Intraction Tim for a givn photon, T I = / c (7) Th lctron numbr-dnsity may b writtn as a fraction, y, of th starting dnsity, i.., 0 y (8) Th fraction, y, has bn drivd as a function of th tmpratur (tim) in Chaptr 8, and th starting dnsity of lctrons is also known in trms of th tmpratur from Equs.(,) abov. Hnc, using Equ. for th cross-sction,, allows th intraction tim, T I, to b found at any tmpratur (tim). Th rsult is as follows:- tim, t Tmpratur, y Intraction T I / t yars o K Tim T I, yars 0,000 8, ,000 0, ,000 5,750, ,000, , ,000, , ,000, , ,000, , ,000, , ,000, ,000.08,000, , ,000 (), ,50, ,000*, () 6 x 0 0 ~0 5,760,000*, () 6 x 0 ~ x 0 7,0,000* () ~0 6 ~ x 0 9 *using Tt / =.5x0 for T<000 o K. Earlir tims us T t =.0x0 0. () Ths vry low y valus ar actually rronous bcaus frz-out of hydrogn rcombination occurs at y ~ 7 x0-5 (as shown in Chaptr 9b), forming a lowr limit for y. () Using T t =.08x0 0 this would b 6,000 yars. Th final column givs th intraction tim as a fraction of th ag of th univrs. Hnc, w s from th abov Tabl that th univrs dos indd bcom transparnt Pag of 9

4 ovr th sam tim priod that hydrogn is rcombining. Th univrs is opaqu at th start of this priod (say, 80,000 yars) sinc th avrag photon will xprinc ~0 intractions during its passag across th univrs. By th nd of th priod of hydrogn rcombination, howvr (at, say, 6,000 yars) most photons will circumnavigat th whol univrs without intraction. Th tim at which half th photons intract in thir circumnavigation is ~00,000 yars (which is 0,000 yars using T t =.08x0 0 ). This is th narst w can judg to th middl of th transition btwn opaqu and transparnt conditions (translucnt?). Hnc, th usual statmnt is confirmd. ot that w can considr th transparncy condition as bing anothr xampl of a raction bing frozn out by th univrsal xpansion, this tim th raction bing lastic collisions btwn lctrons and photons. In this cas w would compar th raction rat with H =/(t), or quivalntly compar th raction tim with t. It can b sn from th abov Tabl that this maks littl diffrnc to th conclusion, th transparncy tim bing dlayd from 00,000 yars (y = 0.057) to about,000 yars (y = 0.0). ot that th intraction tim is proportional to th invrs of th fraction, y, of lctrons rmaining (s Equs.7, 8). Examining th abov Tabl w s that it is th rduction in y that causs th univrs to bcom transparnt. For xampl, btwn 80,000 yars and 50,000 yars, y dcrass by a factor of 00 whras T dcrass only by a factor of around. If y rmaind at unity, th T I /t ratio would not incras abov ~0. and th univrs would rmain opaqu indfinitly. This is ntirly rasonabl sinc it is th rduction in th numbr of fr lctrons which vntually rndrs th univrs transparnt. Howvr, it is th absolut dnsity of fr lctrons that dtrmins whthr th univrs is opaqu or transparnt. As wll as dpnding upon th fraction of th original numbr of fr lctrons rmaining, th absolut dnsity of fr lctrons is proportional to th numbr primordial baryons, i.. invrsly proportional to th photon:baryon ratio which has bn assumd to b.9 x 0 9 in th abov calculation. This raiss th intrsting qustions, (a) If th photon:baryon ratio wr diffrnt, might th onst of univrsal transparncy not coincid with rcombination aftr all?; (b) If th photon:baryon ratio wr diffrnt, might th univrs hav rmaind opaqu indfinitly? Th answr to (a) is ys, as dmonstratd in th nxt Sction. Th answr to (b) is no, th univrs would bcom transparnt vn if th photon:baryon ratio wr changd by up to ordrs of magnitud. This will b dmonstratd in Chaptr 9b.. Rcombination and Transparncy Tims Dpndnc On Photon:Baryon Ratio Firstly w r-calculat th rcombination tim for hydrogn for a rang of diffrnt photon:baryon ratios. Equ.(9) of Chaptr 8 may b writtn for an arbitrary photon:baryon ratio as, y y 0. mc k T B / kt (9) Pag of 9

