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# Adv Micro Electronics PHYS 497

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This 97 page Class Notes was uploaded by Chelsea Gerhold I on Friday October 30, 2015. The Class Notes belongs to PHYS 497 at Texas A&M University - Commerce taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/232401/phys-497-texas-a-m-university-commerce in Physical Science at Texas A&M University - Commerce.

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Date Created: 10/30/15

lmlt Id N Review of observable Qr QQZr i las Basic parameters to compare theory and observations Mass M Luminosity L gt The total energy radiated per second ie power in W 00 L j Lidl 0 Radius R Effective temperature Te The temperature of a black body ofthe same radius as the starthat would radiate the same amount of energy Thus i 471122073 where 039 is the StefanBoltzmann constant 567x 10398 WmZK394 Stellar luminosities vary from 104 108 that of our sun with surface temperatures varying from 2000 50000 K gt 3 independent quantities l depends on distance to star inverse square law V 2 L 47rd iiii ie39jnice if d is Known then L determined 95ij determine distance if we measure Qaralax apparent stellar motion to orbit of earth around Sun Nearest star Proxima Centauri Parallax after 6 month is 0765 9 d 1AU0765 427 ly Since nearest stars d gt 1pc must measure p lt 1 arcsec eg and at d 100 pc p 001 arcsec Telescopes on ground have resolution 1quot Hubble has resolution 005quot gt difficult Tycho 2 satellite measured 106 bright stars with 6p 0001quot gt confident distances for stars with d lt 100 pc 71 ll 3 i allilr rlii Angular diameter of sun at distance of 10pc e 2R 10pc 5gtlt 10399 radians 10393 arcsec 1 Size of Star u Size ol Earth s Orbit Il Size of Jupiter39s Orbit Compare with Hubble resolution of 005 arcsec gt very difficult to measure R directly Radii of 600 stars measured with techniques such as interferometry and eclipsing binaries 9 Radii have to be obtained another way UI Q 39l 9 The rIzrri zjgr39umqjR0552JJ Hggr um 444 108 x 0 M R L and Te do not vary 9 Deneb B393te39geusr Independently 104 0 e o Two major relationships o 790 quota 2 LwithT j10 Vegga 2 H c 7x v LWIthM a we SII IUSA x10 9 1 8 g 79 a Sun The first is known as the E 93 m S 2an HertzsprungRussell H R quot 102 0 quot5 5 9 diagram Q h g Ptocyon B 0 quotJ orthe 10 63K colormagnitude diagram I I I I 30000 10000 5000 3000 2500 Temperature K Colour lndexBV 06 0 06 20 Spectraltype O B A F G K M 212 The Herfzs run Russel din aa39 II o1 I g d a h d1 Typical life paih of a lowmass star 108 Planetary 106 nebula Red r supergiant 104 r o 50 M i G Horizontal Bed 3 102 10 Me O 7 branch g39am E 3 M A 39 a Red g 1 C 0 subgiant l 102 1 Me O Sun 10 4 White 03 MG dwarf 10 6 I I I I I I I 1 I I I I I I I I 1 100000 50000 10000 5000 1000 Effective temperature Te K On main se uencei 1 on the left upper corner a star with L 6 L0 and T 4 TO 9 R 60 R0 2 red cool dwarf stars right lower corner with L 200 L0 and T12TO 9 R01RO Out of main sequence 1 supergiants L 104 L0 and T 12 TO 9 R 400 R0 2 red giants L 102 Loand T 12 TO 9 R 50 R0 3 white dwarfs L 1200 Loand T 2 TO 9 R 150 RO Eliilirldfj l39lv r lui io ri 39r thfe i ew mainsequence stars for h masses are known there is a Massluminosity relation 7 397 Lee M l Where n 3 5 Slope changes at extremes less steep for low and high mass stars aim 425 Haiti i This implies that the mainsequence gliltgl l m 3 M8 on the HR is a function of mass ie from bottom to top of main sequence stars increase in mass We must understand the M L relation and L7 relation theoretically Models must reproduce observations m a a E c E E 3 E c m a E E 3 1eHIIEI 1JJEIIJEI 1EZIIIEI 1EIIJ 10E I11 001 0001 39 01 1n Adapted from data I omp e 1 In Swahili3v S Bessomw 1984 i t 1 itquot h 1 1H 1 urn at m i i 1 I i 1 1 1 Hi i 139 If i 1 tilquot I l39u39laaa Solar masseaj 10 100 1 0 97 More massive stars burn hydrogen more quickly 939 stars with larger masses have higher effective temperatures 1 1 Ae and metallici r sithatwe can measure dilikeitostudystars of same age and chemical composition to keep 55 arameters constant and determine how models reproduce the other observables NGC3603 from Hubble Space Telescope We observe star clusters Stars all at same distance Dynamically bound Same age Same chemical composition Can contain 103 106 stars 1 3 STar39 clusTer39 known as The Pleiades Globular clusTer39 NGC 2808 Selection of Open clusters l l l PIC362362 F leiai eg Cluster HR diagrams are quite similar Age determines overall appearance i In these diagrams BV logTe and MV logL Open MStur n off point varies massively faintest is consistent with globulars Maximum luminosity of stars can get to Mvz 10 Ver y massive stars found in these clusters The differences are interpreted due to age open clusters lie in the disk of the Milky Way and have large range of ages The Globulars are all ancient with the oldest tracing the earliest stages ofthe formation of Milky Way 12gtlt 109 yrs Summary If 9 JT5ZJ JVWUQJ ES Fourfundamental observables used to parameterise stars and compare with models M R L Te M and R can be measured directly in small numbers of stars Age and chemical composition also dictate the position of stars in the HR diagram Stellar clusters very useful laboratories all stars at same distance same 1 and initial 2 We will develop models to attempt to reproduce the M R L Te