Great Ideas in Science
Great Ideas in Science PHYS 2018
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Physics 2018 Great Ideas in Science The Astronomy Module The Big Bang Lecture Notes Dr Donald G Luttermoser East Tennessee State University Edition 10 Abstract These class notes are designed for use of the instructor and students of the course Physics 2018 Great Ideas in Science This edition was last modi ed for the Fall 2007 semester I The Big Bang The Origin of Matter and Structure in the Universe A Hubble s Observations and Hubble s Law 1 Stars are thermonuclear furnaces that emit their own light a Planets on the other hand do not emit their own light they just re ect a star s light b Stars are gravitationally clumped together in bigger col lections known as galaxies see 2 of this module c The Milky Way our home galaxy contains over 200 billion stars 2 In 1931 Edwin Hubble and Milton Humason published a paper that showed that the fainter a galaxy appeared the higher its spectrum was redshifted gt Hubble s Law a It was proposed that the redshift resulted from the Doppler Effect AA g 1 1 Ac 0 l where z is the redshift AA A A0 is the spectral line wavelength shift A is the observed wavelength A0 is the rest or laboratory wavelength UT is the radial ie line Of sight velocity and c is the speed of light Z b Note that Eq 1 1 is valid only if the velocity of the galaxy is small with respect to the speed of light ie UT ltlt c If UT E c then the relativistic form of the Doppler Effect must be used ZA V1UTc1 L2 Ac il vTc ll c d e f Prior to the launch of the Hubble Space Telescope HST HO was not accurately known gt values ranged between 50 to 90 kmsecMpc Mpc mega parsec or million 106 parsecs and 1 parsec 326 light years depending on the technique used to Rewriting this relativistic formula we can express velocity as a function of redshift UT 2 12 1 c z12139 1 3 You will note that there is no way for a galaxy s velocity to exceed that of light when using the relativistic form of the Doppler Effect They also proposed that the fainter a galaxy appeared the farther away it was Hubble and Humason came up with the rst extragalactic standard brightness markers to estimate distance gt distance indicators Hence the more distant a galaxy e fainter galaxies the larger the redshift or recession velocity gt The Universe is Expandingl Hubble s Law mathematically UT Ho d 1 4 where v is the recession velocity d is the distance to the galaxy and H0 is Hubble s constant measure it The primary reason the HST was built was to determine an ac curate value for Ho This is one of the so called Key Projects of HST This HST Key Project has ascertained the following value for Hubble s constant see Freedman et al 2001 Astrophysical Jour nal 553 47 HO 72 l 8 kmsecMpc HST Result 1 5 The WMAP mission to map the cosmic microwave background see below has determined an even higher precision to Hubble s constant HO 71 l 4 kmsecMpc WMAP Result 1 6 see the WMAP website at httpmapgsfcnasagov Example 171 If we see an Fe line at 4800 A which at rest is at 4000 A how far away is the galaxy AA 4800 A 4000 A c 0 A0 4000 A 800 A 7 300 105 k 4000 A l X 18 02 300 x 105 kms 600 x 104 kms UT i 600 X 104 kms d 74 849 102M H0 71 kmsMpc X p0 d 850 Mpc 0T 300 x 105 kms Velocity in Hubble s Law represents the expansion velocity of the Universe as a whole This Hubble velocity is referred to as the Hubble Flow 8 A galaxy s recession velocity will not exactly match the velocity as predicted by Hubble s Law since this law is based on the ve locity distribution as a function of distance of numerous galaxies Any excess velocity that a galaxy retains when the Hubble Flow is subtracted out is called a galaxy s peculiar velocity B Olber s Paradox i Why does it get dark at night 1 During the 19th and beginnings of the 20th centuries the scien ti c viewpoint was that the Universe is unchanging static has no spatial boundaries in nite and had no beginning eternal If the Universe is static in nite and eternal we should light in every direction we look a This statement is referred to as Olber s Paradox b Actually Halley of the comet fame was the rst to pose this question Olber was the rst to formally publish it in 1826 c The inverse square law of light cannot be used as a solu tion to Olber s Paradox for the following reasons i The luminosity ie brightness at the surface of a star is given by L 47TR2F where R is the radius of the star and F is the energy flux energy per unit time of the star at its surface ii The flux of a star falls off as V2 where 7 is the distance to the star f Ri 2F where f is the observed ux iii For a uniform distribution