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# Topics Astrophysics ASTR 321

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Lecture Notes for Astronomy 3217 W 2004 R Frey 1 Stellar Energy Generation Physics back ground 11 Relevant relativity synopsis We start with a review of some basic relations from special relativity The mechanical energy E of a particle of rest mass m moving at speed 1 not including any potential energy is E ymcz where 1 l 41471202 E2 p262m264 or alternatively where p is the momentum p 77711 The particle rest energy is mcz and its kinetic energy is KE7m02 yi1m02 In the non relativistic limit 1 ltlt 0 we can use the binomial theorem Taylor series to expand y in powers of vc2 i 1 1112 314 Y 7 2 02 8 04 In this limit 1 ltlt 0 the kinetic energy reduces to the familiar classical form 1 112 1 K m m02171 inn2 p22m We primarily will use the notion of energy mass equivalence from relativ ity since the kinetic energy of protons in main sequence MS stellar cores is typically very small compared to the rest energy Hence the non relativistic expressions for K are accurate and we will often notationally use E to mean kinetic energy We will borrow the unit system which is customary in atomic nuclear and particle physics Energy is expressed in units of eV where 18V 160 gtlt 1019 J 160 gtlt 1012 erg Most importantly we use E2 p202m204 to allow us to also write rest mass and momentum using the eV So rest mass has units eVc2 more typically MeVcz and momentum eVc In this way we do not usually have to multiply or divide by the numerical value for c 12 Physical data and conversions c 29979 gtlt108ms 18V 160 gtlt10 19 J 160 gtlt10 12 erg 1fm 10 15 m h h27r 1054 gtlt10 34 J s 658 gtlt10 22 MeV s hc 197 MeV fm e 160 gtlt10 19 C 480 gtlt10 10 esu k 133 gtlt10 23 JK 862 gtlt 1011 MeVK G 667 gtlt 1011 N m2 kg2 04 6247T60hc in 81 1137 771502 0511 MeV mpc2 938 MeV mn 7 mp02 1293 MeV 1 u 93149432 MeVcz 1pc 3086 gtlt 1016 m 32621y MD 199 gtlt1030 kg LG 335 gtlt 1026 w 13 Four forces of nature The electromagnetic EM7 strong and weak forces have intrinsic strengths which are equal7 or nearly so7 at very short distances The probed distance d is related to a characteristic momentum or energy for example a collision energy Via the deBroglie relation d A hp Since the ranges of the strong N 1 fm and weak forces lt 01 fm are nite7 these forces become negligible for most purposes at distances greater than a few fm For energies typical of the stellar cores7 the weak force is typically much weaker than the strong force by something like a factor 106 A few tables which summarize basic properties of the four forces and elementary particles are provided from a separate link from the class web age The table below indicates which forces couple to the listed particles All listed objects are subject to gravity7 although it is utterly negligible at the atomic scale or smaller in stellar astrophysics object EM strong weak dogs7 books7 etc Y N N electron7 e Y Y proton7 p Y Y Y neutron7 n N Y Y photon7 y Y N N neutrino7 V N N Y quark7 ud7 etc Y Y Y We need to look at some examples of the weak force7 and to distinguish it from the strong force A familiar example of weak interactions is B decay Here is the same decay process7 shown at increasing levels of sub structure fHagHe pe napei e dauei e In the top line we have used the notation A ZX 3 for an atomic nucleus of element X7 consisting of Z protons atomic number and A 7 Z neutrons The number A7 known as the atomic mass7 is the total number of nucleons protons plus neutrons The second line shows the same process at the nucleon level7 that is at the scale of neutrons and protons The third line gives the same process at the level of nucleon sub structure7 that is at the quark level The quarks are7 according to the Standard Model of particle physics7 elementary particles7 with no sub structure We note that only the weak interaction can transform n to p or p to 71 From the data in Section 12 we see that for a free neutron or proton7 the process 71 a p 5 PE is energetically possible7 but p a n 6 V5 is not How then can we understand the existence of the weak decay EBaiBeeV57 which requires the p a n process The point here is that the binding energies of the B and Be nuclei are different In this case the 2B6 nuclei is more tightly bound larger binding energy Eb than the EB by an amount larger than the difference between p and 71 plus 6 rest energies7 making the process energetically possible Figure 1 shows the binding energy per nucleon for all isotopes as a function of atomic mass A We see that the maximally bound nuclei are in the vicinity of iron Fe 14 A process which gives hydrogen burning We assume initially that the main sequence star under consideration consists entirely of hydrogen7 or at least the presence of other elements can be ignored As we have seen7 a typical temperature in the core of a MS star like the sun is T m 107 K The average kinetic energy of the constituents of a classical gas is F SkT w1KeV 1 for T 107 Since this energy is much larger than the atomic binding energy of the hydrogen atom7 the hydrogen is completely ionized Hence the core is a dense gas of positively charged protons and negatively charged electrons We need to nd a way for the protons to interact and fuse to form heavier elements As Figure 1 shows7 the creation of heavier elements up to iron will allow the binding energy of the product to be released as kinetic energy7 thus providing the stellar energy source moqocxc E m 4 l A75 100 155777777 300 250 A Figure 1 Binding energy per nucleon in MeV as a function of atomic mass A Figure 109 of Carroll and Ostlie The most obvious potential candidate is p 10 Hg He However this state does not exist in nature 7 there is no protoniproton bound state We next consider the process Which actually does take place and is the rst step for all stellar hydrogen burning pp dev 2 or in nuclear notation 1H1HgtfHeV1 3 Where d fH is the deuteron We see that this process involves the inverse 3 decay process mentioned earlier p a 71 6 11 W ere the resulting neui tron forms the deuteron bound state With the surviving proton So although the positive energy for this process results from the d binding energy due to the strong force interaction of nip this initial step utilizes the weak force This is importan since as We shall see beloW the lOW rate for this Weak process guarantees a long life for MS stars Before We consider the reaction 3 and the remainder of the protoniburning sequence in more detail We consider a simple model of the nip bound state the deuteron 5 15 A simple model of the deuteron An approximate form of the strong force for nucleons is the square well potential energy shown in Figure 2 The radius of the well of approximately r0 2 fm and its depth of 36 MeV are based on the results of scattering experiments When the Schrodinger equation is applied to this system with the n and p masses7 one nds that only one bound state results with an energy eigenvalue of 722 MeV This indeed correspnds to the experimental nding that the deuteron consists of only one bound state with a binding energy of 22 MeV Ur MeV Figure 2 Square well model of the deuteron 2 Stellar Energy Generation 7 PPchain physics 21 The PPI chain and energetics Figure 3 shows the main sequences by which hydrogen nuclei protons are converted into helium nuclei The rst step is the we interaction process of Eq 3 This step determines the rate of energy production in MS stars and this property is the focus of the next two sections Here we look at the overall energetics of the processes HH iai fve HullI aaHeaty 69 31 Iquot r Tim s in 3H6 W gt He ZEH gHe 3H3 lBe y PPI quot 997 1quot 113 e39 3M v 132 H EB Y gLi H 39 iigt 23th B ire 233 e vE PP H Be 23 PP III Figure 3 The PP chains Figure 108 of Carroll and Ostlie All three of the PP chains involve conversion of 4 protons to a helium74 nucleus Reference to 1 shows that He is unusually stable relative to isotopes of similar atomic mass In fact EH6 is also known as an alpha particle and due to its large binding energy and stability is a typical byproduct of nuclear reactions such as ssion Figure 3 shows a branching ratio of 69 for process 3 of the PPI chain This re ects the probability of this path However the PPI chain accounts for 85 of the overall rate of He production and hence about 85 of the total energy output of the sun and similar MS stars So we use the PPI chain to illustrate energy production and in so doing we account for the vast majority of the solar luminosity Rewriting the PPl chain HiHa Heue fHfHa Hev He Hea He2iH 4 We note that process 3 requires two reactions each of processes 1 and 2 So in sum there are 6 protons reactants and 2 proton products Hence7 the net effect of the PPl chain in terms of overall energetics is equivalent to 4 H H6262V52v 5 We follow the usual practice in nuclear physics of using the atomic mass unit or u for the rest masses of nuclei By de nition7 1 12 la 7 EM6 C 7 and a conversion to MeV can be made from Section 12 Using the AMU7 the proton mass is 10078 u and the 3He mass is 40026 u So the energy available to the other nal state particles7 both as rest energy and kinetic energy7 is Am 4 gtlt10078 7 40026 00287 u 7 or Amcz 00287 gtlt 93149432 MeVu 27 MeV The positron rest rnass will be available for kinetic energy7 too7 since it will annihilate with an ambient electron So except for the energy given to the neutrinos7 which exit the star without depositing any energy7 the 27 MeV is available to provide the solar lurninosity The neutrinos will be discussed separately in lecture 6 We note that the rst step of the PP chain7 the p p interaction7 provides only about 03 MeV of kinetic energy So for each PPl reaction zle for every 4 protons7 the fraction of stellar mass which is converted to energy lurninosity is except for the neutrinos e 0028740313 m 7 