5 whr is th ionisation potntial of hydrogn (.6V) and y is th fraction of fr lctrons rmaining at tmpratur T, as pr Chaptr 8 and Sction, abov. W considr ordrs of magnitud variations in, dfining th factor of chang, f, by, 9 f x.9 x0 (0) Th tmpraturs at th onst of rcombination (y = 0.99) and th nar-compltion of rcombination (y = 0.0) ar givn in th graph and Tabl blow; Dpndnc of Hydrogn Rcombination Tmpratur on Photon:Baryon Ratio % of fr protons rmain % of fr protons rmain 6000 Rcombination Tmpratur (ok) Factor By Which Photon:Baryon Ratio Is Changd, f f Tmpratur ( o K) for y = 0.99 Tmpratur ( o K) for y = , , Ths tmpraturs may b convrtd to tims as usual using t(sc s) 0.0 x 0 / T, or Tt / =.56x0 for T<000 o K. W can now calculat th avrag photon-lctron intraction tim, Ti, in xactly th sam way as in Sction xcpt that th lctron dnsity in Equ. is factord down by f. W prform th calculation at th two tmpraturs corrsponding to th onst and nd of Pag 5 of 9

6 rcombination (y = 0.99 and 0.0 rspctivly). Thus, for ach diffrnt photon:baryon ratio w can dduc whthr th onst of univrsal transparncy coincids with th onst of hydrogn rcombination (y = 99%) or with rcombination bing 99% complt (y = %). In gnral it dos not, as w s from th following Tabl, p:b ratio factor f tmp K y Ti yars univrs ag, yrs Ti / univrs ag E E-07 opaqu E E-05 opaqu E E-06 opaqu E E-0 opaqu E E-05 opaqu E E-0 opaqu E E-0 opaqu E E-0 opaqu E E-0 opaqu E E-0 bordrlin E E-0 opaqu E E+00 transparnt E E-0 bordrlin E E+0 transparnt E E+00 transparnt E E+0 transparnt E E+0 transparnt E E+0 transparnt E E+0 transparnt E E+0 transparnt E E+0 transparnt E E+05 transparnt W s that changing th photon:baryon ratio by a factor of tn ithr way just prsrvs th coincidnc btwn transparncy and rcombination, although th actual valu (~.9 x 0 9 ) maks th coincidnc most accurat. Howvr, changing th photon:baryon ratio by mor than a factor of 0 ithr way would lad to diffrnt tims (tmpraturs) for transparncy and rcombination. In short, th coincidnc of transparncy and rcombination is an accidnt which dpnds upon th particular (apparntly contingnt) valu for th photon:baryon ratio adoptd by our univrs. Turning this around, w could postulat that an quality of th rcombination and transparncy tims is rquird - which would allow th corrct photon:baryon ratio to b drivd, albit only within about an ordr of magnitud. Quit why th transparncy and rcombination tims should b rquird to b qual is not clar. Pag 6 of 9

7 . Algbraic Exprssions and th Origin of th Coincidnc It is instructiv to r-xamin th coincidnc btwn th hydrogn rcombination tim and th transparncy tim using algbraic xprssions. This clarifis how th coincidnc ariss, and what physical paramtrs it dpnds upon. Thus, w combin Equs.,, 7, 8 abov to giv th intraction tim as, TI.96. () yf kt p whr th following ar all dimnsionlss, f p = th proton:baryon ratio (0.875 from Chaptr 6); = th photon:baryon ratio (.9 x 0 9 from Chaptr ); = th fin structur constant ( / c in cgs units /7); = mc /kt; y = th fraction of th original numbr of fr lctrons lft (that is, as a fraction of thos rmaining aftr hlium rcombination); Both y and vary with tim. For compltnss w not that th cofficint of.96 drivs from th black body quation and th Thomson cross sction (Equ.) and othr factors, giving,.96 x 0.6 and x dx () x and 0.6 is th cofficint that appars in th xprssion for th numbr dnsity of black body photons. Th ag of th univrs can b writtn, from Chaptr Equ., c T t univ () k G whr th numrical cofficint is, ().68 and th trm.68 is th factor by which th thr nutrino spcis incras th radiation nrgy dnsity with rspct to that du to th photons alon. Th lattr is drivd in Chaptr 5 as,.68 8 W display all ths xact xprssions xplicitly as a rmindr of how far w can go in calculating ths faturs of th univrs from first principls. Howvr, at som point rality must impos itslf. W hav alrady notd in Chaptr that Equ.() givs a rmarkably Pag 7 of 9