of stars each succes sive shell radially away from the Earth will contain 47T7 2N stars where N is the areal density cm2 of stars on a shell at surface area 47T7 2 l4 iv The total ux that should arrive at Earth from all of the stars in the Universe is a convolution of these two quantity giving fobsshell 47Wng 47T7 2NR7 2F LN gt the sky should be ablaze with light for an in nitely large eternal Uni versel v Even for a non uniform distribution of stars the inverse square law will fail to account for the dark night sky for an in nite eternal Universe 1 Though not known when Olber s Paradox was published interstellar absorption also cannot be used to explain the dark night sky i Absorption from gas and dust in the interstellar medium ISM and the intergalactic medium lGM would heat the gas and dust ii Over an in nite amount of time this dust and gas would reach an equilibrium temperature that is equal to the radiative temperature of the integrated light from the stars iii Hence the ISM and lGM would shine as bright as the surface of the stars contained in the Universe 2 The solution to Olber s Paradox is that the Universe is expanding hence not static and is not eternal gt Big Bang Theory it had a beginning a Light gets redshifted out of the visible band for stars and galaxies at large distances b 3 How long ago did this happen We will treat this question in a very simpli ed way initially Calculate when all the galaxies As we look out we look back in time We cannot look in nitely far out since sooner or later we will see the Big Bang were at the same position from Hubble s Law w M G Since there is still some uncertainty to the value of Hub ble s constant we will use the following scale factor for malization EL uphkmwwmc324xinmhs4 102x1040hyr4 Ln where h the Hubble constant scale factor 10 if HO 100 kmsMpc and h 05 if HO 50 kmsMpc From Newtonian mechanics the distance to a galaxy is d v t where v is the galaxy s radial velocity assumed constant here and t is the time since the galaxy started at the origin But from Hubble s Law UT HO d so d i d i i Ef d amp tH978gtlt109h 1yr The Universe actually younger than that due to gravity tH 01quot 0 slowing down the expansion over time Hence the time given in Eq 1 8 corresponds to the maximum age of the Universe which is referred to as the Hubble Time i If h 10 HD 100 kmsMpc then tH 978 billion years d ii If h 05 HD 50 kmsMpc then tH 196 billion years iii The best value for Hubble s constant as deter mined by HST and WMAP see Eqs 1 5 and 1 6 is h 071 HO 71 kmsMpc and tH 138 billion years If a galaxy is farther than dobS ctH 978 138 or 196 billion light years depending upon the value of H O away we will never see it since light would not have had enough time to reach us gt dobS is the size of the observable universe C The In uence of Gravitation on the History of the Universe 1 For Hubble s Law to be true galaxies in the Universe have to be distributed homogeneously and isotroplcally see below for de nitions of these terms on a large scale a b Hubble s Law is referred to as the observed kinemath world model of the Universe One question we will come back to later is How large is large 2 EA Milne and WH McCrea 1934 extended this at rst purely kinematic model so as to make it a Newtonian cosmology 3 0 They investigated the motions l e trajectory of a medium of gas particles where the gas particles represent galax les in the Universe le the galaxies are treated as point particles These trajectories can be determine in accordance with Newtonian mechanics if one demands throughout that the I C d e v distribution of gas particles are i Homogeneous The volume number density cm gl of galaxies is constant throughout the Universe ii Isotropic The areal number density cm gl of galaxies on the sky from any point in the Universe is constant note that isotropy results automatically for a homogeneous distribution iii Irrotational The Universe as a whole is not rotating about some axis Consider at time t a galaxy at distance Rt then accord ing to Newton s law of gravitation this galaxy is attracted by the mass Within a sphere of radius R by M 43 R3 pot 19 where pt is the mass density at the instant of time con sidered Thus the equation of motion of this galaxy of mass m is determined by setting the force of motion equal to the gravitational force F Ri GMm quot1 dt2 1 or 121 GM W F 0 3910 where the mass M is constant Multiplying each term in Eq 1 10 by dRdt it is then possible to easily integrate Eq 1 10 and obtain the 18 f g h energy equation 1 dB 2 GM 7 7 7 1 11 2 ltdtgt R K l l where Is is the integration constant or R2 87F kc2 in which we have written