gtlt 103 6 Hence7 the available hydrogen mass can be eventually converted into energy with an ef ciency of 07 22 The pp interaction Coulomb barrier penetration The strong force for the 71 1 and p p interactions are identical This is a symmetry property called isospm But we now have to add to this the Coulomb interaction between the protons 1 62 47TEO r SI Ur Ustrong Ucouloinb 7 Ucouloinb This potential is depicted in Figure 4 Ur MeV 07 W Figure 4 Potential energy model of the proton proton interaction At 7 r0 2 fm we have using 81 units then converting to eV Ucoul 9 x10916 gtlt10 1922 gtlt 1045 gtlt1eV16gtlt 1019 J 07 MeV So while the Coulomb barrier is 07 MeV the average kinetic energy of the protons from Equation 1 is m 1 KeV So classically a proton at this average energy will never be able to get close enough to another proton to engage in strong or weak interactions However quantum mechanics provides the possibility of barrier penetration We now sketch the solution to this cal culation which will be familiar to students who have had an introductory quantum course In the case of a 1 dimensional step potential of height U5 and width L7 a particle of mass m and kinetic energy E has a barrier penetration probability PC in the case where E lt Ub of PG 6 7 where 72 8mLU17 7 Eh2 For our potential of Fig 4 where we assume spherical symmetry7 we have an analogous solution to the 1 dimensional step potential7 except that we now have to integrate over the Coulomb potential So U5 is replaced by a geometrical factor multiplied by the barrier height In the limit E lt 07 MeV7 which is our situation7 the solution for the barrier penetration probability becomes PadE m e VdE 72 EGE 7 where EC7 known as the Gamow energy7 depends on the composition of the charged particle gas In the present case for protons7 this is 62 2 E 2 m 2 12 7 8 a lt u gt lt gt in SI units7 where pm is the classical reduced mass of the p p system7 ip2 The factor 14 comes from the integral over the Coulomb barrier Using combinations de ned in Section 12 gives 2 71771170 2 7 2 7T EC 7 8 hzcz 47mm 938 MeV 7137 049 Mev Therefore7 for an average proton kinetic energy of ng 1 KeV7 we nd the Coulomb barrier penetration probability to be PG m WW1 10 10 9 One can compare the probability above with that for which the protons exceed the classical Coulomb barrier because there is a small fraction which have thermal energies far above the average For a classical7 non interacting gas the distribution of kinetic energies at a temperature T is given by the Maxwell Boltzmann distribution PmbEdE X ElZe EdeE 10 which follows the exponential decay form far above the average For kT 1 KeV7 the probability of nding a proton at the Coulomb barrier is x 6 720 10 compared to the quantum barrier penetration factor which we found to be x 6 22 Clearly the thermal mechanism is insigni cant compared to quantum tunneling in this case A correct determination of the probablility of Coulomb barrier penetra tion should involve the convolution of the two distributions of the form PGE A PqE PmbE i E dE 11 The resulting distribution called the Gamow curve has a maximum at about 5 KeV for a star like our sun 23 Proton survival time We can now estimate the rate of the initial hydrogen burning process given in Equation 3 As stated earlier this process is the slowest in the PP chain and hence determines the rate of energy production in main sequence stars This rate is given by Rpp RcolPGsz 12 where R601 is the pp collison rate PC is the barrier penetration probability of Equation 11 and PM7 is the probability that a barrier penetration results in the weak process p a n 6 V In class we estimated the collision rate using the standard formulation R601 mm where n is the number density of protons 1 is their average speed and 039 is the collision cross section 71 is about 1026 cm g We can nd 1 from ng 7102 E m 1 KeV which gives vc2 QEmcz 2938 gtlt 105 so 1 m 4 gtlt 107 cms And we nd 039 from the proton scattering radius N 1 fm giving 039 10 26 cm2 So then we have R601 m 1026M gtlt 10610 26 4 gtlt 107 s71 Because the weak force has such a short range its strength is very small relative to the strong force for the rather large deBroglie wavelengths A hp associated with the 1 KeV protons in the stellar cores Therefore even after a proton manages to penetrate the Coulomb barrier ofthe other proton it is very unlikely that the weak process we need will occur rather than p p elastic scattering via the strong interaction Hence we need to estimate sz PP n5VPPp ppl 11 We are now going to revise and reformulate the expression in Equation 12 compared to what was done in class in order to ll in the pieces more easily We note that since the collision is dominated by strong p p scattering at these energies then Ppp CO1 Furthermore Ppn TimPW where F is the transition rate transition probability per unit time Hence we can re write Eq 12 as R RcolPGsz NrppPGrmrpp NPGrm 13 where N is the number of protons available for the pp process Now the rate for the weak 6 decay process can be used to evaulate PW It is a standard nuclearparticle calculation see for example Halzen and Martin 1984 riGz EOZE Ezd 14 7 W 0 p lt o 7 gt p lt gt where p and E 4 p2 m are the daughter electron7s momentum and en ergy c E 1 momentarily and E0 is the binding energy difference between nal and initial nuclei This calculation ignores many details but should provide an order of magnitude estimate G is the Fermi or weak constant which is related to the strength of the weak force Ghc3 12 gtlt 10 5 GeV Z An evaluation of the integral in the expression above is shown in Figure 5 as a function of E0 The relativistic approximation p gt mac often shown in texts is clearly not appropriate for binding energies of the p p re action Using EO 03 MeV for the p p process Eq 14 yields F N 10 7 s l Combining this with the Coulomb barrier penetration probability Eq 9 we nd Rm ltNgtlt1047gt Therefore our order of magnitude calculation yields an average lifetime for protons in the core of solar like MS stars of 7p NRpp 10 s or about 10 billion years which is the appropriate time scale Along with the overall energy output for the PPl chain found in the previous section we now have a viable model for MS star energy generation 0 009 0 00s 0 007 0 000 g 0 005 g 0 004 0 003 0 002 O 001 Figure 5 The integral in Eq 14 as a function of E0 24 Other nuclear sequences As discussed in Carroll and Ostlie the temperature dependence of the PP chains is roughly given by Rpp appT1064 where 04 depends on many other parameters Other nuclear fusion processes are also possible One example is the ONO cycle depicted in Fig 6 Typically these processes involve heavier nuclei and the temperature de pendence is much greater than the PP chain For example Rm am0T10620 Two other chains with even greater temperature dependence are depicted in Figs 7 and 8 The latter only becomes relevant for T N 109 K The basic scenario for how these other chains come into play is as follows For solar like MS stars with a core temperature 107 K At this temperature the PP rate dominates the rate as expected The higher A processes have much lower rate at these ternperatures so represent a small correction to energy and el ernent production Now since the equation of state gives a pressure P x T in the steady state of hydrogen burning the PP will continue to dominate 13 39lr39ugmi This cycle was proposed by Bethe in 1938 jllf gt discovery of the neutron 1n the GNU cycle carbon nitro use l as cabalysts being consumed and then regenerated du st as with the pp chain the NO cyrle has trumpeting bra anch culminates with tho production of TEU lNJIkl Z and llnliI 15c H 1N 17 7 731V 4 l C 1 r4 13C l H 7 le 7 d 19Nlu a igo i520 4 11 3N i 5quot 14 13N1H 13c 31 Figure 6 The main branch CNO chain Equation 1051 of Carroll and Ostlie However as hydrogen is depleted the PP pressure is reduced allowing further gravitational contraction in turn requiring larger pressure to maintain me chanical equilibrium With a correspondingly larger T The larger T rapidly increases the rate of the highereA processes So as the lighter nuclei are con sumed the burning of these elements Will continue at layers ar larger radius While the inner core will involve the processes With heavier nuclei These larger rates Will consume the fuels relatively quickly At some point in this evolution electron degeneracy becomes important This Will be discussed in subsequent lectures Igo 2 gm w 8Ne 3H9 130 150 a iiNa pt 3M n i Mg N Figure 7 Another highetemperature fusion process Equation 1059 of Care roll and Ostlie i ngg 2 3H6 fESi gHe 130 130 gt P P 158 n i 5 7 Figure 8 Yet another highitemperature fusion process Equation 1060 of Carroll and Ostlie 3 Solar neutrino physics Nuclear processes in stellar cores produce large numbers of neutrinos of en ergy 1 MeV The feeble interaction of neutrinos with matter ensures that they exit the core and star with near 100 transmission This makes neu trinos a unique probe of stellar astrophysics Here we focus on MS stars7 speci cally the sun The underlying physics7 the story of solar neutrino detection7 and the solar neutrino problem7 and its resolution are brie y discussed in this lecture 31 Neutrinos are elementary particles We discussed neutrinos earlier in our brief introduction to the elementary particles Because the neutrinos couple only to the weak force7 they are very dif cult to detect directly It is not so much that the weak force has a small intrinsic strength 7 the coupling constant is actually slightly larger than the electromagnetic coupling The weak force7 however7 has a very short range So while at an energy 100 GeV7 the strength is comparable to EM7 at low energy7 where the deBroglie wavelength is large compared to the force range7 the effective strength is tiny The range helps us understand why the primary neutrino cross sections obey 039 