relationships alloWed for a sTaTic win Hubble announced 7 o f galaXies ouTside our y ay shoWed ThaT They were cally moving away from us wiTh a Was p roporTional To Their disTance 39f m us The more disTanT The galaxy The fq er iT was receding from us The universe was expanding afTer all jusT as General RelaTiviTy originally predicTed Hubble observed ThaT The lighT from a given galaxy was shifTed furTher Toward The red end of The lighT specTrum The furTher ThaT galaxy was from our galaxy The specific form of Hubble39s expansion law is imporTanT The speed of recession is proporTional To disTance The expanding raisin bread model aT lefT illusTraTes why This is imporTanT If every porTion of The bread expands by The same amounT in a given inTerval of Time Then The raisins would recede from each oTher wiTh exachy a Hubble Type expansion law In a given Time inTerval a nearby raisin would move relaTively liTTle buT a disTanT raisin would move relaTively farTher and The same behavior would be seen from any raisin in The loaf In oTher words The Hubble law is jusT whaT one would expecT for a homogeneous expanding universe as predicTed by The Big Bang Theory Moreover no raisin or galaxy occupies a special place in This universe unless you geT Too close To The edge of The loaf where The analogy breaks down a m 39 o Hubbles discover ix J 9 5 gt r f l gt i r 3 1 f i l 559 nm IMIU I b 341nm 39 13 F Eiuw Hiio22kmismlly 35 5 quot 3 g TEW El hiiimhi s39 orlginal plat Dlslance Mlya Doppler effect 52 1 1 where z121 D09 2 3r hi f39ij Comparison of laboraTor39y To blueshi ed objecT Reference lines from laboratory source Absorption lines from star Comparison of labor39oTor39y To redshi ed objecT 39 Reference lines from laborato source L Absorption lines from star Light from Galaxies are red shifted ie all galaxies move away frOm each other Velocity of recession is proportional to distance 1 ant i 6 Systematic s per 10 y Expansion 5 64 kms per 106 pc an 39 0 or H39114gtlt109y Hoyle39s conclusion Big Bang Turn time arrow around in Hubble s discovery then the matter in the Universe has once been very closely together This hypothetical 39singularity39 was named Big Bang by Sir Fred Hoyle Hoyle together with Taylor also realized that the observed large 4He mass fraction could not have been produced in stars 4 Reeirlim ril iijr rl il The existence of the CMB radiation was first predicted by George Gamow in 1948 and by Ralph Alpher and Robert Herman in 1950 It was first observed inadvertently in 1965 by Arno Penzias and Robert Wilson at the Bell Telephone Laboratories in Murray Hill New Jersey The radiation was acting as a source of excess noise in a radio receiver they were building Coincidentally researchers at nearby Princeton University led by Robert Dicke and including Dave Wilkinson of the WMAP science team were devising an experiment to find the CMB When they heard about the Bell Labs result they immediately realized that the CMB had been found The result was a pair of papers in the Physical Review one by Penzias and Wilson detailing the observations and one by Dicke Peebles Roll and Wilkinson giving the cosmological interpretation Penzias and Wilson shared the 1978 Nobel prize in physics for their discovery Today the CMB radiation is very cold only 2725o above absolute zero thus this radiation shines primarily in the microwave portion of the electromagnetic spectrum and is invisible to the naked eye However it fills the universe and can be detected everywhere we look In fact if we could see microwaves the entire sky would glow with a brightness that was astonishingly uniform in every direction The picture at left shows a false color depiction of the temperature brightness of the CMB over the full sky projected onto an oval similar to a map of the Earth The temperature is uniform to better than one part in a thousand This uniformity is one compelling reason to interpret the radiation as remnant heat from the Big Bang it would be very difficult to imagine a local source of radiation that was this uniform In fact many scientists have tried to devise alternative explanations for the source of this radiation but none have succeeded Isnmqu IF THE EDEMIE MICRDWAVE BACKERDIJNI iT eJSTraighT line conTinuously in a sTraighT line 39 circle wiTh any cenTr e and radius or equal To each oTher e esszhan Two righT angles The Two lines if exTended meeT on ThaT side on which are The angles less Than The Two righT ForThousands of years maThemaTicians Tried To prove asserTion 5 wiTh no success Gauss showed ThaT iT is in facT impossible To prove ThaT In facT iT is possble To show ThaT iT is wrong for cerTain curved spaces Eg lines on The surface of an sphere r39oll iT To a cylinder 9 sum of angles is TnquotclitghT line and iT will r39eTur39n To The same poinT ThaT is 155 euclidean buT has a differenT Topology piaf cledTdxdydz is deTer39mined iTs local geomeTr39y ITs measur39e l dfeTer39mines infiniTesimal disTances EXam39piles for The case of Euclidean geomeTr39y ds2 dxi2 dy2 Car39Tesian coor39dinaTes dsz 2542dX2 dy2 X is measured in inches for some reason while y and s are measured in cm ds2 dr392 r2 d 9 2 polar39 coor39dinaTes wiTh angle 9 in radians 6 l39C ordinafes39 in Spherical geomeTr39y A jcajn be app OXMafeb Transformed inTo asmall region around any poinT For insTance near39 r The quotTaxicab meTr39ic t39lsdy39l J stances driven in ciTies wiTh a grid sTr39eeT plan in The limiT of a