PURPOSE: some cases the number of points assigned to the problem on the quiz is listed

Transcription

1 INSTITUTE OF TECHNOLOGY MASSACHUSETTS Department Physics Early Universe December 2, 26 Physics Alan Guth Prof. REVIEW PROBLEMS FOR QUIZ 3 QUIZ DATE Wedneday, December 7, 26, during normal class time. Lecture Notes 6 (pp. 2 end), Lecture Notes 7 and 8. Problem COVERAGE 7 9; Steven Weinberg, First Three Minutes, Chapter 8 and Af- Sets Barbara Ryden, Introduction to Cosmology, Chapters 9 ( Cosmic terword; Background) and (Inflation and Very Early Universe); Alan Microwave Guth, Inflation and New Era of High-Precision Cosmology, http//web.mit.edu/physics/news/physicsatmit/physicsatmit_2_cosmology.pdf. of problems on quiz will be taken verbatim (or at least One verbatim) from eir homework assignments, or from almost problems from this set of Review Problems. starred problems starred are ones that I recommend that you review most carefully Problems 2,, 5, 7, 8,,, and. se review problems are not to be handed in, but are being made PURPOSE to help you study. y come mainly from quizzes in previous years. available some cases number of points assigned to problem on quiz is listed In cases it is based on points for full quiz. inallsuch addition to this set of problems, you will find on course web page In quizzes that were given in 99, 996, 998, 2, 22, 2, 27, actual 2, and 23. relevant problems from those quizzes have mostly 29, incorporated into se review problems, but you still may be interested been looking at quizzes, just to see how much material has been included in in quiz. coverage of upcoming quiz will not necessarily match each of any of quizzes from previous years. coverage for each quiz coverage recent years is usually described at start of review problems, as I did in here. SESSION AND OFFICE HOURS To help you study for quiz, REVIEW Li will hold a review session on Sunday, December, at 73 pm, in Victor In addition, Victor Li will be holding special office hours on Room December 5, at pm in Room -63 (our regular classroom), and Monday, on Tuesday, December 6, at 5 pm in same room. I (Alan Guth) also unfortunately be out of town (in San Francisco for Breakthrough Prizes will Ceremony and Symposium) until Tuesdaynight, but I will try to answer Award to extent that time allows. s QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 2 INFORMATION TO BE GIVEN ON QUIZ For third quiz, following information will be made available to you DOPPLER SHIFT (For motion along a line) z vu (nonrelativistic, source moving) vu z vu z s +fi (nonrelativistic, observer moving) (special relativity, with fi vc) fi COSMOLOGICAL REDSHIFT SPECIAL RELATIVITY Time Dilation Factor fl p fi 2 a(t observed) a(temitted) ; fi vc Lorentz-Fitzgerald Contraction Factor fl ofsimultaneity Relativity clock reads later by an amount fi`c. Trailing Energy-Momentum Four-Vector p μ E ;~p c p 2 j~pj 2 p 2 j~pj 2 E 2 ; ~p flm~v ; E flmc 2 c 2 (m c) 2 q 2 ) 2 + j~pj 2 c 2 ; (mc OF A HOMOGENEOUSLY EXPANDING KINEMATICS UNIVERSE Law v Hr, Hubble's v recession velocity of a distant object, H Hubble expansion rate, and r distance to distant object. observed +z emitted

2 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 3 Present Value of Hubble Expansion Rate (Planck 25) H 677 ± 5 km-s -Mpc Scale Factor `p(t) a(t)`c ; `p(t) is physical distance between any two objects, a(t) is scale factor, and `c is coordinate distance between objects, also called comoving distance. Expansion H(t) da(t) Hubble Rate dt a(t) Rays in Comoving Coordinates Light rays travel in Light lines with speed straight c dt Horizon Distance `p;horizon(t) a(t) Z t c ) dt a(t a(t). ρ 3ct (flat, matter-dominated), COSMOLOGICAL EVOLUTION H 2 _a a 2ct (flat, radiation-dominated). 2 kc2 8ß3 Gρ ; ä ß 3 G ρ + 3p 2 a. c 2 a; a3 (ti) ρm(t) 3 (t) ρ m(ti) (matter); ρr(t) a (ti) a (t) ρ r(ti) (radiation) a _ρ 3 _a a ρ + p c 2 ρρc ; ρc 3H2 ; 8ßG QUIZ 3 REVIEW PROBLEMS, FALL 26 p. OF A MATTER-DOMINATED UNIVERSE EVOLUTION (k ) a(t) / t 23 Flat Closed (k >) ct ff( sin ) ; 2 + cos > ; ff ß 3 Gρ 2 c Open (k <) ct ff (sinh ) ; 2 + cosh < ; ff ß 3 Gρ 2 c» k> ROBERTSON-WALKER METRIC ds 2 c 2 dfi 2 c 2 dt 2 +a 2 (t) ρ a ff( cos ) ; pk a a a ff (cosh ) ; p» ; 2 dr kr 2 + r2 d 2 + sin 2 dffi 2 ff for k>, we can define r sin ψ p Alternatively,, and n k ds 2 c 2 dfi 2 c 2 dt 2 +~a 2 (t) Φ dψ 2 + sin 2 ψ d 2 +sin 2 dffi 2 Ψ ; ~a(t) a(t) p k.for k<we can define r sinh ψ p,andn k ds 2 c 2 dfi 2 c 2 dt 2 +~a 2 (t) Φ dψ 2 + sinh 2 ψ d 2 +sin 2 dffi 2 Ψ ; ~a(t) a(t) p k. Note that ~a can be called a if re is need to relate it to a(t) that appears in first equation no above. SCHWARZSCHILD METRIC 2 c 2 dfi 2 2GM ds rc 2 dt 2 + 2GM 2 c + r 2 d 2 + r 2 sin 2 dffi 2 ; rc 2 pk 3 p» 3 2 dr