x E2 for E lt 100 GeV7 which is the range of interest for us A neutrino interaction was not directly observed until 1956 by Reines and Cowan However7 their existence was inferred and expected beforehand In 1932 Pauli predicted their existence He based this on the observed energy spectra of electrons or positrons from beta decay processes such as EB a iBe 6V5 If the nal state really consisted only of 2 bodies7 then E5 is a constant7 depending only on the masses involved lnstead7 a continuous distribution was observed for E5 Pauli reasoned that either energy is not conserved in such decays or an undetected particle was present in the nal state the neutrino There are 3 species of neutrinos7 V5 VM and VT one in each of the 3 generations of elementary particles However7 with notable exceptions7 most processes we consider in astrophysics involve only generation 1 this is the lightest7 so we mostly encounter V5 and is Neutrinos were originally assumed to be massless However we now know their masses to be nite7 in 16 part due to the solar neutrinos discussed below Their mass values are not known7 but they are small7 likely to have m lt 1 eVcz 32 Predictions for neutrino production A simple calculation provides a good estimate of the expected ux of solar neutrinos at the earth By rst approximation the PPl chain provides the solar power we observe on earth As discussed in lectures 4 and 57 every iteration of this sequence results in 2 V5 and 27 MeV of kinetic energy7 which we observe eventually as luminosity This luminosity corresponds to a power of 137 mWcmz 853 gtlt 1011 MeVcmzs Hence7 we expect the neutrino ux at earth to be 853 gtlt 1011272 64 gtlt 1010 Vescmz s Figure 9 is the state of the art prediction of the overall solar neutrino ux at earth from Bahcall7 et al The estimate above is indeed rather close to the calculated PP ux The peaking of the curves near the maximum energy is due to the parity violating nature of the weak interaction We note that although the PP neutrinos dominate the ux7 their energy is limited to below 04 MeV The neutrinos from EB decay on the other hand extend to about 15 MeV 33 Solar neutrino detection Detection of neutrinos in general is dif cult In high energy Le elementary particle physics7 neutrino beams can be produced and used as a very incisive probe of the weak force However7 this is only practical because the interac tion probability cross section inreases rapidly with energy7 as noted earlier Hence7 for a neutrino in a beam with energy N 100 GeV7 the interaction rate is large enough so that an experiment at Fermilab called NuTeV was able to collect a few million neutrino events over about a year of data collection These were the pictures I showed in class of a huge splat77 of energy in the center of large quantity of iron The experimentalist hoping to measure solar neutrinos does not have the advantage of such high energy7 although the ux of neutrinos is large7 as seen above Figure 10 summarizes the principal detectors used to measure solar neu trinos over the last few decades The eld was pioneered by R Davis7 who 17 Solar neu ino speckum Flux cm z s 1 1 Neutrino energy 41 MeV Figure 9 Predicted solar neutrino spectra 7 the ux at earth as a function of neutrino energy From Bahcall came up With a viable detector consisting of a large vat of cleaning uid deep underground in a mine at Homestake SD The experiment is sensitive to neutrinos in the reaction 15 37Cl a 6 37Ar The resulting argon gas bubbles out and is extracted Since this isotope of argon is unstable its quantity is determined by measuring the decay curve All solar neutrino experiments are deep underground so that cosmic ray particles are ltered out by the overburden The Davis experiment ran for about tWo decades With a measured ux Which on average Was 34 i 3 of the predicted ux before the result Was con rmed by other experiments In the meantime there Was a great deal of debate about the result Davisls ex periment Was primarily sensitive to 8B neutrinos since its threshold is above the PP neutrinos Could the solar theory be trusted for nonipp sequences Where there are such large temperature dependences see lecture 5 The disr crepancy betWeen theory and experiment Was known as the solar neutrino problem Experiment Reaction Thrmhold MeV Observedexpech rate SAGE GNO cc 7 Gave7 ce 02 053 i 004 HOMESTAKE cc 37C1v c37Ar 03 034 l 003 SNO cc ve 2 H a 13 p e 5 035 d 003 SUPERK ES v 4 e a v 4 e 5 046 i 001 SNO ES v e a v e 5 047 1 005 SN0 NC v2H gtpnv s 10114112 for vu v and Via NC and CC fwavc Figure 10 Main solar neutrino detectors and measured uxes From Perkins The next big breakthrough Was the gallium experiments SAGE and GNO The technique Was similar to the Davis experiment except using ale lium as the target Which has a threshold of only 02 MeV in the process indicated in Fig 10 T ese experiments Were sensitive to the PP neutrinos and their detected ux Was roughly half that expected by theory Meanwhile 19 the Super K experiment in Japan had also con rmed the missing high energy ux using a technique which allowed them to actually image the direction of the neutrinos con rming their solar origin The detectors using water as a neutrino target rely on the following principle The solar neutrinos collide with an atomic electron sending it out into the water as a free particle with a speed close to c This is faster than the speed of light in the water which has speed cn where n is the index of refraction This results in a shock wave phenomenom known as Cherenkov radiation which can be detected by sensitive light detectors photo multiplier tubes installed on the periphery of the water tank The problem was nally resolved a few years ago by the SNO exper iment Recall that we know that there are 3 species of neutrino V5 VM and VT The Davis and gallium experiments could only measure the V5 type Since only V5 are produced in the nuclear reactions in the sun this may not seem to be an issue However we note that the Super K measurements were slightly sensitive to the other two types and they measured a higher rate The SNO breakthrough was that its detection target material is heavy water that is H20 where the H is largely deuterium As we said in lecture 4 the deuteron is very weakly bound Hence a neutrino of a few MeV energy can disassociate the deuteron The resulting charged proton can be observed by the Cherenkov technique This process um2H pnum7 is equally likely for any of the 3 neutrino species x The SNO result for this reaction is consistent with the predicted solar ux This means that the problem was not a problem with the solar theory but rather indicated a new property of the neutrinos themselves as discussed brie y below The V5 type produced in the solar core were turning into the other species with some probability on their journey to the earth The other detectors only saw the remaining yes but SNO saw them all Hence SNO not only con rmed the origin of the problem but also con rmed that the solar theory indeed predicted the correct neutrino ux Davis shared the 2002 Nobel Prize in physics 34 Neutrino oscillations and mass To illustrate the point we discuss only two neutrino species The general ization to three is qualitatively the same but is more complicated Let the 20 neutrino species have small7 but nite masses7 with values eigenstates m1 and mg The quantum state for m1 will in general be a linear combination of V5 and VM The state begins as ye as determined by the weak interaction process in the solar core Now7 quantum mechanics requires that the mixed state evolve in time according to 7W w06 AE E 7 where AE is the energy difference of the states Relating this to the mass eigenstates and tto the distance travelled7 the probability for transition from one type of neutrino to the other7 called neutrino oscillation7 over a distance L in km in vacuum becomes PM 1321 sin220 sin2 127Am2 LE 15 at an energy E in GeV7 and A7712 mg 7 in is in eVZcz The parameter 6 determines the linear combination ibl cos ibe sianM and is called the mixing parameter One additional detail While the neutrinos do not get signi cantly absorbed by the sun on their outward trek7 there is an interesting resonance effect where the neutrino mass difference matches the ambient mass density The interaction is analogous to light passing through transparent material with index of refraction gt 1 The effect is called MSW enhanced oscillations7 and can easily result in maximal mixing of the neutrino states7 ie tant9 1 In this case we would also expect the ratio of V5 to total ux at earth to be with some energy dependence from Equation 15 The data are consistent with both of these predictions Hence7 the resolution of the solar neutrino problem requires nite neutrino masses This effect has also been seen in other types of neutrino experiments7 so it is known that all 3 types mix with each other While only A7712 is directly measured7 the implications from all the data are quite strong that the individual masses are very small7 m lt 1 eVcz If this is indeed the case7 as we shall see later7 this makes neutrino mass irrelevant for cosmology 21 4 Type II Supernova Physics We consider the physics and phenomenology of supernovae energetic explo sions of stars focussing on the so called Type ll type of supernova which involves an evolved massive star which is not necessarily a member of a bi nary system As discussed in the next lecture a supernova event produces a shock wave of rapidly expelled stellar material which is bombarded by a huge neutron ux Neutron capture events result leading to the produc tion of elements heavier than iron and nickel which we havent been able to produce in the stellar processes discussed so far Hence heavy elements are produced exclusively by supernovae and the development of earth and its life forms depend on at least one cycle of heavy star evolution and subsequent supernovae 