ls k02 2 in anticipation of later comparison with relativistic calculations We can de ne the current Hubble constant as Ho E PLORD 1 13 where we denote present time t to by a subscript o For a complete characterization of a model universe we need besides H o a second variable that describes the de celeration of the Universe due to its mass M This is the so called deceleration parameter R0 R0 27 R0 i47TGpo qquot R0 Re ROHET 3H3 using Eqs 1 9 1 10 and 1 13 114 This formula relates the acceleration R0 to a uniform ac celeration which would lead to the observed velocity ROHO at distance R0 in the Hubble time tH to H 0 1 starting from zero velocity The solution of the above equations leads to world mod els which from a starting point singularity of in nitely great density either expands monotonically total energy M Is 2 0 or oscillates periodically between R 0 and an Rmax if ls lt 0 Static models are not possible within the framework of Eq 1 10 3 Newton actually realized this which is why he wanted the Universe to be in nite in size Relativistic cosmologies are based on the general theory of rel ativity instead of Newtonian mechanics Here we use a spacetirne description of the Universe in cosmology gt we lay a coordinate grid across the spacetime manifold a An event is de ned as a point in spacetime speci ed by 4 coordinates xi t time coordinate f l 0 1 2 F spatial position coordinate 3 8888 l J b A world line is the locus of successive events in a parti cle s history i Spatial distance in spacetime d3 ii Separation in spacetime d7 ln spacetime the separation or line element is de ned by the tensor equation 3 172 E Z gij dxzdx 1 15 ij0 meanwhile the spatial distance is given by 3 ds2 E Z 9W dye dx 1 16 MyV1 where 239 j u and V are summation labels and not exponents The various metrics or con guration coe icients are given below 110 a Minkowski metricspacetime re special relativity 1 0 0 0 0 l2 0 0 gij 771739 0 06 612 0 0 0 717 in a local inertial coordinate system dr2 gij dxi dxj 1 18 1 it gm dg dz 1 19 dr2 it 72 1 20 i dr is invariant gt another inertial observer 0 measuring dr gets the same answer ii Note that since dr 1 dt we can write 1 dt2 d72dt2 2 C 121 01quot dr W where dr is the proper time interval One recog nizes this equation immediately as the time dilation equation from special relativity see Dr Gardner s module dt 122 iii Principle of Equivalence At every spacetime point in an arbitrary gravitational eld it is possi ble to choose a locally inertial coordinate system such that the laws of nature take the same form as in a non accelerating coordinate system in the absence of gravity lll b Frame f m5 mg Jim 123 and in f a frame with acceleration g7 md g Fm m6 13m 124 where an is a non gravitational force Hence lo cally we can say that gij 1717 A line element in spacetime with spherical symmetry has a metric tensor of 67 t 0 0 0 0 L 0 0 gij 0 06 T 0 1 25 0 0 0 L 01quot 1 d7 en t it 7 n 15 d7 2 r2 162 7 2 sin 0 6 1 26 where e7 t and f7 t are functions to be determined from the boundary conditions of the problem in question i For a static line element in spherically symmetric spacetime Schwarzschild metric ZGM 1 67 02 7 1 i 1 27 r lt gt where 7 8 2GJWc2 is the Schwarzschild radius Al so 1 f7 1 W 128 in this metric 112 ii The line element in spacetime then becomes 2 Vs 2 d7 lt1 i dt 129 7 1 2 7 d7 r2 102 7 2 sin was c 1 rsr which is the spacetime equation that is used for a black hole c The RobertsonWalker metric concerns a spherical symmetric homogeneous spacetime lts metric tensor is 1 0 0 0 R2 0 0 0 g i 0 W2 0 0 0 0 r2 S1190 1 30 01quot d7 it R2515 lt W r2 102 7 2 sin was c 1 M2 1 31 where 7 is the coordinate distance Rtr is the metric distance and 120 is known as the unitless scale factor of the metric d This Robertson Walker metric spans a 3 space of constant curvature K k K K t 7 1 32 lt gt R2 lt gt 1 positive curvature k 0 at positive space zero curvature l negative curvature 5 On a curved surface the shortest and longest paths ie the extrema between two points in spacetime are called Geodesics I13 a On a spherical ie positive curvature surface one can have both a minimum and a maximum geodesic between two points A and B b On a at ie zero curvature or hyperbolic ie negative curvature one can only have a minimum geodesic The interval between two events in spacetime T TAB is de ned by B T TAB A d7 l 33 where dT V dT2 is the spacetime line element ie separation We can de ne one of three outcomes for a given geodesic a T is real events have timelike separations or time like geodesics Events can lie on a world line of a material particle since material particles have