3 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 5 GEODESIC EQUATION or d ds d dfi ρ ρ gij j gμν ν ff 2 (@ igk`) k ` ds ds ds ff 2 (@ μg ff) dfi BLACK-BODY RADIATION g g Λ and g ß2 u 3 dfi (kt) 3 (energy density) (μhc) ff dfi p 3 u ρ uc2 (pressure, mass density) g Λ (3) n 2 (kt) 3 ß g 2ß2 s 5 (μhc) 3 (number density) T 3 k 3 ; (entropy density) (μhc) ( per spin state for bosons (integer spin) 7/8 per spin state for fermions (half-integer spin) ( per spin state for bosons 3/ per spin state for fermions, (3) ß QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 6 gfl g Λ fl 2; 7 gν z } 8 Fermion factor Λ ν 3 g z } Fermion factor e + e 7 g z } 8 Fermion factor Λ e + e 3 g z } Fermion factor 3 species 3 νe;νμ;νfi 3 species 3 νe;νμ;νfi Species Species 2 Particle antiparticle 2 Particle antiparticle 2 Particle antiparticle 2 Particle antiparticle Spin states Spin states 2 Spin states 2 Spin states 2 ; 9 2 ; 7 2 ; 3 OF A FLAT RADIATION-DOMINATED EVOLUTION UNIVERSE kt ρ ßGt 3 c 5 5μh 3 gg 6ß p mμ 6MeVfl kt fl me 5 MeV, g 75 and For n kt MeV p g sec) (in t t After freeze-out of electron-positron pairs, Tν Tfl COSMOLOGICAL CONSTANT 3 ρvacc 2 Λc uvac ; 8ßG

4 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 7 ρvacc 2 Λc pvac 8ßG GENERALIZED COSMOLOGICAL EVOLUTION Age of universe t H H q H m;x +rad; +vac;x +k;x 2 ; x dt kc2 k; 2 (t)h 2 a Z Look-back time tlook-back(z) z Z H a(t) x +z ; a(t) m; rad; vac; x p m;x +rad; +vac;x +k;x 2 dz + z) p m;( + z) 3 +rad;( + z) +vac; +k;( + z) 2 ( dz + z ) p m;( + z ) 3 +rad;( + z ) +vac; +k;( + z ) 2 ( PHYSICAL CONSTANTS G 667 m 3 kg s cm 3 g s 2 k Boltzmann's constant joulek h μh 55 3 joule s 2ß erg s ev s c m/s 38 6 ergk evk QUIZ 3 REVIEW PROBLEMS, FALL 26 p cm/s μhc 973 MeV-fm; fm 5 m yr356 7 s ev62 9 joule 62 2 erg 9 ev kg ( c ) GeV 2 g 783 Units Planck length `P, Planck time tp, Planck Planck mp, and Planck energy Ep are given by mass `P tp mp EP r Gμh m ; c cm ; r μhg s ; c r μhc kg ; G 5 g ; 277 r 5 μhc G 22 9 GeV EQUILIBRIUM CHEMICAL topic was not included in course in 23, but formu- (This are noneless included here for logical completeness. y las not be relevant to Quiz 3. y are relevant to Problem 3 will se Review Problems, which is also not relevant to Quiz 3. in enjoy looking at se items, or enjoy ignoring m!) Please Ideal Gas of Classical Nonrelativistic Particles ni gi (2ßmikT) 32 (2ßμh) 3 e (μi mic2 )kt ni number density of particle gi number of spin states of particle mi mass of particle μi chemical potential any reaction, sum of μi on left-hand side of For equation must equal sum of μi on right-hand reaction Formula assumes gas is nonrelativistic (kt fi mic 2 )and side. (ni fi (2ßmikT) 32 (2ßμh) 3 ). dilute

5 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 9 PROBLEM LIST. Did You Do Reading (27)? (Sol 2) *2. Did You Do Reading (29)? (Sol 26) 3. Did You Do Reading (23)? (Sol 28) *. Number Densities in Cosmic Background Radiation... (Sol 3) *5. Properties of Black-Body Radiation (Sol 3) 6. A New Species of Lepton (Sol 33) *7. A New ory of Weak Interactions (Sol 36) *8. Doubling of Electrons (Sol 2) 9. Time Scales in Cosmology (Sol ) *. Evolution of Flatness (Sol ) *. Sloan Digital Sky Survey z 582 Quasar (Sol 5) 2. Second Hubble Crossing (Sol 5) 3. Neutrino Number and Neutron/Proton Equilibrium... 