41 Massive star core collapse sequence Consider an evolved star one which has evolved off the main sequence and has an evolved mass M which may be signi cantly smaller than its zero age main sequence ZAMS mass As discussed in previous lectures at some point the pressure due to electron degeneracy becomes important This is the pressure which results from electrons being forced to higher energy states due to the Pauli exclusion principle which applies to all fermions spin h particles such as electrons or neutrons Now if M lt 14MG then the electron gas is non relativistic and the electron pressure is x 7153 where n is the electron number density and the star is stable under gravitational collapse since the gravitational pressure is x 7143 The fate of such a star is a white dwarf as discussed before The mass 14M is known as the Chandrasekhar mass However if the evolved mass is gt 14MG then the electron gas is rela tivistic the electron pressure is x 7143 and the star is in unstable equilibrium with the gravitational pressure Eventually an unstable equilibrium always moves toward a stable equilibrium at with lower potential energy con gura tion A massive evolved star would eventually have a core temperature of 5 gtlt 109 K and a density of 3 gtlt 10L0 kgm3 The average thermal kinetic energy of nucleons is then 1 MeV and the Boltzmann distribution extends to suf ciently high energy to allow fusion of nuclei up to nickel and iron see Figure 1 from Lecture 4 After iron it is energetically unfavorable to go to higher atomic number so the massive star develops an iron core with silicon 22 burning in the next sub shell outward in radius The silicon eventually ends up as iron7 too7 but after about one day this fuel is exhausted Our mas sive star is now in unstable mechanical equilibrium7 supported only by the electron degeneracy pressure The following describes what is thought to be a typical sequence of events for the collapse of a massive star7 resulting in a supernova event However7 keep in mind that rotation and magnetic elds7 which we have ignored so far7 may have an important role7 although the outcome is unlikely to change signi cantly 1 After fusion processes zzle out7 there is some gravitational collapse which causes heating to T N 10L0 K This is suf cient to trigger two processes i Photo disintegration of iron and the subsequent photo disintegration of the products7 eventually leading to complete inverse fusion7 with products p and n 56Felt gtgt134He4n 4Hegt2p2n 16 v v A x lnverse beta decay When electrons have K gt 37 MeV then the following can occur 6 56Fe a 56Mn V5 and by time we get down to nucleons eip nye 17 Note that the Fermi energy of the degenerate electron gas is m 4 MeV at a density of 1012 kgmil7 so there are plenty of electrons which can trigger these inverse beta decays D The processes above are endothermic that is they remove kinetic en ergy and hence uid pressure from the core The unstable core now quickly collapses C40 The iron core is now in free fall The time of fall to a much smaller equilibrium radius is N 100 ms 23 4 During collapse the neutronz39zatz39on processes in Eqs 16 and 17 proceed rapidly Neutrinos result as well and these mostly exit the star This neutrino emission represents about 1 to 10 of the total emission the remainder resulting from subsequent steps 9 The collapse ends when the core reaches nuclear density Actually the density exceeds nuclear brie y by what is estimated to be a factor of 2 to 3 This is discussed with some more detail in Section 43 below CT The core now strongly bounces back to nuclear density from the super nuclear density We can think of the protons and neutrons as bags of quarks bound together by very strong springs spring constant N 10 GeVfmz which are compressed by the collapse but then spring back to equilibrium thus the bounce I The bounce sends a shock wave outward at high velocity blowing off the remaining stellar atmosphere in the process One the shock reaches the outer atmosphere the photons emitted by recombination powered by the shock itself and by subsequent nuclear decays become the visible supernova explosion 00 The core will radiate away its huge energy content in neutrinos as discussed below and the remnant core will settle down into a neutron star The radius is something like 15 km depending on initial core mass but has a mass of 14 to about 3 MG The neutron rich shock meanwhile will induce creation of elements heavier than iron by neutron capture This will be discussed in the next lecture in more detail 10 The shock continues into interstellar space at speeds of 010 For example the crab neubula resulting from the 1054 AD supernova is large and still expanding 11 The neutron star may become visible in radio as a pulsar depending on rotation and magnetic elds The crab7s neutron star is indeed a very loud pulsar faithfully producing a radio burst once per revolution every 333 ms The visible light luminosity of atypical supernova is roughly 1042 J with a peak power of 1036 Js 1036 W This is about a factor 10L0 greater than the 24 solar luminosity which is comparable to an entire galaxy The visible light decays exponentially as unstable nuclei decay so the supernova is visible for weeks depending on its location This is discussed more in the next lecture However as impressive as the light output is it represents only about 1 of the total energy output as we shall see in Section 44 We follow up now with some detail 42 Energy from collapse We saw earlier that the total gravitational energy released in collapse is the change in potential energy given by GM2 5 B when the nal radius R is much smaller than the initial In our case the ratio is 103 Using M 15MG that is just over the Chandrasekhar limit then with R 15 km we get E grav Egrav 6 gtlt1046 J 4 gtlt1059 MeV 18 This is a big number It will feed into the energy release of the supernova 43 A very large nucleus As mentioned above the collapse proceeds to nuclear density where it is halted by the strong nulcear force It takes collision energies of 10 GeV to partially break up a nucleon neutron or proton As we shall see momentar ily the average energy from the collapse is still 2 orders of magnitude from this So the nucleons predominantly neutrons at this point get squeezed by the collapse to sub nuclear volume then rebound giving rise to the super nova bounce and the return to nuclear density So we are justi ed in treating the collapsed core as closely packed nucle ons essentially at nuclear density and consisting primarily of neutrons As we saw with the electrons the neutrons will also form a degenerate gas77 since they are fermions We return to this brie y at the end In any case the collapsed core will settle down into what is termed a neutron star Nuclear density pN we can estimate by the average density of a proton or neutron pN mp47Tr23 2 gtlt10 lL7 kgmg 25 where we have used To 12 fm for the proton radius based on lab scattering measurements With pN as the average nuclear density then the total mass of this proto neutron star is Mm Nmp pN47TR33 where N is the total number of nucleons neutrons Combining these and using Mm 15M yields RroN13 15 km N 15MGmp 2 x1057 19 Combining Eqs 19 and 18 gives an estimate for the average energy per nucleon of 3 gtlt 10592 gtlt 1057 m 100 MeVnucleon This number provides two lessons o It is a large kinetic energy but far short of what is required to break up the nucleons 0 Recall from Fig 1 of Lecture 4 that iron had the largest binding energy per nucleon at 8 MeVnucleon The collapse provides more than enough energy to completely break up even iron into constituent nucleons as we had presupposed 44 The rst seconds of the proto neutron star Because the proto neutron star is so dense a qualitatively new feature is manifest Namely the vast majority of the collapse energy of Eq 18 can not easily escape To see this we calculate the mean free path of the most weakly interacting particles the neutrinos As we saw in Lectures 5 and 6 the collision rate is given by 7101 where n is the number density and 039 is the interaction cross section Hence the time between collisions is the inverse of this and the mean distance be tween collisions the mean free path is Z vmm 1011 In this case n N47TR33 and 039 is the neutrino nucleon cross section The so called charged current cross section for the process 71 VS a p 6 is given by 58 2 2 N 745 E 2 2 U007 WGFE N10 lt1MeVgt m7 where Gp is the weak Fermi coupling constant encountered in Lecture 5 If we use EV 20 MeV this gives Z x 2 m l The mean free path for neutrinos attempting to escape the collapsed core is only a few meters 26 As it turns out7 the neutral current scattering process V1 n a V1 n is also possible here7 where z represents any of the 3 neutrino species A similar calculation a bit more complicated for this process gives Z x 10 m Hence we arrive at the following conclusions 0 The collapse energy is locked in the proto neutron star and can only be carried away by neutrinos Estimates are that about 99 of the total energy of collapse is carried by the neutrinos o The neutrinos are radiated from a thin few meter thick skin on the outside of the core With Z N 10 m7 the energy transport via neutrinos is a random walk7 diffusion problem We can estimate the diffusion time to be t N RZCZ 1 s This compares to a speed of light ight time of Rc 10 4 s 0 There are effectively 6 neutrino populations in thermal equilibrium with the super hot core 7 3 species7 with both neutrinos and anti neutrinos These all share the energy transport via the neutral current process Hence we expect the 150 MeVnucleon to be carried away with an average energy per neutrino of 1506 20 MeV Thus7 along with the visible supernova7 we expect a huge neutrino ux equal to the collapse energy 1059 MeV carried by about 1058 neutrinos in all 3 species and V plus 9 with average energy 20 