n39i iTely small blocks This shows Tha r quadr39aTic meTr39ics which are The only ones we use in physics are JusT a special case In cosmology we have a 4D spaceTime which according To gener39al r39ela riviTy is locally equivalenT To The Minkowski metric of special r39elaTiviTy dsZ ch2 dxz dyz dz l directions 9 There is a quotnaturalquot time V mallframes with comoving observes q moment in the life of the universe for instance Thu space is finite and the universe is said to be closed 39 E1211 spaCe Conventional geometry of Euclid k 0 This can be considered as the limit of the other two cases for infinite radius of curvature Because it is balanced between the other two this is sometimes called a critical universe For true Euclidean geometry the topology is also open meaning that space is infinite in all directions It is also possible to have compact topologies e g the 3torus in a at space which have finite volume and so are closed the hardest case to imagine as it is not 7 r quoti onutan negrativeurvature a pseudosphere in i 39 v local regions of negative curvature e g jquot am eSttopological case is when the universe is in nite 011 to he hwjl t in nitequot than the Euclidean case in the sense that at a given llquot quot more space than we would expect from Euclidean geometry As 39 are compact tOpologies with negative curvature which are closed is unaffected by Topology which simply restricts the possible the coordinates The RW metric is encapsulated in the equation Zci 2 sin2 9 d 2 The evolution of the Universe is governed by the scalefactor Rl the term in square brackets is the square of the distance 0 12 and the physical distance between two points is just Rd At the moment R is increasing with time which is another way of saying that the Universe is expanding The geometry of space at a given time is de ned by the function 5km which is just a neat way 39of writing the 3 different functions needed for the 3 values of k k Ska Curvature l sinoa Positive 0 X zero 1 sinhOd Negative The geometry of space cannot change with time in a homogeneous universe if the curvature is negative to start with it stays that way The same applies to topology a fundamental observer39s neighbours cannot suddenly or gradually change with time The Schwartzschild metric For spherically symmetric objects rsr ltltl except close to black holes or for neutron stars 2 7 2 2 drz 2 2 2 2 2 ds l S 0 dt 1 A72 r d6 r s1n6 d r For the universe RobertsonWalker 39 2 is 1 r 5jczdt2 1dr 2 r2d62 r2sin WOW I I 30 Klt1 9 III Kgt calar curvature guv the metric tensor 5 a o LAW 1 mg 39 ten or 639 the gravitational constant imam TI39Re dugce to Newton39s law of gravity by using both the weakfield approximation and the slowmotion approximation The cosmological constant Introduced by Einstein to allow for a static universe ie one that is not expanding or contracting quotM y biggest buna er f Einstein EXPANSION OF THE UNIVERSE 1 open Universe w I where kde rermines The geometry of The Universe at Universe k gt 0 closed Universe Big Crunch closed Universe Relative size of the universe N l k 0 at Universe expands forever k lt 0 open Universe Big Chill Billions of Years quotFriedman39s equations can be Newfonian equations This is due To a a force in spherical symmetry The of dlas frquot7i39buion is zero Then one can easily AS I5 given from The conservation of energy V2 GM LR 2 G47rpR3 R 2 3R E 5 Compare 1390 Friedmann equa l39ion The equaTion is The same as ThaT of The EinsTein equaTions if we idenTify The ToTal energy EwiTh The curvaTure k The piece of The Universe we examine will expand forever if E 0 k J or E 0 k 0 open Universe and iT will recollapse if Elt 0 k I closed Universe NewTon could have done all The cosmology ThaT was done in The 2039s if he was willing To consider an expansion of iTs Universell mm 3MWM3 Mkmg P go m E hem ke39dl ye n 39 39 39 l0 T 100 years ago We he nearesT big galaxy y million years ago ng disTanT galaxies wiTh e ce Telescope can see Them as ere only a few billion years afTer The ng MosT cosmologisTs believe ThaT The s beTween 12 and 14 billion years o The GMB radiaTion was emiTTed only a few hundred Thousand years afTer The Big Bang long before sTars or galaxies ever exisTed Thus by sTudying The deTailed physical pr OperTies of The radiaTion we can learn abouT condiTions in The universe on very large scales since The radiaTion we see Today has Traveled over such a large disTance and aT very early Times One of The profound observaTions of The 20Th CenTury is ThaT The universe is expanding This expansion implies The universe was smaller denser and hoTTer in The disTanT pasT When The visible universe was half iTs presenT size The densiTy of maTTer was eighT Times higher and The cosmic microwave background was Twice as hoT When The visible universe was one hundredTh of iTs presenT size The cosmic microwave background was a hundred Times hoTTer 273 degrees above absoluTe zero or 32 degrees FahrenheiT The TemperaTure aT which waTer freezes To form ice on The EarTh39s surface In addiTion To This cosmic microwave background radiaTion The early universe was filled wiTh hoT hydrogen gas wiTh a densiTy of abouT 1000 aToms per cubic cenTimeTer When The visible universe was only one hundred millionTh iTs presenT size iTs TemperaTure was 273 million degrees above absoluTe zero and The densiTy of maTTer was comparable To The densiTy of air aT The EarTh39s surface AT These high TemperaTures