time like geodesics In that case dT is the proper time between events b T is imaginary spacelike separations The events cannot lie on a world line of a material particle cm is the proper separation if we were at a particular event This would be the regime one would be in if one could travel faster than light c T 0 lightlike separation or null geodesics gt this is the way light propagates World lines of photons are null geodesics These various separations can be seen in Fig 1 1 Here we de ne the lightcone which are double hyper cones joining an event P to past and future events Time like geodesics all lie in the cone Events and geodesics outside the cones ie space like can never communicate to events inside the cones i 6 our Universe ll4 Communication ossible within Timelike Geodesics v lt c s ace Spacelike Geodesics v gt c NULL Geodesics V c Figure l 1 Light cone diagram of event P occurring in spacetime In the general theory of relativity Einstein used the concept of geodesics to derive the Field Equations of the Universe 1 87F G Rij E gij R 762 a R are the components of the Riemann curvature ten sor which are related to the scale factor of the Universe Note that R 9 RH b Tij are the components of the energymomentum ten sor c Essentially the eld equations are nothing more than the conservation of energy and momentum in a 4 D spacetime The tensor eld equations as shown in Eq 1 34 produce non static metrics as was shown by de Sitter and others Einstein didn t like this and added an additional constant term to keep 115 the Universe static 87TG T39 02 Z l Ru E gij R 02 A 92739 13935 a A has been coined the cosmological constant b This constant term 02 A 92 effectively represents a nega tive gravity in order to keep the Universe static We will have more to say about this later in this section c After Einstein heard of Hubble s results that the Universe is expanding he claimed that adding this constant to his equations was the biggest blunder of his life He then reset A 0 1 But is A really zero We will keep this constant in mind while developing the formalism of these equations e At this point we will go no further with the tensor form of the eld equations since it s too advanced for this course However we can still develop a differential equation for the change in scale factor of the Universe from an alge braic de nition of curvature We will introduce this in the Structure of the Universe subsection D The Big Bang Theory 1 The Universe started in a hot high density state which began to expand and cool a Hubble Time ago gt the Big Bang Theory a The Big Bang occurred everywhere in space not just at one location gt we are in the Big Bang b Galaxies were Lot thrown apart gt the fabric of space itself is expanding and the galaxies move apart as a result as they ride along on the fabric of spacetime 116 2 We shall be developing equations that describes the evolution of the Universe with time In these equations we will start with two postulates called the Cosmological Principle a Homogeneity matter is uniformly distributed in space on a very large scale d gt 100 Mpc b Isotropy the Universe looks the same in every direc tion In addition two additional assumptions are typically included when describing the Big Bang a Universality physical laws and constants are the same everywhere in the Universe 10 Cosmological Redshifts redshifts are caused by the expansion of the Universe through the Doppler Effect We see the Big Bang reball in every direction as microwave blackbody radiation 3 K Cosmic Microwave Background CMB radiation a When this light was emitted some 300000 years after the Big Bang when the Universe was around 3000 K b As the Universe expanded this light was redshifted until today it is microwave light 2 1100 the farthest quasar is at z 49 c PenZias and Wilson discovered this background radiation in the early 1960 s con rming the theoretical predictions of the Big Bang Theory made by Dicke and Peebles Pen Zias and Wilson later won a Nobel Prize for their discov ery ll d The COBE spacecraft was launch in the early 1990 s to investigate this background radiation i Found that the Universe radiates as a perfect black body after the Earth s Sun s Milky Way s and Local Group s motions are subtracted at a tem perature of 2726 K ii Small variations in the thermal distribution of this radiation on the sky on the order of 1 part in 100000 ic the intrinsic anisotropy show that by the time this radiation was emitted inhomo geneities in the mass energy of the Universe had begun which would later form the galaxies E The Structure of the Universe 1 We will now develop the equations that describe