MeV 45 SN1987a In 1987 supernova was observed at earth located in the large magellenic cloud LMC7 a small galaxy adjacent to the milky way7 at a distance of about 60 kpc Luckily7 two underground proton decay experiments were taking data These detectors can observe neutrino events using the water technique discussed in Lecture 7a In fact7 one of the detectors7 Kamiokande7 later became the Super K detector encountered in 7a The two detectors simultaneously observed a burst of neutrino events about 7 hours before the optical supernova was observable The predominant detection process was Dep ne 27 Energy Mew Figure 11 Neutrinos detected from SN1987a With the 1MB and Kamiokande detectors The neutrino data is reproduced in Fig 11 We can noW compare the observations With our expectations The mea7 sured neutrino energy is indeed about 20 MeV The apparent decline With time can be used to determine neutrino mass The limit turns out to not be better than terrestial experiments The neutrinos are indeed spread out in time by N 1 s Again7 nite neutrino mass can disperse this somewhat The number of detected events Was 20 Only 1 Were observed7 so the total ux at earth Was 20 X 601 m 1010cm27 Where an is the detection cross sec tion times e iciency Translating this ux to the distance of the LMC gives a total energy of 2 X 1059 MeV7 in good agreement With our expectations from the previous section It is interesting to note that the neutrino burst from the collapse may in fact transfer enough energy to the supernova shock front to keep it from falling back onto the collapsed core This may7 in fact7 require all 3 neue trino species although the calculations are di icult7 in Which case We Would require the 3 species to make heavy elements and life on earth 28 46 Further collapse and black holes We discussed collapse stoppage and bounce above in terms of the strong nuclear force However since we form a core of mostly neutrons we might expect neutron degeneracy to also play a role And it does As the proto neutron star radiates away its collapse energy we might expect it to undergo further collapse if not for the degeneracy pressure Carrying out the same calculation we did earlier for electrons results in an equivalent Chandrasekhar mass of about 56M If the collapsing core mass exceeds this then we might expect the collapse to proceed to a black hole If less then it remains a neutron star However the observed transition from neutron star to black hole is closer to 3M9 We might have expected that this would not be so simple Among the complicating factors all dif cult to calculate but important in this regime are 0 repulsion due to the strong nuclear force 0 neutron degeneracy o non linear gravity that is strong gravitational effects as predicted by General Relativity large angular velocity 0 large magnetic elds On the note of strong gravity we nish by mentioning the Schwarzschild radius R5 predicted by GR as the event horizon77 where proper time inter vals go to zero and light cannot escape It is given by Rs 2GMc2 For M 5M9 RS 15 km comparable to the size of our neutron star but with larger mass Adding mass to a neutron star would result in a black hole If the initial collapsing core had mass 5M or more the collapse might proceed directly to a black hole although many researchers believe there would be a stopping bounce followed by infall back onto the collapsed core leading then to a black hole 29 5 Cosmological expansion In this lecture we discuss Hubble7s observation of galactic redshifts in the context of general relativity The cosmological principle provides simplifying assumptions7 allowing to an equation of motion77 for the scale factor B known as the P riedmann equation This equation will be a starting point for many of our discussion of physics of the early universe 51 Spacetime in Relativity In special relativity7 a key concept is the invariant spacetime interval dsz dsz c dt2 7 ltd2 dyz d22gt which separates two points in spacetime The interval do is the same for any observer in an inertial reference frame It can also be written equivalently as dsz Zgwdxpdxy where gm is the metric tensor and dxp ct7z7y72 are 4 vectors For the at spacetime of special relativity7 gm is the Minkowski metric7 represented by a 3 gtlt 3 matrix with diagonal elements 17 71 71 71 and all other elements zero In spherical coordinates7 then the interval in at spacetime becomes d5 7 c dt2 7 dr2 7 r2 d62 7 sin2 was This can be compared with the Schwarzschild interval which is a solution of the equations of general relativity for the case of spherical symmetry at a distance r from a mass M 2GM 2 cdt2 7 17 QGM T62 d5 7 1 7 T 7 72 do2 7 sin26d 2gt 20 rc The proper time ClT measured by a clock at rest at a distance r from the mass is given by 2GM dt 2 he gt where dt is the time measured in a distant inertial frame We see that as one approaches the distance r 2GMc2 E RS7 the proper time interval CdT2 17 30 approaches zero This distance R5 is the Schwarzschild radius encountered before7 representing the horizon of a black hole Finally7 under the assumptions that the universe is isotropic and horno geneous7 the FLRW interval result for general relativity is drz d2 dtLthi s c o 17W rzd z 72 sin2 0d 2 21 where Rt is the scale parameter and K represents the type of spacetirne curvature The equations can be re scaled so that K conventionally can take on one of three discrete values 0 K 0 Flat spacetirne The 2 d analog is a at sheet 0 K 1 Positive curvature The surface of a balloon in 2 d 0 K 71 Negative curvature A saddle point in 2 d The scale pararneter Rt represents the fractional expansion of the dis tance between two points due to cosrnological expansion see next sections So if To is a distance between two objects7 then at a later tirne7 the true coor dinate distance between the objects has become rt Rt r0 To is known as the cornoving coordinate distance Note that Rt is only a function of time 52 Hubble s Observation and its interpretation The redshift 2 measures the fractional shift of the observed wavelength X of a spectral feature7 eg an emission line7 relative to the wavelength A of the feature measured in a proper frarne So 2 AAA X 7 AMA or X A1 2 22 Hubble observed redshifts for tens of relatively nearby galaxies for which their distance was independently known The relativistic Doppler shift pro vides an explanation for a shift of wavelength due to the motion 1 of the source with respect to the observer 12 i M 1 1 110 31 XA where the approximation is valid only for 1 lt c Combining this with Eq 22 gives 2 oc which Hubble used to then arrive at his hypothesis iH0r H is the Hubble parameter and the subscript refers to its value at our current epoch The presently accepted value of H0 is about 70 krnsMpc The interpretation of the galactic redshifts in terms of a Doppler shift fails to be consistent with the data at larger values of 2 In addition it implies that the Earth or at least our galaxy is in a unique position in the universe It violates the cosmological principle that the universe at suf ciently large distance scales is isotropic and homogeneous Consistency with this principle implies that spacetirne itself is expanding along with its contents and that the expansion was initiated a singular event the big bang As we will see this hypothesis is consistent with all well de ned predictions to date including the redshifts and has successfully predicted a large number of diverse observations It is natural to describe the cosmological expansion using general relativ ity The scale pararneter R introduced above now becomes the key quantity used to describe the expansion of spacetirne Two important relations follow immediately The scale parameter at a time t after the big bang correspond ing to an observed distant object at redshift z is related to the current scale R by Rt PLO1 z 23 And Hubble7s hypothesis can be re written as R HR 24 where we have introduced the notation dfdt We note that if H were a constant is is not then Eq 24 becomes drR Hdt which is integrated to give Rt olt th In this case the size of the universe increases by a factor e in a time t1H014Gyr 25 This time 1H0 is known as the Hubble time Finally we note that the cosmological principle does not mean that the universe can not have dramatic structure 7 in fact it does 7 only that such structure averages out over suf ciently large ie cosrnological distance scales to look the same in all directions and at all locations 32 53 The Friedmann equation and a useful analog The eld equations of general relativity can be written in tensor form GM 87TGTM A9 26 where TM is the stress energy tensor which contains the description of the mass and energy densities GM is the Einstein tensor which gives the re sulting spacetime curvature and A is the infamous cosmological constant We will not use this equation directly in which many ordinary equations are embedded Envoking the cosmological principle and assuming matter and radiation elds can be described as ideal frictionless uids more on this part later Friedmann found the following solution to Eq 26 in terms of the scale parameter R i 2 R 87TG K02 H2 Ptot W 27 Here pm is the total energy density and K is the curvature parameter in troduced above We can gain signi cant intuition about the Friedmann equation by show ing its equivalence to a more familiar model Let us assume that p is dom inated by ordinary non relativistic matter and that this density is uniform We put a test mass m at the edge of a sphere of radius Rt Only the mass within the sphere M 47TR3p3 contributes to the gravitational force acting on m This Gauss7s Law for gravity also holds in general relativ ity Hence the potential energy of m is iGMmR and the total energy E EK U is 1 2 2mR R Substituting M 47TR3p3 and multiplying through by 2mR2 then the left hand side is identical to Eq 27 And the right hand side the total