The hydrogen was compleTely ionized inTo free proTons and elecTrons at mm 31 I J I L r I SPECTRUM IF THE Eusmu MICROWAVE BACKGROUND Frequency GHz 100 200 300 400 500 T 2725 i 0001 K ing To 39The Big Bang I 300 7 s rum of The fCMB should g orm This was indeed measured 2 r y by The FIRAS experimenT 5 200 Dre shows The predicTion of The Big Bang E r The energy specTrum of The cosmic 100 m hsured The specTrum aT 34 equally spaced poinTs 0 along The blackbody curve The error bars on The daTa poinTs are so small ThaT They can noT be seen under The predicTed curve in The figure There is no aITernaTive Theory yeT proposed ThaT predicTs This energy specTrum The accuraTe measuremenT of iTs shape wasanoTher imporTanT TesT of The Big Bang i i 1 02 M 007 005 Wavelength cm m E am lgme e mm Y mile 7 at 39 kn hundredTrh iTs 7 gave background very weakly wiTh neuTral phoTons moving Th rough The se s analogous To The propagaTion of h T Through The EarTh39s aTmOsphere s in a cloud are very effecTive aT qugh clear air Thus on a cloudy day we can lookt Through The air ouT Towards The clouds buT can noT see Through The opaque clouds Go smologisTs sTudying The cosmic microwave background radiaTion can look Through much of The universe back To when H was opaque a view back To 400000 years afTer The Big Bang This wall of lighT is called The surface of lasT scaTTering since H was The lasT Time mosT of The CMB phoTons direchy scaTTered off of maTTer When we make maps of The TemperaTure of The CMB we are mapping This surface of lasT scaTTering shjown aboVe one of The mosT sTriking 39feaTUresabouT The cosmic microwave background is iTs uniformiTy Only wiTh very sensiTive insTrumenTs such as COBE and WMAP can cosmologisTs deTecT flucTuaTions in The cosmic microwave background TemperaTure By sTudying These flucTuaTions cosmologisTs can learn abouT The origin of galaxies and large scale sTrucTures of galaxies and They can measure The basic parameTers of The Big Bang Theory Big Bang We Can only see the surface of the cloud where light 137 Billion Veal was last scattered when Big Bang The osmk miaowave battkgruund Ranialion39s mum oi Im wiler isznalngou5 w me llgm ning hmth lhe loudslo our eyg on a dandy nayl 3 7 o l afj39iilci 39flill rl b iiiblll39l39ll Triangle In The early hoT dense universe flucTuaTions slosh ed back and forTh We can calculaTe JusT how far iT could slosh in The Time from BB To when we see iT This seTs a disTance and defines a Triangle Qolt1 Sloshing size gt Distance to background A ri12 2005 GEOMETRY OF THE UNIVERSE CLOSED Fluctuations largest on haltdegree scale Fluctuations largest on Fluctuations largest on 1degree scale greater than 1degree scale David Wilkinson Universify of Michigan Alum Wilkinson Microwave AnisoTr39opy Probe FIRAS showed that the spectrum of the CMB radiation was that of a blackbody with T2725 K consistent with prediction that Universe cooled from initial hot big bang FIHAS data wlth 4000 errorbars 2725 K Blackbody DMR showed fluctuations of 1 part in 100000 once the motion of the sun and the emission from the galaxy are subtracted off The fluctuations are regions slightly hotter or colder than the average temperature of 2725 Intensity MJysr 2006 Nobel prize in physics awarded to George Smoot and John Mather for measuring temperature of CMBR precisely at 27 K with COBE 5MB MariSLJr Zmzrrrta Univzr is39a l3 Fliri w p If universe is closed hot spots appear larger than actual size J t 3945 b If universe is flat hot spots appear actual size c If universe is open hot spots appear smaller than actual size f mll 5 7 l ar39e ab0uT 1011 1012 solar masses in a galaxy and The e 39 jldiFSTance beTween galaxies is abouT lMpc Thus The Universe ear39i noa bie Too far39 away from The cr39iTical densiTy From WMAP obser39vaTion we know ThaT The Universe39s densiTy agr39ees 39wiTh The cr39iTical densiTy wiThin an uncer39TainTy of 2 We will in The following assume ThaT The Universe is FlaT SOILI HQV i o Fr ieclirnqm erllJcz fi0quotu Friedmann equa139ion fork O To solve one needs pR There are Two important cases p R4r39adia139iondomina139ed Univer39Se Then Here The index 39039 refers To The values Today As usual we have set RUG 1 u i l H r13 r ac arl39r ClJB ZFfZrl C3 The high redshifT Type Ia supernova survey implies ThaT Universe acceleraTes now The WMAP Boomerang measuremenTs of The CMB power specTrum supporT inFlaTion and Q 1 general fiT assume k 0 20 102 007 005 100 230 0023 i 0004 0022 i 0003 9140 031 i 006 033 i 005 QA 074 007 011 068 004 006 h 060 i 009 066 i 005 age 162 i 25 X 109 y 140 i 05 X 109 y All energy densiTies are measured in uniTs of The criTical densiTy Wha l ls UI The soa irwloc czsl CCHBTCJHT The cosmological consTanT corresponds To an energy density inher39en l39 To The expanding Universe Its presence modifies 139he Friedmonn equa139ions 139o 872G kc2 TpchMatter F Expected from InFla l39ionar39y Univer39Se 4 6 i o cal consTanT CW ro Hub les ConsTanT man TTefrquotambdaquot L Theory of general 7 arm general re laTiviTy un 39erse musT eiTher expand or ThoughT The universe was sTaTic n maThemaTician realized ThaT e fix like balancing a pencil on and proposed an expanding universe model now called The Big Bang Theory When Hubble39s sTudy39 of nearby galaxies showed ThaT The universe was in facT expanding EinsTein regreTTed modifying his eleganT Theory and viewed The Cosmological consTanT Term as his quotgreaTesT misTakequot Many