how the Universe has evolved in time We will assume the cosmological principle as postulates to the global structure of the Universe which allows us to use the Robertson Walker metric see Eq 1 30 and 1 31 to describe the geometry of spacetime 2 Since the Universe is assumed homogeneous isotropic and irro tational the 6 and b coordinates are constant and we can ignore them in our description of spacetime The separation in events then becomes dr2 2 2 R2 dr dt t 621 6762 136 3 As described in Eq 1 32 the constant k is a measure of the global curvature K of spacetime a Note that for plane polar coordinates K 0 gt space is at ll8 b c d 4 Using the Robertson Walker metric it can be shown with ad For a spherical surface K 1 R2 For space only curvature x1 and x2 are spatial coordi nates where x1 J 062 However for spacetime curvature x1 t and x2 7 in this equation Hence K K t is a function of time vanced calculus that the curvature formula becomes a b C 01 R05 K t 7 Ra In this equation Rt is the scale factor of the Universe and is the second time derivative ie the accelera tion of this scale factor Matter in the Universe is described by its mass density pt and is the cause of the curvature of spacetime though A can give curvature without matter With this in mind we can use a dimensional analysis type of argument to describe K t in terms of a linear proportionality to this mass density pt We guess at an equation K0 amt GE cm constant 138 where the constant is inserted to allow for the possibility that empty spacetime p 0 might be curved It follows from Eq 1 37 that K t must have dimensions of time 2 lf oz is dimensionless this implies that the powers of E and m to which G and c are raised are 1 and 0 respectively When taking the Newtonian limit see below oz 47T 3 l19 137 e D Q m A curved empty universe requires that the cosmological constant introduced in Eq 1 35 be nonzero As such we choose the constant in Eq 1 38 to be equal to A3 for consistency with the eld equations Therefore the cosmological constant A has the same dimensions as K05 or time 2 The curvature K05 now becomes K0 47Tpt G A 3 3 139 Equating this with Eq 1 37 gives the equation of motion for 120 as 47Tpt G A M0 4447R 7R 3 3 1 40 Precisely this same formula follows from Einstein s general relativity eld equations If A gt 0 A acts like a negative density p Since self gravitation of matter acts to slow down the expansion of the Universe a positive A must act to accelerate it For this reason ARB is sometimes called the cosmic repulsion term If A 0 Eq 1 40 follows exactly from Newtonian me chanics i To see this consider the galaxies lying inside the comoving sphere with radial coordinate 7 ii A galaxy on the surface of the sphere will acceler ate inwards under the attraction of the mass within the sphere I20 iii If the galaxy has a mass m Newton s second law gives ma mrl t Fgal GmM WWW Gnm4w3nmwirRaM3 WWW where Fgal is the gravitational force on the galaxy and M is the mass that lies inside the sphere iv Thus quot 47T 47T RU Gm Rt GpR 1 41 hence our choice of oz in Eq 1 38 Note that when ever cosmological parameters are listed without a subscript ID the parameter is to be taken as a vari able function of time k Even though Newtonian mechanics seems to work quite well in describing the evolution of the Universe general relativity still must be used since expansion velocities quickly exceed the speed of light if one looks out far enough under the assumptions of classical mechanics Since we don t know pt throughout the entire history of the Uni verse we need to eliminate this term from Eq 1 40 We will use the conservation of mass energy here Within a comoving sphere the mass remains constant where the volume is proportional to R3t ie pt R305 p0 Pu constant thus MMRWJim plt i 7 a 142 121 where pO is the current mass density of the Universe e a mea surable quantity and R0 is the current scale factor of the Uni verse Eq 1 40 now becomes 47TpO G R3 ARt R O 7 1 43 Multiplying this equation by and integrating we obtain 47Tpo G R3 AR RR quotR 7R 3132 T 3 dR i 47TpoGR R72 dRAR dR dz 7 3 dz 3 dz 4 OGR3 A RdR R2dR Rd1 z 47TpOGR 2 A RdR TR dR RdR 1 2 i 47TpOGR Pf1 A R2 2R 7 3 1 3 2 const 87TpOGR3 A 2 i o i 2 R t i 3Rt 3 R t const 8 A g G pt R205 g R205 const 1 44 where we have made use of mass energy conservation U 6 p0 Pu pt R3t in the last step of this derivation The constant must be proportional to 02 since it is the only combination of the quantities c G and A which have the required dimensions of velocity The eld equations of general relativity give its exact value as k02 where k is the curvature index of the Robertson Walker