2 (Sol 53) *. Event Horizon for Our Universe (Sol 56) QUIZ 3 REVIEW PROBLEMS, FALL 26 p. PROBLEM DID YOU DO THE READING? (25 points) following problem was Problem, Quiz 3, in 27. Each part was worth 5 points. (CMB basic facts) Which one of following statements about CMB is not correct After dipole distortion of CMB is subtracted away, mean temperature (i) averaging over sky is ht i 2725K. (ii) After dipole distortion of CMB is subtracted away, root mean square temperature fluctuation is D ffit T 2 E 2 3. dipole distortion is a simple Doppler shift, caused by net motion of observer relative to a frame of reference in which CMB is isotropic. In ir groundbreaking paper, Wilson and Penzias reported measurement of an excess temperature of about 3.5 K that was isotropic, unpolar- and free from seasonal variations. In a companion paper written by ized, Peebles, Roll and Wilkinson, authors interpreted radiation Dicke, to be a relic of an early, hot, dense, and opaque state of universe. (CMB experiments) current mean energy per CMB photon, about 6 ev, is comparable to energy of vibration or rotation for a small such as H2O. Thus microwaves with wavelengths shorter than molecule ο 3 cm are strongly absorbed by water molecules in atmosphere. To CMB at <3cm, which one of following methods is not a measure solution to this problem? feasible (i) Measure CMB from high-altitude balloons, e.g. MAXIMA. (ii) Measure CMB from South Pole, e.g. DASI. Measure CMB from North Pole, e.g. BOOMERANG. Measure CMB from a satellite above atmosphere of Earth, e.g. WMAP and PLANCK. COBE, (Temperature fluctuations) creation of temperature fluctuations in CMB variations in gravitational potential is known as Sachs-Wolfe effect. by Which one of following statements is not correct concerning this effect? A CMB photon is redshifted when climbing out of a gravitational potential (i) and is blueshifted when falling down a potential hill. well, At time of last scattering, nonbaryonic dark matter dominated (ii) density, and hence gravitational potential, of universe. energy

6 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. large-scale fluctuations in CMB temperatures arise from gravitational effect of primordial density fluctuations in distribution of nonbaryonic dark matter. peaks in plot of temperature fluctuation T vs. multipole l are to variations in density of nonbaryonic dark matter, while due contributions from baryons alone would not show such peaks. (Dark matter candidates) Which one of following is not a candidate of (d) dark matter? nonbaryonic (i) massive neutrinos (ii) axions matter made of top quarks (a type of quarks with heavy mass of about GeV). 7 WIMPs (Weakly Interacting Massive Particles) (v) primordial black holes (Signatures of dark matter) By what methods can signatures of dark matter (e) detected? List two methods. (Grading 3 points for one correct answer, be points for two correct answers. If you give more than two answers, your 5 will be based on number of right answers minus number of wrong score answers, with a lower bound of zero.) Λ PROBLEM 2 DID YOU DO THE READING? (25 points) This problem was Problem, Quiz 3, 29. ( points) This question concerns some numbers related to cosmic microwave background (CMB) that one should never forget. State values of numbers, to within an order of magnitude unless orwise stated. In all se question refers to present value of se quantities. cases (i) average temperature T of CMB (to within %). speed of Local Group with respect to CMB, expressed as a (ii) vc of speed of light. ( speed of Local Group is found fraction measuring dipole pattern of CMB temperature to determine by velocity of spacecraft with respect to CMB, and n removing motion, orbital motion of Earth about Sun, Sun spacecraft galaxy, and galaxy relative to center of mass of Local about Group.) QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 2 intrinsic relative temperature fluctuations TT, after removing anisotropy corresponding to motion of observer relative to dipole CMB. ratio of baryon number density to photon number density, nbarynfl. angular size H, in degrees, corresponding to what was Hubble (v) ch at surface of last scattering. This answer must be within distance a factor of 3 to be correct. (3 points) Because photons outnumber baryons by so much, exponential of photon blackbody distribution is important in ionizing hydrogen tail after ktfl falls below QH 36 ev. What is ratio ktflqh when well fraction of universe is 2? ionization (i) 5 (ii) 5 3 (v) 5 (2 points) Which of following describes Sachs-Wolfe effect? Photons from fluid which had a velocity toward us along line of sight (i) redder because of Doppler effect. appear Photons from fluid which had a velocity toward us along line of sight (ii) bluer because of Doppler effect. appear Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. redder Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. bluer Photons traveling toward us from surface of last scattering appear (v) because of absorption in intergalactic medium. redder Photons traveling toward us from surface of last scattering appear (vi) because of absorption in intergalactic medium. bluer (d) ( points) For each of following statements, say wher it is true or false Dark matter interacts through gravitational, weak, and electromagnetic (i) forces. T or F? virial orem can be applied to a cluster of galaxies to find its total (ii) most of which is dark matter. T or F? mass, Neutrinos are thought to comprise a significant fraction of energy density of dark matter. T or F?