CosmologisTs advocaTe reviving The cosmological consTanT Term on TheoreTical grounds Modern field Theory associaTes This Term wiTh The energy densiTy of The vacuum For This energy densiTy To be comparable To oTher forms of maTTer in The universe iT would require new physics The addiTion of a cosmological consTanT Term has profound implicaTions for parTicle physics and our undersTanding of The fundamenTal forces of naTure The main aTTracTion of The cosmological consTanT Term is ThaT iT significanle improves The agreemenT beTween Theory and observaTion The mosT specTacular example of This is The recenT efforT To measure how much The expansion of The universe has changed in The lasT few billion years Generically The graviTaTional pull exerTed by The maTTer in The universe slows The expansion imparTed by The Big Bang Very recenle iT has become pracTical for asTronomers To observe very brighT rare sTars called supernova in an efforT To measure how much The universal expansion has slowed over The lasT few billion years Surprisingly The resulTs of These observaTions indicaTe ThaT The universal expansion is speeding up or acceleraTing While These resulTs should be considered preliminary They raise The possibiliTy ThaT The universe conTains a bizarre form of maTTer or energy ThaT is in effecT graviTaTionally repulsive The cosmological consTanT is an example of This Type of energy Much work remains To elucidaTe This mysTery If The cosmological mo s39T of The energy en The exTrap olaTed us much larger Than iT would in Term which helps avoid The The rexiTrapolaTed age of The u er Than s0me of The oldesT 7 gt e A cosmological consTanT Term To The inf laTionary model an exTension of g Bang Theory leads To a model ThaT iapptears To be consisTenT wiTh The observed largescale disTribuTion of galaxies and clusTers wiTh COBE39s measuremenTs of cosmic micrOWave background flucTuaTions and wiTh The observed properTies of Xray clusTers The InflaTion Theory proposes a period of exTremely rapid exponenTial expansion of The universe shorle afTer The Big Bang While The Big Bang Theory successfully explains The shape of The cosmic microwave background specTrum and The origin of The lighT elemenTs iT leaves open a number of imporTanT quesTions Why is The universe so uniform on The largesT lengTh scales Why 39is The physical scale of The universe so much larger Than The fundamenTal scale of graviTy The Planck lengTh which is one billionTh of one TrillionTh of The size of an aTomic nucleus Why are There so many phoTons in The universe WhaT physical process produced The iniTial flucTuaTions in The densiTy of maTTer The InflaTion Theory developed by Alan GuTh Andrei Linde Paul STeinhardT and Andy AlbrechT offers answers To These quesTions and several oTher open quesTions in cosmology IT proposes a period of exTremely rapid exponenTial expansion of The universe leading To The Big Bang expansion during which Time The energy densiTy of The universe was dominaTed by a cosmological consTanT Term ThaT laTer decayed To produce The maTTer and radiaTion ThaT fill The universe Today The InflaTion Theory links imporTanT ideas in modern physics such as symmeTry breaking and phase TransiTions To cosmology The InflaTion Theory makes a number of imporTanT predicTions a The densiTy of The universe is close To The criTical densiTy and Thus The geomeTry of The universe is flaT b The flucTuaTions in The primordial densiTy in The early universe had The same ampliTude on all physical scales c There should be on average equal numbers of hoT and cold spoTs in The flucTuaTions of The cosmic microwave background Prooar as of Cosmological lJ39n Jvawa RmR U DeceleraTion parameTer of The Universe 0 IT can be shown ThaT A lt gt QA Using The WMAP resulTs QMamr 03 and A 07 one finds qo lt 0 The Universe acceleraTes Today However GraviTaTion has a braking effecT Thus The cosmological consTanT corresponds To a subsTance wiTh negaTive pressure SolluTion for a Universe wiTh A AsympToTically for large values of RT The cosmological consTanT dominaTes and The Friedmann equaTion reduces To When The Universe is dominaTed by a cosmological consTanT The expansion raTe grows exponenTially 4 9 V p l Temperature Grand U i il IICCiTlQl ofuniverse 1032 K 1027 K 1015 K 1013 K 3K Strong nuclear force Electromagnetic force Time after 1043 s 10 35 s 1012 s 10 6 5 5x1077 5 Big Bang now kT uni l39ed forces big bang Time 102 GeV elec rromagne ric weak 1013 GeV elec rromagne ric weak 10 35 s strong 1019 GeV elec rromagne ric weak 10 43 s strong gravi ry Planck Time The separation of forces happens in phase Transitions wi139h spon l39aneous symmetry breaking Strong bras weak f0 me 31 actroweak at 6 om g 3W 3U per fume GUT force relative strength of force gravity I 1015 1019 1023 temperature K WhaT Triggered inf aTion When The universe was very young very hoT and very dense all The forces of naTure sTrong weak elecTromagneTic graviTaTionaI were probably unified AT around T 1025 K graviTy may have decoupled from The GUT grand unified Theory force To power a brief bursT of ianaTion The UT Erma From The Planck er39a To abouT 103935 seconds is The Grand UnificaTion Theory er39a Forces gr39aviTy sTr39ongweakelecTr39omagneTic MaTTer39 forms a sea of free quarks phoTons and some oTher39 parTicles These flucTuaTions are small buT as The Universe expands They eT large enough To be The seeds 0 galaxies UI J Irifla iiori tillage The TransiTions are