energy term becomes 2EmR2 Hence we can make the identi cation E iKcz2m or EK U E X 7K 54 Synopsis of matterdominated solution Under our assumption of a matter dominated universe the analogy presented above can be related to the familiar classical mechanics problem of a mass 33 subject to a spherically symmetric gravitational eld for which E 0 repre sents the dividing line between in being gravitationally bound or unbound So we have 0 K gt 0 i E lt 0 i bound closed R returns to 0 0 K lt 0 i E gt 0 i unbound open R goes to in nity 0 K0 at E0 RaooandRa0astaoo These solutions for Rt as a function oft are summarized in Figure 12 The last solution for a at universe corresponds in our analog to the solution for the classical escape velocity where the velocity of the test mass goes to zero at in nity We note that K 0 is quite strongly favored by current observations We also note that in this case the condition E 0 means that two large numbers K and U the total kinetic and potential energies of the universe have to cancel to large precision We will discuss this seemingly unlikely situation when we discuss in ation It is interesting to evaluate the time evolution of the three cases above For K 0 it is easy to show that the Friedmann equation with p M47TR33 becomes RlZdR xQGM dt which gives Ron 3 and the age of the universe as 23H0 Furthermore the critical density which enables this E 0 solution see homework can straightforwardly be shown to be 3H3 i 87TG 39 For K gt 0 one can show using a similar analysis that a big crunch occurs at a time 27TGMc3 10 Gyr after the big bang p 28 34 8 Open I 0 39 Hal 20 b w a V 4 a 5 RH x H3 E 2 Closed K l 05 H C units of ZwGMcz Figure 12 Time evolution of the scale parameter for the three possible types of spacetime curvature7 K 7 assuming nonirelativistic matter dominates the energy density 6 Energy densities We begin with a brief overview of the energy density contributions to the cosmological evolution of spacetime Using our de nition of the critical den 2 sity in Eq 28 p0 3 we can re write the RHS of the Friedmann equation Eq 27 87TG K02 K02 H2Ptot72H2PPc 1 7 or K 2 c PP0W1 29 Now we de ne 9 Eppc and we write out the contributions to p due to non relativistic matter radiation ie relativistic particles and vacuum energy see below P Pm Pr Pu Finally we divide through this expression by pc to arrive at 1Qm979v9k 30 where we have made the de nition of an equivalent energy density due to spacetime curvature K02 RZHZ 39 Current observations of the present epoch give the following best values 9k 0 9k 0 cosmic Mwave background observations especially the most recent results from WMAP Q m 105 0 9m x 03 All of normal baryonic matter is 005 the remainder is dark matter 9v 07 This would be the contribution due to dark energy 7 more on this later 36 Table 1 Properties of cosmic uids R t p eqn of state matter R x t2 3 p x Pig P p02vc2 radiation R x tlZ p x Pf pcz vacuum R x 6 p const pCZ P P The three types of cosmic uids which we have input to our general relativistic description of matter and energy are listed in Table 17 along with a few oftheir important properties The time dependence t23 for matter was determined in the previous section Figure 15 shows the cooling of the constituents due to cosmic expansion as a function of time7 for the radiation and matter dominated eras The transition between the radiation and matter eras occurs at the decoupling time7 tdec 3 gtlt 105 yr 1013 s Before this time7 photons7 electrons7 and protons were in thermal equilibrium7 with e plt gtH At tdec cooling has gone suf ciently below the ionization energy for hydrogen 136 eV that the reaction above goes primarily from left to right7 thus producing predominantly neutral matter which is suddenly transparent to photons These photons are what we currently observe as the cosmic microwave background CMBR with a blackbody temperature of 27 K This cooling corresponds to a redshift in the wavelength of 1100 So the decoupling time is tdec 3 gtlt105 yr 1 zdec 1100 31 This last relation implies that the size of the universe was about 103 smaller at decoupling than today Finally7 a useful relation is that between R and T ch 1T 32 37 End of elecImweak V uni canon 1039 1 Te Quark hadron mxnsilion n 1 GOV Bi ang 390 nucleosymhcsis 1 MEV f 0390 Neulrinos decouple Decoupling m manor and radiaxionwmms I07 U 6 39 forcpr HY T K lcV 10 l ch m 10quot 1M 1 106 1013 3x10 Time 15 a 39 Figure 13 Energy kT per particle Versus time Radiation dominated the energy density until the time of photon decoupling7 tdec m 101 s 7 Vacuum energy and In ation In Section 5 we introduced Einstein7s eld equations Eq 26 including the comsological constant A term Since gm represents at spacetime this term is uniformly distributed For the Friedmann equation Eq 27 this term gets included as A3 on the RHS Since this term has no R dependence then for suf ciently large B a non zero A will eventually dominate the RHS of Eq 27 In this case the Friedmann equation becomes p ltA3gtR e Melt e Rlttgt 0lt v lt33 where 04 E m as in Table 1 Now we make the connection between this funny A term in Einstein7s equations and the idea of vacuum energy from quantum eld theory QFT In quantum mechanics the uncertainty principle states that there can be a non conservation of energy AE over a time At such that AEAt 2 252 ln QFT this must occur with particle pairs such as ee appearing and disappearing consistent with the uncertainty relation above On average there is a net contribution which is non zero This represents the vacuum energy So the vacuum is not a state of nothing it is simply the lowest energy level or ground state Hence Einstein7s general relativity will couple to this vacuum energy In addition in QFT we expect the vacuum to have a uniform energy density just like the A eld Hence we identify vacuum energy with a non zero A Hence this allows us to possibly connect episodes of spacetime evolution in uenced by A with fundamental elementary particles and elds Such is the case for in ation discussed below We complete the identi cation of A with vacuum energy by including it as a contribution to p in the Friedmann equation A p 87TG39 We used pv in this form in the previous section We will not get into speci c particle physics manifestations for in ation We will motivate it using the observations of the CMBR and then discuss the properties the in ationary era must possess 39 71 The Horizon problem of the CMBR Refer to Figure 14 which shows the main features ofthe major epochs going back to very early times From the CMBR observations we have a very good measurement of tdec and zdec as shown in Eq 31 If we extrapolate this to earlier times using the t12 scaling appropriate for this relativistic phase then we can see from Fig 14 that the universe at tdec is too big A more precise way to see this is the horizon problem7 discussed below The horizon is the distance over which two objects are causally connected that is they can exchange energy Since such processes occur at speed c or less then in a static universe the horizon distance is simply LH ct where t is the age of the universe In an expanding universe LH is somewhat increased In general for R x t then 7 ct flin39 LH So for the radiation era n 12 and LH 2ct Therefore at the decoupling time we can expect that regions of radius 2ctdEC were in causal contact in thermal equilibrium and so at uniform temperature Outside this volume the hot plasma had developed independently and so there it should not be expected to be at the same temperature We can easily relate the LH for the CMBR to an angular range for our current CMBR observations Since the decoupling time a volume containing a single causal horizon has expanded with the general cosmic expansion by a factor 1 zdec Since such a region under current observation is at a light distance of Cto 7 tdec where to is the present time then one CMBR horizon subtends an angle for current observations of i QCtdec l Zdec i Cto 7 tdec On the contrary the CMBR observations indicate a remarkably uniform tem perature over the entire sky at the level of ATT 105 This departure from naive expectation is known as the horizon problem It must have been the case that our currently observable universe was entirely within one hori zon at some early time However during the radiation era R x tlZ whereas LH x t thus it is not easy to see how to make R N LH at some early time without some period of rapid expansion of R lndeed that is now thought to be the case This period of rapid increase is known as in ation o m 2 34 40 72 In ationary scenario We saw above Eq 33 that in principle some kind of vacuum energy if it dominates the total energy density can drive spacetime into a period of exponential expansion For various particle physics reasons it is generally assumed that this in ationary period begins at t N 10 34 s In the usual scenario a spin zero particle moves from a local minimum to a global mini mum the true vacuum at this time ln ation terminates when the true vacuum is attained What is required for the in ationary period Let us extrapolate back from the decoupling time to the in ationary era throughout which we expect R x tlZ Now the current Hubble radius is about 1026 m so that at decoupling the size of the universe was this divided by 1 zdec or about 103 So then at 10 34 s the size was 102510 341012 2 1m whereas the horizon was LH 201 10 26 m Hence in ation must make up these 26 orders of magnitude so that horizon can become comparable to the size of the universe thus providing a uniform CMBR So from H RR we get for the in ationary expansion ratio R2R1 CHAt 1026 or HAt needs to be 26 ln10 60 We can write this in terms of a vacuum energy density using the Friedmann equation 87TG H2 T p H mm The basic scenario is as shown in Figure 14 In addition to the horizon problem in ation also provides a solution for the flatness problem