7 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 3 Magnetic monopoles are thought to comprise a significant fraction of density of dark matter. T or F? energy Lensing observations have shown that MACHOs cannot account for (v) matter in galactic halos, but that as much as 2% of halo mass dark could be in form of MACHOs. T or F? PROBLEM 3 DID YOU DO THE READING? (35 points) This was Problem of Quiz 3, 23. (5 points) Ryden summarizes results of COBE satellite experiment for measurements of cosmic microwave background (CMB) in form of important results. first was that, in any particular direction of three spectrum of CMB is very close to that of an ideal blackbody. sky, instrument on COBE satellite could have detected deviations from FIRAS blackbody spectrum as small as fflffl ß n, n is an integer. To within ±, what is n? (5 points) second result was measurement of a dipole distortion of CMB spectrum; that is, radiation is slightly blueshifted to higher tem- in one direction, and slightly redshifted to lower temperatures in peratures opposite direction. To what physical effect was this dipole distortion at- tributed? (5 points) third result concerned measurement of temperature fluctuations after dipole feature mentioned above was subtracted out. Defining ffit T T ( ; ffi) ht i ( ; ffi) i ht ht i 2725 K, average value of T, y found a root mean square fluctuation, ffit * T to some number. To within an order of magnitude, what was that number? equal (d) (5 points) Which of following describes Sachs-Wolfe effect? Photons from fluid which had a velocity toward us along line of sight (i) redder because of Doppler effect. appear Photons from fluid which had a velocity toward us along line of sight (ii) bluer because of Doppler effect. appear ; ; QUIZ 3 REVIEW PROBLEMS, FALL 26 p. Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. redder Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. bluer Photons traveling toward us from surface of last scattering appear (v) because of absorption in intergalactic medium. redder Photons traveling toward us from surface of last scattering appear (vi) because of absorption in intergalactic medium. bluer (5 points) flatness problem refers to extreme fine-tuning that is needed (e) at early times, in order for it to be as close to today asweobserve. in with assumption that today is equal to within about %, one Starting that at one second after big bang, concludes j jt sec< m ; m is an integer. To within ± 3, what is m? (5 points) total energy density of present universe consists mainly of (f) matter, dark matter, and dark energy. Give percentages of each, baryonic to best fit obtained from Planck 23 data. You will get full according if first (baryonic matter) is accurate to ±2%, and or two are credit accurate to within ±5%. (5 points) Within conventional hot big bang cosmology (without inflation), (g) is difficult to understand how temperature of CMB can be correlated it angular separations that are so large that points on surface of last at was separated from each or by more than a horizon distance. Ap- scattering what angle, in degrees, corresponds to a separation on surface proximately scattering of one horizon length? You will get full credit if your answer is last right to within a factor of 2. PROBLEM NUMBER DENSITIES IN THE COSMIC BACK- Λ RADIATION GROUND temperature of cosmic microwavebackground radiation is 27 ffi K. Today number density of photons in this radiation. What is number Calculate density of rmal neutrinos left over from big bang?