metric given in Eq 1 31 Thus we have R205 3 14220 kc2 8m pz A 1 45 122 a b C By setting A 0 we can rewrite Eq 1 45 as 1 dB 2 8 if 7G lltR it 3 p which is known as the Friedmann equation R2 k02 146 With the cosmological constant Friedmann s equation be comes 1 81 R dt 3 which I shall refer to as the modi ed Friedmann equation Gp if R2 k02 1 47 We will developing solutions to both forms of these Fried mann equations in the subsection titled Big Bang Models The solutions will be described by various parameters as described below i Hubble s constant is the proportionality con stant between the rate of change of the scale factor 239 6 the expansion velocity to the scale factor Rt at the current time to see Eq 1 13 Rota R0 L48 ii The deceleration parameter was de ned by Eq 1 14 RtoRto R2050 o R050 Rto H02 1 49 Note that for a universe with A 0 we can use Eq 1 41 in Eq 1 49 to give the deceleration pa rameter as i 47TG pO qr 3 H339 1 50 l23 iii The cosmological constant can be determined from p0 Ho and q0 by using Eq 1 49 in Eq 1 40 which gives A Alma 3qu3 1 51 iv The curvature index also can be expressed in terms of pa Ho and q0 by using Eq 1 48 in Eq 1 45 which gives R k 2 O 02 MG0 H3 qo 1 1 52 v The quantities pO current mass density qO cur rent deceleration parameter and H0 are measur able so that both A and k can be determined from these observable quantities vi If the cosmological constant is zero the density parameter is de ned as Pa 90 p6 l 53 where pc is called the critical density see next page of the current Universe gt a universe with 90 1 has k 0 gt Euclidean at space vii For A y 0 the current density parameter has two components 90 2m QA 1 54 where 2m is the density parameter due to matter sometimes this is listed as Q for baryon density given by Eq 1 53 and QA sometimes given as l24 2m the vacuum energy density is given by GA E 155 3H3 when A is expressed in units of time 2 some sci entists de ne A expressed in units of length 2 for A02 3Hgl that case QA E F Observational Constraints in Modeling the Universe 1 Current Mass Density a What is the current mass density of the Universe Are there enough galaxies e mass to stop the expansion The mass density of the Universe indicates its geometry Here we will assume for the time being that A 0 i If the Universe s mass density p0 is less than a critical density p lt pc 0 lt 90 lt 10 lt qO lt 12 gravity will not halt the expansion gt the Universe will continue to expand forever a so called Open Universe ii If p0 gt pc 90 gt 1 qO gt 12 gravity will halt the expansion and cause a contraction down to a Big Crunch gt a Closed Universe iii The Universe may be able to rebound Big Bang and start over again in such a universe gt an Os cillating Universe If pa pg 90 1qO 12 gravity will halt the expansion after an in nite amount of time gt iv a Flat Universe 125 b c Table l 1 Structure of the Universe Density Deceleration Type Geometry Curvature Parameter Parameter Age Closed Spherical Positive 90 gt 1 go gt r to lt 71Ho Flat Flat Zero 90 1 go g to Hyperbolic Negative 0 lt 90 lt 1 0 lt qo lt 1Ho lt to lt 1HO No Matter Hyperbolic Negative QC 0 go 0 o 1 v If there were no matter in the Universe then p po0 QC 0andqo0 see Table 1 1 also an open universe The current critical density is given by the expression i 3H3 p6 T 87TG which ranges from 470 X 10 30 gmcm3 for H0 50 kmsMpc to 188gtlt 10 29 gmcm3 for H0 100 kmsMpc For the WMAP value for H o the current value of the crit ical density is 948 X 10 30 gmcm3 188 x 10492 gmcm3 1 56 We can now ask what is pO or 90 of the Universe i Galaxy counting amount of luminous matter 211mm m 001 gt 100 times too small in order to close Universe ii Galaxy dynamics motion of galaxies used to de duce mass amount of matter in galaxy clusters out to 30 Mpc includes both bright and dark mat ter anl 025 l010 25 211mm gt 4 times too small for closure iii Deuterium 2H abundance baryons Q 0071 004 7 211mm 03 anl gt only 30 of dark matter is composed of baryons We will discuss the reason for the current deuterium density giving the 126 baryon density in the History of the Universe sub section below iv Light photons pmd 111162 and Trad 273 K pmd 65 X 10 34 gmcm3 nmd 7 x 105 prad ltlt pl gt today we line in a matter dominated universe Radiation dominated at earlier times see History of the Universe subsection v What is the identity of the non baryonic matter that causes anl 025 o Neutrinos outnumber photons in the Universe by a factor of 109 Neutrinos were originally thought to be massless like photons However one out come of the theory of quantum chromodynam ics suggest that neutrinos have mass and that they oscillate