This is related to our observation that the total energy E of the universe is very nearly zero requiring the cancellation of two large numbers N 1070 J the kinetic and potential energies of the universe Recall that E 0 corresponds to K 0 Now the curvature term in the Friedmann equation is KCZRZ Hence in ation this contribution suddenly by a factor R2R12 1052 Hence after in ation when pv no longer dominates then from Eq 29 we see that the extreme atness requires that 9 ppc 1i10 52 7 35 or in other words 9 1 to 1 part in 1052 41 5 20 m kTMFL 20 w a 10 J o logw R m l 93 lngm krch 40 k cxpml decoupling 720 730 3 w New l 4 I 740 30 WED 710 0 10 20 long 5 Figure 14 R and T VS time using expanded logarithmic scales This shows the in ationary era at about t 10 34 s 42 8 Fluids and 2nd Friedmann equation Recall that we have discussed matter7 radiation7 and vacuum within the context of general relativity as ideal cosmic uids Along these lines7 we now come up with a second relation for the time evolution of R which is also very useful Our ideal uid obeys conservation of energy in the form of the First Law of Thermodynamics for an isolated system no heat ow dE iPdV where E is the internal energy Letting p02 be the total energy density7 then this becomes dp02R3 7PdR3 After carrying out the variations with respect to time and some algebra7 we can solve for dpdt 2 pa P i 73 7 R 36 p lt R02 gt If we now differentiate both sides of the Friedmann equation Eq 27 with respect to time7 this gives 87TG 3 If we substitute for p from Eq 367 this gives the so called second Friedmann equation 4 GR R p 31302 37 A number of consequences spring from this result First7 for positive values of p and P7 then R gt 0 immediately implies that R 0 at some early time the big bang Next7 we can use the equations of state for the uid components from Table 1 to determine the resulting acceleration of the scale factor Einstein7s original idea for using A was to make p 31302 07 which is possible since A gives a negative pressure Note that in general7 however7 the equation of state for vacuum energy guarantees in fact that p 31302 lt 07 resulting in positive acceleration This must have been the case for the in ationary period7 and7 according to recent evidence7 we may just now be entering a new phase of positive acceleration due to the dominance of vacuum energy This will be a topic of discussion in upcoming lectures 2BRlt gtR2p2pRR 43 9 Particle dark matter We discuss some issues surrounding neutrinos as potential sources of dark matter Although we will reject this hypothesis7 neutrinos represent a proto type for other possible sources of particle dark matter7 generically known as WlMPs We will then discuss some possibilities for WlMPs and nish with some the dif cult task of detecting WlMPs 91 Neutrino decoupling Figure 15 was also shown earlier in an earlier lecture In this case we high light the point at which neutrino decoupling occurs Figure 16 gives another history of the universe7 showing the main quantities of interest as powers of ten At times prior to decoupling7 the neutrinos are in equilibrium with the rest of the particle and radiation of the expanding universe according to equilibrium reactions such as ylt gt667lt gtZ lt gtVP where y is a photon and the Z0 particle is the neutral carrier of the weak interaction see the particle physics primer The large mass of the Z0 about 91 GeVcz accounts for the short range ofthe weak interaction Since neutri nos only couple to the weak force7 they cannot directly interact with photons As long as the neutrinos are frequently exchanging energy in reactions like this7 then they will remain in thermal equilibrium As the universe expands and cools7 at some point the particles will no longer stay in thermal contact 7 this is decoupling We encountered this earlier in the discussion of the decoupling of photons7 thus giving rise to the cosmic microwave background We now try to determine the point of decoupling We can use a simple relation to understand the point where decoupling oc curs As discussed previously7 the particle collision rate is given by P n01 where n is the particle density of the interacting particles7 039 is the interaction cross section7 and 1 is the relative speed of the colliding particles We simply compare this collision rate with the rate of expansion of the universe7 which is H RR So within factors of order 17 we nd P mm gt H i equilibrium 38 P 7101 lt H i decoupling 39 44 my I V 11 Qualkrlmdlon uammon mu lGrV BigBang nuueumu m Radznhon dnmmmcd 0 1M v m MM 9 Dccmlphng m manor 1 and mmwun mm 07 1 keV H Tum av 10 l m l Nev Vlagi lr11mmuml 7 K t w v i NW 10quotquot Mquot I W 10 x lo mama 39 Figure 15 Energy kT per particle versus time Note the indicated time of neutrino decoupling We can noW input the known physics to determine the factors above When We do We nd that F H at kT m 1 MeV When the age of the universe Was approximately 1 s 92 Cosmic neutrino background In class We simply looked at the temperature dependence of each term of Eqs 38739 to determine that FH X T3 40 So the temperature of decoupling varies rapidly With collision rate for ex am le We can use these ideas to qualitatively see hoW neutrino decou lin in uenced formation of light nuclei Which began occuring slightly later When t m 1 hr Since We saW earlier that X 9 2TZ W ere 9 increases With t e number of available particle species then a larger number of neutrino types 45 WW v n C a39 w Figure 1e Myalhmm humy nuke unwmse NW Increases H end mam by Ba 0 me damplmglempemlure m7 deem demuplmg Occurs eexllm This In turn means um the eqummum m ninemmn 7 mm mm Mme WWW 1 3 MeV 15 1319 eldemuplmg Thaleng My mum 011m mm a 1319 mm cf neulxmmdq nualm In pmmm amp 1313 NV gves use m awedmed HeH rem Wm 15 1319 m quenmeuve mm cf m above exgxmmls Wm mg m the 1980 s xnfneumnc species In swam ehoulNy s A x as 89 sandman mm m m es m 2 pexuds m theme guy 2 ms 1 nae mmeme whm me m mm enany equeummess mime 2 91 cewca es shown In g 17 Thewld q aim quantum smedepmds an menumhexm pmmhls e decay modes and hence N The current experimental result is N 300 i 001 Hence the prediction based on cosmic astrophysics Was con rmed my 2quot znyllimg 6 mm Emma r 7 Figure 17 Accelerator experiment result showing the excitation of the Z resonance The curves shoW the theoretical expectations for different as sumptions for the number of neutrino species The neutrinos Which decoupled from the cosmic reball at t m 1 s in analogy to the 273 K photon background should still be around As dis cussed in the previous lecture the predictions are that the density of these neutrinos is n 113 X N m 340 our3 With an average thermal temperature noW of 195 K 7 93 Neutrinos as dark matter We saW in Lecture 7 that there is noW good evidence that all 3 neutrino species have a small but nite rest mass my N 10 3 eVc2 But since there 47 are a lot of them around us from the decoupling7 then perhaps this gives an appreciable contribution to the estimated dark matter mass density We nd that to make a critical density from cosmic neutrinos requires that the mass of the 3 species total 47 eVc27 that is pl p0 pl 340 gtlt 47 eVcm3 41 We would require about 30 of this in order for neutrinos to give the righ 77 contribution to dark matter The data suggests it is about 103 short of this So it seems that neutrinos contribute rather insigni cantly to dark matter There is one other reason why it was not expected that neutrinos would contribute to this Structure formation companion lecture demands that the dark matter be cold that is7 that the particles be non relativistic during the early era of structure formation Recall that the netrinos decouple with an average energy kT 1 MeV7 much larger than the rest energies lndeed7 at the dawn of structure formation7 the neutrinos would have only cooled to 10 eV7 still relativistic Therefore7 neutrinos are not only too light7 but also too hot77 to be an important contributor to dark matter 94 WIMPS We didn7t discuss this in much depth in class But this is an important topic for both astrophysicscosmology and elementary particle physics aka HEP An important issue in HEP theory is that basic calculations become untenable at energies which are about 10 times higher than the reach of our current accelerators7 which currently probe elementary particle interactions at energies of 200 GeV One solution to this is a theory called supersym metry aka SUSY A prediction of SUSY is that there is a stable7 neutral7 weakly interacting particle7 usually called the neutralino7 xquot For our present purpose we can simply think of these as heavy neutrinos Generically7 such dark matter candidates are called WlMPs7 for weakly interacting massive particle Just as with neutrinos7 we can evaluate WlMP decoupling using Eqs 38 and 38 However7 because they are massive7 they will be non relativistic at decoupling7 and hence will contribute to cold dark matter CDM The decoupling temperature is about MczkT 25 i 5 for a WlMP of mass M The WlMP energy distribution will follow a Boltzmann function after decoupling Generically7 the WlMP energy density is found to follow pumpp0 1025 cmgs ll mm 48 Figure 18 gives this critical density as a function of WIMP mass M basically re ecting the denominator in the equation above Qualitatively we see three branches in this gure The rst at low mass correspnds to light relativistic WIMPs Neutrinos fall on this branch but off scale at low mass The next branch at intermediate mass is for WIMPs which are non relativistic at decoupling For these rnasses the weak cross section varies as 039 x GgMz The form of the weak cross section changes for rnasses near the mass of the weak force quanta the W and Z0 particles which have masses of about 90 GeVcz to UOltG 7 M2 M2 M2l which gives the third branch The shaded region is the range of cosrnological interest for dark matter This third branch is