8 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 5 PROBLEM 5 PROPERTIES OF BLACK-BODY RADIATION (25 Λ points) following problem was Problem, Quiz 3, 998. answering following questions, remember that you can refer to formulas In at front of exam. Since you were not asked to bring calculators, you may leave your answers in form of algebraic expressions, such asß 32 p 5 (3). (5 points) For black-body radiation (also called rmal radiation) of photons at temperature T,what is average energy per photon? (5 points) For same radiation, what is average entropy per photon? (5 points) Now consider black-body radiation of a massless boson which has zero, so re is only one spin state. Would average energy per particle spin entropy per particle be different from answers you gave in parts and? If so, how would y change? and (5 points) Now consider black-body radiation of electron neutrinos. se (d) are fermions with spin /2, and we will assume that y are massless particles have only one possible spin state. What is average energy per particle and this case? for (5 points) What is average entropy per particle for black-body radiation (e) neutrinos, as described in part (d)? of PROBLEM 6 A NEW SPECIES OF LEPTON following problem was Problem 2, Quiz 3, 992, worth 25 points. calculations describing early universe were modified by including Suppose an additional, hypotical lepton, called an 8.286ion ion has roughly same properties as an electron, except that its mass is given by mc 2 75 MeV. - of this question require numerical answers, but since you were Parts told to bring calculators, you need not carry out arithmetic. Your answer not be expressed, however, in calculator-ready" form that is, it should be an should involving pure numbers only (no units), with any necessary conversion expression included. (For example, if you were asked how many meters a light pulse in factors travels in 5 minutes, you could express answer as ) vacuum (5 points) What would be number density of 8.286ions, in particles per a) meter, when temperature T was given by kt 3 MeV? cubic (5 points) Assuming (as in standard picture) that early universe is b) described by a flat, radiation-dominated model, what would be accurately QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 6 of mass density att sec? You may assume that 75 MeV fi value fi MeV, so particles contributing significantly to black-body kt include photons, neutrinos, e + -e pairs, and 8.286ion-anti8286ion radiation Express your answer in units of g/cm 3. pairs. (5 points) Under same assumptions as in, what would be value of c) in MeV, at t sec? kt, (5 points) When nucleosynsis calculations are modified to include effect d) 8.286ion, is production of helium increased or decreased? Explain of your answer in a few sentences. (5 points) Suppose neutrinos decouple while kt fl 75 MeV. If e) are included, what does one predict for value of TνTfl today? 8.286ions Tν denotes temperature of neutrinos, and Tfl denotes temperature (Here of cosmic background radiation photons.) PROBLEM 7 A NEW THEORY OF THE WEAK INTERACTIONS Λ points) ( This problem was Problem 3,Quiz 3,29. a New ory of Weak Interactions (NTWI) was proposed, which Suppose from standard ory in two ways. First, NTWI predicts that differs interactions are somewhat weaker than in standard model. In addition, weak ory implies existence of new spin- 2 particles (fermions) called R+ R, with a rest energy of 5 MeV ( MeV 6 ev). This problem will and with cosmological consequences of such a ory. deal NTWI will predict that neutrinos in early universe will decouple a higher temperature than in standard model. Suppose that this decoupling at place at kt ß 2 MeV. This means that when neutrinos cease to be takes coupled to rest of matter, hot soup of particles would contain rmally only photons, neutrinos, and e + -e pairs, but also μ +, μ, ß +, ß,andß not along with R + -R pairs. ( muon is a particle which behaves particles, identically to an electron, except that its rest energy is 6 MeV. pions almost lightest of mesons, with zero angular momentum and rest energies of are MeV and MeV for neutral and charged pions, respectively. ß + and 35 are antiparticles of each or, and ß is its own antiparticle. Zero angular ß momentum implies a single spin state.) You may assume that universe is flat. ( points) According to standard particle physics model, what is mass ρ of universe when kt ß 2 MeV? What is value of ρ at density temperature, according to NTWI? Use eir g/cm 3 or kg/m 3. (If you this you can save timeby not carrying out arithmetic. If you do this, wish,