in state eg electron neutrinos to muon neutrinos to tau neutrinos In the 1990s Los Alamos have detected muon neutrinos transmuting into electron neutrinos The amount of oscillations place a mass range of 05 50 eV for the neutrino Supernova 1987A that exploded in the LMC sent a lO second burst of 19 electron anti neu trinos that was detected by the various solar neutrino detectors around the world The neu trino event preceded the rst sightings of the IQ 2 supernova s light by 3 hours Con rms super nova theory that it should take about 3 hours for the core collapse shock to propagate to the stellar surface The fact that the neutrinos ar rived nearly as quick as the photons allowing for the shock time delay indicates that the mass of the neutrinos must be less than 3 eV Assuming a mass of 1 eV 18 X 10 33 gm for a neutrino gives a universal mass density that is over 10 times the critical density of the Uni versel Other hypothetical non baryonic matter has been speculated on including weakly interactive parti cles WIMPS and massive magnetic monopoles These strange particles have never been detected however vi From all of the matter measurements along with the best estimates of the non baryonic mass 2m m 14 13 Cosmological Constant 3 Prior to the launch of the Hubble Space Telescope the best estimate for the size of the cosmological constant based upon the measured density see below Ho and go gave A 091l093 X 10 20 yrs The negative value for A indicates a cosmic attraction on top of gravity However recently the value of A has been scrutinized in a much more detailed way l28 b C d e Modern eld theory now associates this term with the energy density of the vacuum hence the 2m mentioned above If the cosmological constant today comprises most of the energy density of the Universe then the extrapo lated age of the Universe is much larger than the current maximum age given by Hubble s Law Adding a cosmological constant term to the in ationary model an extension of the Big Bang Theory see be low leads to a model that appears to be consistent with the observed large scale distribution of galaxies and clus ters with COBE s measurements of the cosmic microwave background uctuations and with the observed properties of X ray clusters A few groups of astronomers has ascertained with large uncertainties from the brightness of Type la supernovae at high redshift that the Universe is currently expanding faster today than it was 5 to 7 billion years ago i This acceleration called a de Sitter universe see next section is predicted from general relativity and Newtonian cosmology for A gt 0 ii The supernova data implies that M m 23 34 giving 90 m 1 gt a at universe iii This supernova data given an age for the Uni verse at l42ll7 Gyr which is consistent with the WMAP value The Compton Gamma Ray Observatory CGRO carried an instrument called the Burst and Transient Source Experiment BATSE which monitored gamma ray bursters l29 3 during its 8 year lifetime BATSE ascertained that these gamma ray bursts are cosmological and in 1994 also showed evidence that the Universe is expanding faster at present times than in the past Hubble s Constant 3 b As discussed at the beginning of this section the HST Key Project reports this parameter to be 7218 kmsMpc see Freeman et at 2001 Astrophysical Journal 553 47 The WMAP mission determined HO 71 l 4 kmsMpc from the cosmic microwave background radiation Deceleration Parameter a b This is perhaps the most dif cult parameter of cosmology to measure though it can be determined from the other parameters previously mentioned in Eq 1 51 One of the most accurate techniques in the determination of qO from an angular size of extragalactic radio sources as a function of redshift These measurements give a value of 01 for the deceleration parameter G Big Bang Modeling 1 The solution to the eld equations indicate the geometry of the Universe By knowing the value of Do we will know the overall shape i 6 curvature of the Universe in 4 dimensions and know the nal fate of the Universe as shown in Figure 1 2 a A positive curvature is a spherical space As an analogy assume a 2 D surface being bent into a third dimension in the shape of a sphere s surface The Universe is said to be closed if spacetime has a curvature where k 1 130 Figure l 2 The three possible geometries that spacetime can have Top negative curvature open universe Middle zero curvature at universe Bottom positive curvature closed universe Note that the graphs displays the scale factor on the vertical axis and time on the horizontal axis b A negative curvature is a hyperbolic space With our above analogy assume a 27D surface