the one where one would expect the SUSY particle 95 WIMP detection The best and clearest way to test the WIMP particle dark matter solution is to look at data from the next generation of particle accelerator such as the LHC in Europe and the proposed linear collider These accelerators will probe exactly the energy range given by the intersection of the third branch and the shaded region in Fig 18 The SUSY version of this solution would be that the WIMP be the neutralino xquot In the accelerator interactions one could test if the observed WIMP had the right properties including stability against decay to be a real manifestation of cold dark matter On the other hand one would also like to directly observe the relic WIMPs left over from cosrnic decoupling In an earlier lecture we showed the diffculty of measuring neutrino interactions in accelerators WIMP detection is even more challenging since the kinetic energy and hence the interaction rate with matter is much much smaller A generic estimate for the rate R per unit detector mass for WIMP interactions in a detector utilizing a medium of atomic mass A is roughly R 104AMeventskgday for a WIMP of mass M This is experimentally very challenging 49 1 m2 10 10quot I0 103 1012 10 47 0V 3 McV I GCV 1 TeV Mva Mass M eV Figure 18 mm pwimppc VS WIMP mass eV From Perkins 1 Particle dark matter We discuss some issues surrounding neutrinos as potential sources of dark matter Although we will reject this hypothesis7 neutrinos represent a proto type for other possible sources of particle dark matter7 generically known as WlMPs We will then discuss some possibilities for WlMPs and nish with some the dif cult task of detecting WlMPs 11 Neutrino decoupling Figure 1 was also shown earlier in an earlier lecture In this case we highlight the point at which neutrino decoupling occurs Figure 2 gives another history of the universe7 showing the main quantities of interest as powers of ten At times prior to decoupling7 the neutrinos are in equilibrium with the rest of the particle and radiation of the expanding universe according to equilibrium reactions such as ylt gt667lt gtZ lt gtVP where y is a photon and the Z0 particle is the neutral carrier of the weak interaction see the particle physics primer The large mass of the Z0 about 91 GeVcz accounts for the short range ofthe weak interaction Since neutri nos only couple to the weak force7 they cannot directly interact with photons As long as the neutrinos are frequently exchanging energy in reactions like this7 then they will remain in thermal equilibrium As the universe expands and cools7 at some point the particles will no longer stay in thermal contact 7 this is decoupling We encountered this earlier in the discussion of the decoupling of photons7 thus giving rise to the cosmic microwave background We now try to determine the point of decoupling We can use a simple relation to understand the point where decoupling oc curs As discussed previously7 the particle collision rate is given by P n01 where n is the particle density of the interacting particles7 039 is the interaction cross section7 and 1 is the relative speed of the colliding particles We simply compare this collision rate with the rate of expansion of the universe7 which is H So within factors of order 17 we nd P mm gt H i equilibrium 1 P 7101 lt H i decoupling 2 my I V 11 Qualklmdlon uammon mu lGrV mgsmg nuueumu m Radznhon dnmmmcd 0 1M v m MM 9 Dccmlphng m manor 1 and mmwun mm 07 1 keV H Tum av 10 l m l Nev vmmwmmm 7 K 1 Wm v r NW 10quotquot Mquot I W 10 x lo mama 39 Figure 1 Energy kT per particle versus time Note the indicated time of neutrino decoupling We can noW input the known physics to determine the factors above When We do We nd that F H at kT m 1 MeV When the age of the universe Was approximately 1 s 12 Cosmic neutrino background In class We simply looked at the temperature dependence of each term of Eqs 12 to determine FH olt Ta 3 So the temperature of decoupling varies rapidly With collision rate for ex ample We can use these ideas to qualitatively see hoW neutrino decoupling in uenced formation of light nuclei Which began occuring slightly later When t m 1 hr Since We saW earlier that H X giZTZ Where 9 increases With the number of available particle species then a larger number of neutrino types N increases H and therefore by Eq 3 the decoupling temperature in Swan my day s u a w m w 39439ka Figure 2 1an historyn he umme cream demuplmg occurs mm This m m means me the equilibrium mm mm 7 pm mm n erw where Q m 7 my 1 3 MeV 5 131g el demung m 191g My mum 011m my 9 131g mm a mumm nualm In penwulsx amp rem w lsxgxl hgvesmemawedmed HeH s g m m vev a mmswmummmmaovs decay modes and hence N The current experimental result is N 300 i 001 Hence the prediction based on cosmic astrophysics Was con rmed my 2quot znyllimg 6 mm Emma r 7 Figure 3 Accelerator experiment result showing the excitation of the Z resonance The curves shoW the theoretical expectations for different as sumptions for the number of neutrino species The neutrinos Which decoupled from the cosmic reball at t m 1 s in analogy to the 273 K photon background should still be around As dis cussed in the previous lecture the predictions are that the density of these neutrinos is n 113 X N m 340 our3 With an average thermal temperature noW of 195 K 7 13 Neutrinos as dark matter We saW in Lecture 7 that there is noW good evidence that all 3 neutrino species have a small but nite rest mass my N 10 3 eVc2 But since there 4 are a lot of them around us from the decoupling7 then perhaps this gives an appreciable contribution to the estimated dark matter mass density We nd that to make a critical density from cosmic neutrinos requires that the mass of the 3 species total 47 eVc27 that is pypc py340gtlt47 eVcmg 4 We would require about 30 of this in order for neutrinos to give the righ 77 contribution to dark matter The data suggests it is about 103 short of this So it seems that neutrinos contribute rather insigni cantly to dark matter There is one other reason why it was not expected that neutrinos would contribute to this Structure formation companion lecture demands that the dark matter be cold that is7 that the particles be non relativistic during the early era of structure formation Recall that the netrinos decouple with an average energy kT 1 MeV7 much larger than the rest energies lndeed7 at the dawn of structure formation7 the neutrinos would have only cooled to 10 eV7 still relativistic Therefore7 neutrinos are not only too light7 but also too hot77 to be an important contributor to dark matter 14 WIMPS We didn7t discuss this in much depth in class But this is an important topic for both astrophysicscosmology and elementary particle physics aka HEP An important issue in HEP theory is that basic calculations become untenable at energies which are about 10 times higher than the reach of our current accelerators7 which currently probe elementary particle interactions at energies of 200 GeV One solution to this is a theory called supersym metry aka SUSY A prediction of SUSY is that there is a stable7 neutral7 weakly interacting particle7 usually called the neutralino7 xquot For our present purpose we can simply think of these as heavy neutrinos Generically7 such dark matter candidates are called WlMPs7 for weakly interacting massive particle Just as with neutrinos7 we can evaluate WlMP decoupling using Eqs 1 and 1 However7 because they are massive7 they will be non relativistic at decoupling7 and hence will contribute to cold dark matter CDM The decoupling temperature is about MczkT 25 i 5 for a WlMP of mass M The WlMP energy distribution will follow a Boltzmann function after decoupling Generically7 the WlMP energy density is found to follow pumpp0 1025 cmgs ll mm 5 Figure 4 gives this critical density as a function of WIMP mass M basically re ecting the denominator in the equation above Qualitatively we see three branches in this gure The rst at low mass correspnds to light relativistic WIMPs Neutrinos fall on this branch but off scale at low mass The next branch at intermediate mass is for WIMPs which are non relativistic at decoupling For these rnasses the weak cross section varies as 039 x GgMz The form of the weak cross section changes for rnasses near the mass of the weak force quanta the W and Z0 particles which have masses of about 90 GeVcz to UOltG 7 M2 M2 M2l which gives the third branch The shaded region is the range of cosrnological interest for dark matter This third branch is the one where one would expect the SUSY particle 15 WIMP detection The best and clearest way to test the WIMP particle dark matter solution is to look at data from the next generation of particle accelerator such as the LHC in Europe and the proposed linear collider These accelerators will probe exactly the energy range given by the intersection of the third branch and the shaded region in Fig 4 The SUSY version of this solution would be that the WIMP be the neutralino xquot In the accelerator interactions one could test if the observed WIMP had the right properties including stability against decay to be a real manifestation of cold dark matter On the other hand one would also like to directly observe the relic WIMPs left over from cosrnic decoupling In an earlier lecture we showed the diffculty of measuring neutrino interactions in accelerators WIMP detection is even more challenging since the kinetic energy and hence the interaction rate with matter is much much smaller A generic estimate for the rate R per unit detector mass for WIMP interactions in a detector utilizing a medium of atomic mass A is roughly R 104AMeventskgday for a WIMP of mass M This is experimentally very challenging 1 m2 10 10quot I0 103 1012 10 47 0V 3 McV I GCV 1 TeV Mva Mass M eV Figure 4 0w pwimppc VS WIMP mass eV From Perkins

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