conTrolled by scalar fields sf which can have negaTive pressure Psf usf where usf is The energy densiTy of The fields If usf gtgt pmd Friedmann equaTion reads The soluTion is Model predicTs Tim 10 34 s InflaTion lasTs as long as The energy densiTy associaTed wiTh The scalar fields is Transformed inTo expansion of The Universe If This lasTs from T1 10 34s To T2 10 32s Then expATTinf e100 1043 AfTer This exponenTial increase The Universe develops as we discussed before UI DJ Irrr lq iio ri Solves pr39obleirri 39 q hrizjf Fr39iedmann OIquot wiTh one has Thus during The inflaTion period The energy densiTy of The Universe comes very very close To The cr39iTical value Inflation Solvzs pr oblemji horizon The CMB is found to very smooth dTT 105 This suggests that the different parts of our universe communicated before they equilibrated Light can only travel a finite distance during the finite age of Universe and photons of the CMB coming from opposite directions and observed today can have no information from each other However inflation allows every part of the Universe to have been in close proximity before the inflation phase In ationary Epoch FL 1 1a 40 1 g 30 standard Model Had in 3 o Obsewable 10quot Unmarse 10 40 in mm l l 1 l I 1045 1035 1025 1045 10 5 10v w s frijgr iivfl bobbin The Universe looks The same everywhere in The sky ThaT we look yeT There has noT been enough Time since The Big Bang for ighT To Travel beTween Two poinTs on opposiTe horizons This remains True even if we exTrapoIaTe The TradiTionaI big bang expansion back To The very beginning So how did The opposiTe horizons Turn ouT The same eg The CMBR TemperaTure AT T 103935 s The Universe expands from abouT 1 cm To whaT we see Today 1 cm is much larger Than The horizon which aT ThaT Time was 3 x 103925 cm 3 x 0 35 cm Zi39i39lri iri iii bri Space expands from 3 x 103925 I cm To much bigger Than The I V Universe we see Today 5 39 Flg i rlaaa Problem Why does The Universe Today appear To be near The criTical dividing line beTween an open and closed Universe I DensiTy of early Universe musT be correcT To 1 parT in 1060 in order To achieve The balance ThaT we see InflaTion flaTTens ouT spaceTime The same way ThaT blowing up a balloon flaTTens The surface Scale Factor 211 I I I I I I I I I I Densiiy 1 n5 after BB Since The Universe is far bigger Than we can see The parT of iT ThaT we can see looks fIaT UI W quot 10 sec0nd electromagnetic and weak forces separate 39 10396 second free quarks condense into protons and neutrons 39 1 second Universe becomes transparent to neutrinos 1035 gt 1025 1020 015 1010 Temperature K I O U l and Or End of Planck time gravity freezes out Strong force freezes Cinf39nerknfnt out inflation begins 0 War 5 Weak and Universe E electromagnetic transparent E I forces freeze out to neutrinos Synthesis of gt a E 4 primordial E E i 8 5 helium e I I 3 c l i a E E Universe E E g transparent 5 E 3399quot E l l l t0 Photons m 2 l i I I I 3 I l H O E l i i l I Now I S I E E I E I 3 IE I 10 50 1040 10 30 10 20 1010 1 1010 1020 Time after Big Bang 5 UI QA 1 14 Il u a u 99 I u 99 cxpuudxigggg v QM w7dummwxmemmM Flul quot6amp0 if J y 0 Univcrsc 906 9 0 a y I s39 H 0 1 2 i mifstry 187Re 9 187Os tlZ 40 billion years To 12 17 238 gdecay t1245 billion years 16quotE billion years Ageof oldest star clusters measure luminosity of brightest star relies on stellar evolutionary model To 11 13 billion years 4 v Oldest white dwarfs measure luminosity of faint white dwarfs to determine how long they have been cooling To 1213 billion years S icvlclqr cl Bic Balm lllloclel 1 In its earliest phase The Universe was hot enough 139ha139 all particles present have been in equilibrium 2 The physical laws as we know Them have been valid For39 The following discussion i139 is sufficient 139o assume Thai The Universe has been ho Her39 Than 1012 K 5 rs posiTive This requires ThaT baryon oldTed in a very early phase of The Universe aT KL l 1 imparflcles which we know have exisTed aT The Time of inuoleosynThesis 7 v 17 ex n p All oTher parTicles had l an earlier phase efosmolog39ical principle is valid ie The Universe is isoTropic and homogeneous General RelaTiviTy is The correcT Theory of graviTaTion From These assumpTions one can reconsTrucT The Thermal hisTory of The UniVerse aT The Time of nucleosynThesis The fundamental euafions I eng rh scae39r39adius39 p energy densi ry G gravi ra rional consfan r P pre33ure reactions eg up ue uma u v Thegchemicol potentials can be deduced from The following densi ries NQ charge densi39fy NB baryon number densi ry N a elec rron Iep ron number densi ry NH muon Iep ron number densi ry General Ni N up He Uve UV Chemical o ren riuls 39c1 illeNi musf be odd func rions of pi H 0 for39 all i 9 Particle disfr39ibu rions depend only on T Par Ticle disTribuTions and v 1339 V 4 872395k4 a a 7 7 15h c 71 6 7 Thermal equilibrium In Thermal equilibrium one has energy densi139y Mechanical pressure of a relativistic gas Zi sum over all particles presen i in The equilibrium l39e39rriye r q ilir erRquiLi r elq iiori 2quotd law of Thermodynamics in equilibrium dSVTdUPdVdlpeqTVlIqTdV lt1 5SVT 1 6SV T V dp T T P T gt Z L W Tpeq 81 67 T 52501 62SVT Usmg gives W W dPTTpeqTzqT 2 Together with energy conservation R3 PdiR3Pp follows I t R73lpeqTPeqTl 0 3 A giiaba i ig Expq r1319 1 By inser39139ing Eq 2 info 1 we find dSV T Valqu qu Peg a V V dpeq Pet VFpeq Pet yT From This follows SVTgt peqnzqltn