9 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 7 you should give answer in calculator-ready" form, by which I however, an expression involving pure numbers (no units), with any necessary mean factors included, and with units of answer specified at conversion For example, if asked how far light travels in 5 minutes, you could answer end m.) ( points) According to standard model, temperature today of neutrino background should be () 3 Tfl, Tfl is tempera- rmal of rmal photon background. What does NTWI predict for ture of rmal neutrino background? temperature ( points) According to standard model, what is ratio today of density of rmal neutrinos to number density of rmal photons? number What is this ratio according to NTWI? ( points) Since reactions which interchange protons and neutrons involve (d) se reactions freeze out" at roughly same time as neutrinos neutrinos, At later times only reaction which effectively converts neutrons decouple. protons is free decay of neutron. Despite fact that neutron decay to a weak interaction, we will assume that it occurs with usual 5 minute is lifetime. Would helium abundance predicted by NTWI be higher mean lower than prediction of standard model? To within 5 or %, what or NTWI predict for percent abundance (by weight) of helium in would universe? (As in part, you can eir carry out arithmetic, or leave answer in calculator-ready form.) information proton and neutron rest energies are given by mp c 2 Useful MeV and mn c MeV, with (mn mp)c 2.29 MeV mean lifetime for neutron decay, n! p + e +μνe, is given by fi 886 s. Λ PROBLEM 8 DOUBLING OF ELECTRONS ( points) following was on Quiz 3, 2 (Problem ) that instead of one species of electrons and ir antiparticles, suppose Suppose was also anor species of electron-like and positron-like particles. Suppose re new species has same mass and or properties as electrons and that If this were case, what would be ratio TνTfl of temperature positrons. today of neutrinos to temperature of CMB photons QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 8 PROBLEM 9 TIME SCALES IN COSMOLOGY this problem you are asked to give approximate times at which various In events in history of universe are believed to have taken place. important times are measured from instant of big bang. To avoid ambiguities, are asked to choose best answer from following list you 3 sec. 37 sec. 2 sec. 5 sec. sec. mins.,, years., billion years. 2 billion years. 5 billion years. billion years. 3 billion years. 2 this problem it will be sufficient to state an answer from memory, without For events which must be placed are following explanation. beginning of processes involved in big bang nucleosynsis; end of processes involved in big bang nucleosynsis; time of phase transition predicted by grand unified ories, which place when kt ß 6 GeV; takes recombination", time at which matter in universe converted (d) a plasma to a gas of neutral atoms; from phase transition at which quarks became confined, believed to (e) when kt ß 3 MeV. occur cosmology is fraught with uncertainty, in some cases more than one answer Since will be acceptable. You are asked, however, to give ONLY ONE of acceptable answers. Λ PROBLEM EVOLUTION OF FLATNESS (5 points) following problem was Problem 3, Quiz 3, 2. flatness problem" is related to fact that during evolutionof cosmological model, is always driven away from. standard

10 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 9 (9 points) During a period in which universe is matter-dominated (meaning only relevant component is nonrelativistic matter), quantity that as a power of t. Show that this is true, and derive power. (Stating grows right power without a derivation will be worth 3 points.) (6 points) During a period in which universe is radiation-dominated, quantity will grow like a different power of t. Show that this is true, and same power. (Stating right power without a derivation will again be derive 3 points.) worth each part, you may assume that universe was always dominated by In form of matter. specified PROBLEM THE SLOAN DIGITAL SKY SURVEY z 582 Λ ( points) QUASAR following problem was Problem, Quiz 3, 2. April 3, 2, Sloan Digital Sky Survey announced discovery of On was n most distant object known in universe a quasar at z 582. what explain to public how this object fits into universe, SDSS posted on To website an article by Michael Turner and Craig Wiegert titled How Can An ir We See Today be 27 Billion Light Years Away If Universe is only Object Years Old?" Using a model with H 65 km-s -Mpc,m 35, and Billion Λ 65, y claimed that age of universe is 3.9 billion years. that light that we now see was emitted when universe was.95 billion old. years that distance to quasar, as it would be measured by a ruler today, is billion light-years. 