being bent into a third dimension in the shape of a saddle The Universe is said to be open if spacetime has a curvature where k7l c A at curvature is a Euclidean space The 27D analogy is simply a at plane surface that is not curved into a third dimension The Universe is said to be at if spacetime has a curvature where k O 2 Solution to the Eriedmann Equation A O a The Integral Equation Solution i The easiest way to solve the Eriedmann equation see Eq L46 is to isolate the alt differential on one side of the equation and integrate the alt side and 131 the dR side ii All of the Friedmann universes has a big bang origin R 0 at t 0 As such Eq 1 46 can be written as the following integral R 11 t 0 t W 3957 V kc iii If we de ne 87TGp07 3 26106 Rm E k 0 302 lToqO l32 lg R3H2 62 k0 1 58 we can simplify Eq 1 57 to 159 IiiRa dR 0 c anR ki iv This equation has the following solutions for the three different values of k RTmsin l 1 Igj n Igj k1 H B z 234 Sc Rf lt1 s1nh 1lt Bi k 1 1 60 v One can now easily see why the k 1 case is called a hyperbolic universe vi Figure 1 3 shows the histories of these three types of universes 132 30 3 t plt 9 25 20 RRo 153 39D I 39D a 10 05 00 0 20 40 60 80 t Gyr Figure l 3 Histories of Hiedmann model universes A 0 Note that the present size of the Universe hence present time is indicated by R 1 b Closed universes k 1 90 gt 1 i From the rst equation in Eq 1 58 the scale factor would reach a maximum size Rm assuming qO m 1 for our Universe assuming our Universe is closed at 2 c Rm m HO Using the WMAP value for Hubble s constant this maximum would be Rm 26 X 1028 cm 28x 1010 ly 85 x 109 pc 85 Gpc 161 ii By setting 0 in the Friedmann equation this maximum size would be reached in time tm given by i WRm tm i 7 20 or 44 X 1010 yr 44 Gyr after the Big Bang Since 162 I33 the Universe is currently about 14 Gyr old we would still have another 30 Gyr before the collapse would begin iii At this point e tm such a universe starts to collapse back down to a big crunch at t Ztm The Big Crunch for our Universe would occur 88 Gyr after the Big Bang or 74 Gyr into the future we have nothing to worry about iv We also can calculate the current age of the Uni verse to by setting R R0 in Eq 1 60 for k 1 2qO 2gO 1 1 M2 0 1 Ho2qo 132 qo 2qo q l 1 63 Once again using qO 1 with the WMAP value for H0 gives to 28 X 1010 yr 28 Gyr which is a factor of 2 bigger than the best estimate for the to sin 1 current age of the Universe 14 Gyr obviously if the Universe is closed qO lt 1 to give a more reasonable value for to v Since the solution for k 1 presented in Eq 1 60 is fairly complicated one can introduce a link parameter call it 06 between R and t and write this one equation into two parametric equations here we will give two separate equations for each para metric equation and by making use of the density parameter 47er0 R 7 31662 1 cosx 1 64 1 90 E 90 1 1 cosx 1 65 and t 2762 06 sinx 1 66 l 90 271110620 132 x where x 2 0 has no speci c meaning associated with it other than linking R to t sin 06 1 67 c Flat universe k 0 90 1 i With such a universe the expansion velocity would just reach zero after an in nite amount of time has passed ii We can easily invert Eq 1 60 for k 0 to de rive Rt after realizing that pO pc for such a universe R 67TGpC13 193 168 3 23 23 lt7 lt3 169 2 tH iii Taking the time derivative of this equation gives 2 R g 67TGpC13 1713 170 As can be seen from this equation when t gt 00 R gt 0 as previously stated iv Once again we can calculate the current age of the Universe to by setting R R0 in Eq 1 60 for k 0 32 t 7 15 m For the WMAP value of Ho the current age of the Universe is 92 X 109 yr 92 Gyr Since the o 135 oldest globular star clusters are on the order of 12 13 Gyr old our Universe cannot have both k 0 and A 0 1 Open universes k 1 0 lt 90 lt 1 i With such a universe the expansion velocity re mains greater than zero after an in nite amount of time has passed ii For such an open universe the current age of the Universe to is found by setting R R0 in Eq 1 60 for k 1 This gives the following solution I i 2 xl2qo 1 Ho2qo 132 2g 7 2610 h 1 39 1 72 sin 2 l N Ho H39 Using the WMAP value for H0 gives to 14 X 1010 yr 14 Gyr which is consistent with the age of the oldest stars seen in the Milky Way iii Since the solution for k 1 presented in Eq I 60 is fairly complicated like the case for the closed universe one can introduce a link parameter call it 96 between R and t and write this one equation into two parametric equations here we will give two separate equations for each parametric equa tion and by making use of the density parameter 47er0 R Blklcg coshx l 1 73 I36