lt4 If we set V R30 Then Eq 3 implies The conservation of entropy S in The volume V R30 S SR3T R73 peqT Peg T consl The expansion is adiaba l39ic Ultrar39elativistic articles 1E g1 gigs when energy and pressure are 11 Emma fg r y 4 dT 3T T4 The constant depends on which particles dominate peg T tram Eq 3 above d R34 401 R3T3 0 dzT3peq 3dz or R3 T 3 const from which follows 7 1 7 1 llhic pgr39ficlas are EJF39ZL ZKLT M eilldlilibr itlm Only particles with masses m lt kT are present in equilibrium with noticeable number densities For T x 15 1012 K m1T 140 MeV these are pi e v WVWVWTE v u39s e39s Fermi distribution photons Planck distribution neutrinos Fermi distribution When the temperature decreases particles might fall out of equilibrium depending on their masses Thus their number densities are reduced by a Boltzmann factor exp mkT This affects first the muons with mass mu 105 MeV 7 gr m id f6 reactions mediaTed TraTune When The Universe WQ I a gS fi maTe around which Temperafur39e This will m 3 E i irriq re of The iiiadvFreezzml i ie rrirgterq rt Ira Cross section for weak processes 2 74 2 0W z gwh kT 3 Number densrhes for muons elec139rons n kT h l N Reacfion ra re for weak processes per ep139on 0W I ll g jf7 kT5 gW 14gtlt10 49 ergcm3 kT 3 Total energy densi ry 0 N E 3971 N kT 39 Expansion ra l39e H m g z Gp z Jawmy T 3 1010K Then 0W 397 H For T 1012 K muon and electron neu139rinos are still in equilibrium wi139h matter 7 4 Weak freezeOUT ldboi39aTe calculations yield Tfo z 8 109 K s ses cannoT keep pace wiTh The expansion of euTr39inos decouple from The r39esT of The Tur39 sas our r39aTe esTimaTe has To be cor39r39ecTed by a Boszmann facTor39 lkT for39 These species If one Takes This inTo accounT u neuTr39inos d 39 oup e aT TemperaTur esar39ound 3 1011 K For a shor39T period The decoupling has no consequences NeuTr39inos and maTTer39 phoTons e are ulTr39ar39elaTisTic par39Ticles wiTh T UP and TV T rne 511 keV z 4 109 K AT T gt39 me e e 69 phoTons AT T lt me e e 9 phoTons Thus elecTr39on posiTr39on annihilation heaTs The phoTon baTh BuT This occurs afTer39 weak fr39eezeouT Thus The neuTr39inos did noT noTice As a consequence neuTr39inos TV and phoTons T have differenT TemperaTures afTer39 The e e annihilaTion We will use The conser39vaTion of enTr39opy To r39elaTe The TV and T a mw 39 ar39ly Unti ver39Se gf i Z pa 21 an 07 Z Gi lj 4 If We39 36 The number of neu l39r39ino genera rions 1390 n 3 we have If peg aT4 Ralcr rincl T and TV For39 T lt Tfo only photons e1 ei relativistic for T gt 5 109 K 4R3 n epeip7aRT3 For39 T lt 5 109 K e e annihilate for39 T lt 109 K only photons pr39esent 4R3 4 3 S 707 aRT 6 As 5 const follows from 5 and 6 RTTlt109K Ejm RTTgt5x109K 4 D Relct rinclj CUlCl TV The ei annihilaTion heaTs The phoTon baTh buT has no influence on The decoupled neuTrinos For neuTrinos T lR afTer freezeouT RTV Tlt109K 1 RTV Tgt5x109K Thus neuTrinos and phoTons have differenT Temperatures since T lt 109 K One finds 1 1 13 T 21 TV 14TV The phoTon TemperaTure has been observed as The Cosmic Microwave Background Observing TV is sTill a holy grail NoTe For 109 K lt T lt 5 109 K et are noT uTrareaTivisTic and The cacuaTion of The enTropy is more elaboraTe leading To correcTions 3 13 SEER 3 TV 1 73 me 3 T CT 11 CT EvolLr39riori 01quot CI r ctrllica iiorClomi 39 Kl39l d l ribW39 For 5 109 K lt T lt 1012 K The Universe is dominated by ul rrarela l39ivisTic particles 7 ei v Thus 1 T T4 peq R4 I Wi139h The help of The Friedmann equa l ion 872G z 435R4 45 4 eq peg R R 3 which is solved by 3 2 1010K 2 consl z 109 sec consl 3272Gqu For 109 K lt T lt 5 109 K numerical solution necessary Note R139 139 2 for radiafiondominafed Universe J kgi39m3 Density 1010 1015 10 20 1025 1030 Crossovar point MATTER DOMINATED RADIATION DOMINATED Matter density Radiation density iIIIiIiIil 1 102 104 105 108 1010 Time since Big Bang yr The early universe was a soup of lighT and parTicles As The early universe expanded The densiTy of maTTer was dropping rapidly MaTTer and The radiaTion were coupled The radiaTion era for The firsT 2000 years afTer The big bang The densiTy of radiaTion exceeded The densiTy of maTTer The maTTer era as The early universe expanded The densiTy of maTTer overTook radiaTion Today The universe is maTTerdominaTed AT abouT 100000 years The maTTer densiTy was low enough To allow radiaTion To escape This radiaTion is whaT see Today as The cosmic microwave background h T quot1 4Tv However around form This reduces The cross secTion A becomes TransparenT for phoTons nd radiaT ion decouple quot there are 3 differenT Temperatures in The UniverSe TV TV which BpTh scale Iike UP and Tma er The Cosmic Microwave Background CMB is wiTness of The decoupling of radiaTion and maTTer However iT has been exTremer redshifTed since Then due To The conTinued expansion of The Universe Today The CMB has a TemperaTure of T 271 K 39s1quotia ma er eg nucleons wi139h e es 39imafed as pNO mNnNO iIVerse This energy densi ry scales V 160 Timmafm ar v mNnN9TYTY03 wi rh TY0 271 K V 20quotny radiation has been p aT4 for TY lt 109 K 1 he two energy densi39ries one defines a cri rical Tempera l ure Tu Univer Se changes from radiation 139o ma l39Ter domina139ion One z 3000 16000K The age of The Universe was about 300 000 y when if Transformed from radiation 139o ma l39Ter domina l39ion era Cd alums a 9 a 6 0 a I 3 gt 391 399 B a quot3 a i Q 15mm a a a a I 0 e B DfJK EL F E 92 won K From 3 