8 good prdiction of th cosmic microwav background tmpratur for a first principls drivation (.5 o K). Howvr, it is bttr to us th accuratly known valu of th CMB tmpratur (.78 o K) to tun th cofficint in Equ., i.. w put instad, T 5 0. c 0 o t univ.0 x 0 K sc (b) k G Hnc, quating th ag of th univrs from Equ.b to th photon intraction tim from Equ., our dfinition of transparncy, w can simplify th rsulting xprssion to yf p 0.09 M Planck m, whr c M Planck (b) G Hnc, givn th ratio of th Planck mass to th lctron mass (. x 0 ), and givn th ratio of th dnsitis of th rmaining fr lctrons to photons prvailing at th tim (th lattr bing yf p / ), w can driv th valu of, which yilds th prvailing tmpratur, and hnc tim, of transparncy. Thus, w can chck consistncy with th rsults of Sction,.g. noting that by intrpolation th Tabl in Sction givs quality of T I and t univ for y = Substituting this valu for y into Equ. togthr with th usual valus of th othr paramtrs, as summarisd at th start of this Sction, w gt =.85 x 0 6 and hnc a transparncy tmpratur of,00 o K, in agrmnt with Sction. Th rcombination tmpratur from Chaptr 8 or Equ.9 abov may b writtn, y y 0.6 f p mc kt / kt (5) whr th numrical cofficint drivs from 0.6 = /[0.6 x ( ) / ] and 0.6 drivs from Equ.. ow th ionisation potntial of hydrogn is just th ground stat nrgy, and hnc can b writtn, mc Hnc kt (6) So Equ.5 bcoms, y y 0.6 f p / (7) which can b r-arrangd as, yf p.8 y y / (8) Again w may chck that this is consistnt with th numrical rsults in th Tabl in Sction (but noting that th RHS is xtrmly snsitiv to small changs in ). Pag 8 of 9

9 W may now xplor th consquncs of assuming that th rcombination tim and th transparncy tim ar th sam. In othr words, w quat Equs.b and 8 for th sam valu of th paramtr : M Planck m.8 y y / (9) Th valu for y which w insrt into Equ.9 dpnds upon our assumption as to whn rcombination can b said to hav happnd. If th transparncy tim is supposd to align with a tim whn rcombination is largly complt, thn y will b a smallish fraction. Howvr, it is maninglss to insrt zro sinc this is clarly only going to rsult in an infinit tim (zro tmpratur). An assumption y = 0.06 is as rasonabl as any. Solving Equ.9 for with y = 0.06 givs =.77 x 0 6, and hnc T = o K at a tim 96,000 yars. ot surprisingly, this is compatibl with our prvious rsults in Sctions and. Howvr, in this cas w hav mad no assumption for th photon:baryon ratio sinc this dos not appar in Equ.9. Rathr, having found th valu of, w can us Equ.b or Equ.8 to prdict th photon:baryon ratio that rsults from th hypothsis that th rcombination tim and th transparncy tim ar th sam. Th rsult, assuming th usual proton:baryon ratio f p = 0.875, is ~ x 0 9. In conclusion, th hypothsis that th rcombination tim and th transparncy tim ar th sam provids a mans of calculating th rquird photon:baryon ratio. Th rsult is in rmarkabl agrmnt with th obsrvd valu, ~ x 0 9, although w not that thr is an arbitrarinss in th dfinition of th rcombination tim (dfind hr as y = 0.06, which ffctivly tuns th rsult for ). Pag 9 of 9

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Brief Notes on the Fermi-Dirac and Bose-Einstein Distributions, Bose-Einstein Condensates and Degenerate Fermi Gases Last Update: 28 th December 2008

Brief Notes on the Fermi-Dirac and Bose-Einstein Distributions, Bose-Einstein Condensates and Degenerate Fermi Gases Last Update: 28 th December 2008 Brif ots on th Frmi-Dirac and Bos-Einstin Distributions, Bos-Einstin Condnsats and Dgnrat Frmi Gass Last Updat: 8 th Dcmbr 8 (A)Basics of Statistical Thrmodynamics Th Gibbs Factor A systm is assumd to