27 that distance to quasar, at time light was emitted, was. (d) light-years. billion that present speed of quasar, defined as rate at which distance (e) us and quasar is increasing, is.8 times velocity of light. between goal of this problem is to check all of se conclusions, although you are course not expected to actually work out numbers. Your answers can be of expressed in terms of H, m, Λ, and z. Definite integrals need not be evaluated QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 2 that m represents present density of nonrelativistic matter, expressed Note a fraction of critical density; and Λ represents present density of vacuum as expressed as a fraction of critical density. In answering each of energy, questions, you may consider answer to any previous part wher following answered it or not as a given piece of information, which can be used in your you answer. (5 points) Write an expression for age t of this model universe? (5 points) Write an expression for time te at which light which we now from distant quasar was emitted. receive ( points) Write an expression for present physical distance `phys; to quasar. (5 points) Write an expression for physical distance `phys;e between us and (d) quasar at time that light was emitted. (5 points) Write an expression for present speed of quasar, defined as (e) rate at which distance between us and quasar is increasing. PROBLEM 2 SECOND HUBBLE CROSSING ( points) problem was Problem 3, Quiz 3, 27. In 26 we have not yet talked about This crossings and evolution of density perturbations, so this problem would Hubble be fair as worded. Actually, however, you have learned how to do se calculations, not so problem would be fair if it described inmore detail what needs to be calculated. Problem Set 9 (27) we calculated time th( ) of first Hubble In for a mode specified by its (physical) wavelength at present time. crossing this problem we will calculate time th2( ) of second Hubble crossing, In time at which growing Hubble length ch (t) catches up to physical which is also growing. At time of second Hubble crossing for wavelength, of interest, universe can be described very simply it is a radiation- wavelengths flat universe. However, since is defined as present value of dominated evolution of universe between th2( ) and present will also wavelength, relevant to problem. We will need to use methods, refore, that allow for be matter-dominated era and onset of dark-energy-dominated era. As both Problem Set 9 (27), model universe that we consider will be described by in WMAP 3-year best fit parameters expansion rate H 735 km s Mpc Hubble mass density m.237 Nonrelativistic mass density vac.763 Vacuum temperature Tfl; K CMB

11 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 2 mass densities are defined as contributions to, and hence describe mass of each constituent relative to critical density. Note that model density exactly flat, so you need not worry about spatial curvature. Here you are not is to give a numerical answer, so above list will serve only to define expected that can appear in your answers, along with and physical constants symbols μh, c, and k. G, (5 points) For a radiation-dominated flat universe, what is Hubble length ch (t) as a function of time t? `H(t) ( points) second Hubble crossing will occur during interval 3 sec fi t fi 5; years, mass density of universe is dominated by photons and neutrinos. when this era neutrinos are a little colder than photons, with Tν During 3 Tfl. total energy density of photons and neutrinos toger () be written as can utot g ß 2 3 (ktfl) 3 (μhc) is value of g? (For following parts you can treat g as a given What that can be left in your answers, wher or not you found it.) variable ( points) For times in range described in part, what is photon Tfl(t) as a function of t? temperature (5 points) Finally, we are ready to find time th2( ) of second Hubble (d) for a given value of physical wavelength today. Making use of crossing, previous results, you should be able to determine th2( ). If you were not to answer some of previous parts, you may leave symbols `H(t), able g, and/or Tfl(t) in your answer. 