minutes To 3801000 AT abouT T380000 years The elecTr39ons Year39s The un39Verse W05 0 509quot 9 combined with pr39oTons and nuclei To form s0UP 01 eleCTr39ons and UUCIe39 aToms Because fr39ee elecTr39ons wer39e noT mostly meons and hel39um Them To scaTTer39 radiaTion The lighT could Udeo 39 a fog 01 V39S39ble I39ghT sTr39eam ouT We observe Thcn lighT Today as thns The CMB Dark J Act39i i zr39 In 1930 Fr39iTz Zwicky discovered ThaT The galaxies in The Coma clusTer39 were moving Too fasT To remain bound in The clusTer39 SomeThing else ThaT cannoT be seen musT be holding The galaxies in The clusTer39 In 1970 Ver39a Rubin discovered ThaT The gas and sTar39s in The ouTer39 par39Ts 39 z of aIaXies wer39e movin Too fasT 150 l n39n 121131111221111 9 g I 1 I This implies ThaT mosT of The mass in E 100 I The galaxy is ouTside The region 2 where we see The sTar39s Since we do noT see ighT from This Humilullnlll maTTer39iTiscaed 0 0 1o 20 30 Dark MaTTer39 Rikpc NGC 3198 Hwi ms iri Galaxyig iu iier Measure the mass of light emitting maTTer39 in galaxies in The clusTer39 sTar39s 1 Measure mass of hoT gas iT is 35 Times gr39eaTer39 Than The mass in sTar39s CalcuIaTe The mass The clusTer39 needs To hold in The hoT gas iT is 5 10 Times more Than The mass of The gas plus The mass of The sTar39s Dcl l zlcl rizr Halo The rotating disks of the spiral galaxies that we see are not stable Dar39k matter39 halos provide enough gravitational force to hold the galaxies together39 The halos also maintain the rapid velocities of the outermost stars in the galaxies COMPOSITION OF THE COSMOS Neutrinos Free Hydrogen and Helium 4quot0 Dark Matter 25 Dark Energy 70 Todaytu t 15 billion years Life on earth 7 39 39 39 Solar system Quasars Galaxy lormation Epoch ofgravlta onal collapse H ec om bin ation Relic radiation decouples CBH Ill atter domination Onset of gravitational instabilth N ucleosynthesis Light elements created 7 D He Li Qu ark hadron transition H adrons form protons 1 neutrons Electroweak phase transition Electromagnetic 8t weak nuclear forces become differen ated SU3XSU2XU1 gt SUE3XU1 The Particle Desert Axions supersymmetry Gran d u nilioation transition G gt H gt SU3x3U2xU1 Inflation baryo enesis rnon opoles cosmic strings etc The Planck epoch The quantum gravity barrier The B39gchg Timeline 31T In atm Era Maiur Evants Time Since Big Bang Since Big Bang ma a Humans at nhsanm E f galaxies lhE cusmna G haul and nliurslara 3 IE as made af alums and Ihillim EMT fr FIrsI I m years alums and a a 39 lama wm39 Era al 1 star39s Atoms En in j Ast an arm p ans m mam and beam yum I Enigma cl d gnwwnd a 7 If i 39 uquot f A J V r 39 y mgal l 335 w I L V 5 I hanium murals 39 39 I plug gl ms Fusion mans am ms r N 5 g a M mrmalmaltuls 39 39 E I quot39 quot 39 A r paintans nauthms 75 WW M II I 393 I quot 139 5 r 5 39 I abactmnsmaulrinus 95 hammr 53quot uc easyni eals U i 9 H i v 0 w 1 w tanninmu mm mass 439 quotquot 39 A o 7 Manar annihilalas alum ginands 39 I ff r 39 J 39 J o A 53995quot me uid anllnmllar Pealick Era no a I g 9 739 E quot antlrnranar ww mud 539 g a39 i 39 u w cammun Ehctramagnetln and wash Elm k E I s Emmmw Imus became dlsllncL 39 39 quotM m a 4 Parti lea 51mm Inn hummus 1W sacands mang maps GUT Era Hamlary talus n 39 an m de amides unlwrsn Planck Era 39Fquot mullan alnzlmn W an an mmquot E o rming am mmn g anilelactrans quarks I anyrigh l El Addison Wesley 97 1 Big Bang Nucleosyn rhesis L 3 Cosmic Microwave Background CMB 2 Hubble Expansion L 9 3 Problems 1 The criTicaI energy densiTy for a fIaT universe is p 3H2 or 8726 Show ThaT This value is 97 X 103927 kgm3 and 5400 eVcm3 Use H 72 kmsMpc 2 If The presenT day cosmic microwave background has a TemperaTure of 27 K show ThaT The number densiTy of phoTons is 400 cm393 3 If The neuTrino background has a TemperaTure F2 2 411 Ty Show fhaf The neufrno number densiTy is 110 cm393 4 If The baryon To phoTon raTio is I 3 6 x 1010 from The measured deuferZm abundance use parT your resuITs from problems 1 and 2 To cacuaTe The fracTion of The criTica densiTy lb 397 baryons 516 is r nikamilr r39e 39 Triii jca l density in neutrinos Qv K aquot mum 05 0 Temperature rv c4JJI3739y fncfhe dnTSlTy39nCm3 cryon To phoTon raTio is I A 6 x 1010 from 7 7e measured abundancejuse parT your resulTs from problems 1 and 2 To ftex rhe fraction of The critical density Qb m baryons 6 Assume nonbaryonic dark maTTer makes up a fracTion 026 of 7 7e cr ca dens7y If The dark maTTer is made up of 1 Te V mass weaky 3977 erac7 3979 massive par ces WIMPs calculaTe Their number densiTy 9 Using The Friedmann equaTion and The radiaTion energy densiTy formula 7 4 pyZZT show ThaT The relaTion beTween age and TemperaTure in The early universe is lMerz S t z 13 1 Here T is The TemperaTure of The phoTons For your convenience a 756 103915 erg cm393 K394 k 862 X 10395 eV K l 10 When The universe is sufficienle hoT and dense The raTio of neuTrons To proTons is mainTained aT The Thermal equilibrium value 3 e QkT Q mp62 12934 MeV p neguTr ons To proTons or vice l universe The reacTion raTe is a id once The Time for a parTicIe To g Ts longer Than The age of The universe I F39QTIQ 39 freeze s ouT An approximaTe Time per parTicIe in The relevanT TemperaTure seTTing This equal To The expansion Timescale T from problem 9 derive a value for The freezeouT TemperaTure TF and The corresponding freezeouT value of The np raTio

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