3 NEUTRINO NUMBER AND THE NEUTRON/ PRO- PROBLEM EQUILIBRIUM (35 points) TON following problem was 998 Quiz, Problem. This would NOT be a fair for 26, as this year we have not discussed big bang nucleosynsis at problem level of detail. But I am including problem anyway, as you might find it this interesting. standard treatment of big bang nucleosynsis it is assumed that at In times ratio of neutrons to protons is given by Boltzmann formula, early nn np e EkT ; () QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 22 k is Boltzmann's constant, T is temperature, and E 29 MeV is proton-neutron mass-energy difference. This formula is believed to be very but it assumes that chemical potential for neutrons μn is same as accurate, chemical potential for protons μp. ( points) Give correct version of Eq. (), allowing for possibility that 6 μp. μn equilibrium between protons and neutrons in early universe is sustained by following reactions mainly e + + n ψ! p +μνe νe + n ψ! p + e μe and μν denote chemical potentials for electrons (e ) and electron Let (νe) respectively. chemical potentials for positrons (e + ) and neutrinos neutrinos (μνe) are n μe and μν, respectively, since chemical anti-electron potential of a particle is always negative of chemical potential for antiparticle.* ( points) Express neutron/proton chemical potential difference μn μp terms of μe and μν. in black-body radiation formulas at beginning of quiz did not allow for of a chemical potential, but y can easily be generalized. For example, possibility formula for number density ni (of particles of type i) becomes ni g Λ i (3) 2 (kt) 3 ß (μhc) 3 eμikt ( points) Suppose that density of anti-electron neutrinos μnν in early was higher than density of electron neutrinos nν. Express universe equilibrium value of ratio nnnp in terms of E, T, and eir rmal μnνnν or antineutrino excess n μnν nν. (Your answer may also ratio contain fundamental constants, such as k, μh, and c.) (5 points) Would an excess of anti-electron neutrinos, as considered in part, (d) or decrease amount of helium that would be produced in early increase universe? Explain your answer. This fact is a consequence of principle that chemical potential of a * is sum of chemical potentials associated with its conserved quanti- particle while particle and antiparticle always have opposite values of all conserved ties, quantities.

12 8.286 QUIZ 3 REVIEW PROBLEMS, FALL 26 p. 23 THE EVENT HORIZON FOR OUR UNIVERSE (25 PROBLEM points) following problem was Problem 3 from Quiz 3, 23. have learned that expansion history of our universe can be described We terms of a small set of numbers m;, present contribution to from in matter; rad;, present contribution to from radiation; vac, nonrelativistic present contribution to from vacuum energy; and H, present value of expansion rate. best estimates of se numbers are consistent with a Hubble universe, so we can take k,m; +rad; +vac, and we can use flat flat Robertson-Walker metric, ds 2 c 2 dt 2 + a 2 (t) dr 2 + r 2 d 2 + sin 2 dffi 2 Λ (5 points) Suppose that we are at origin of coordinate system, and that present timet we emit a spherical pulse of light. It turns out that re at a maximum coordinate radius r rmax that this pulse will ever reach, no is how long we wait. ( pulse will never actually reach rmax, but will matter all r such that < r < rmax.) rmax is coordinate of what is called reach event horizon events that happen now at r rmax will never be visible us, assuming that we remain at origin. Assuming for this part that to a(t) is a known function, write an expression for rmax. Your answer function be expressed as an integral, which can involve a(t), t, and any of should defined in preamble. [Advice If you cannot answer this, you parameters should still try part.] ( points) Since a(t) is not known explicitly, answer to previous part difficult to use. Show, however, that by changing variable of integration, is can rewrite expression for rmax as a definite integral involving only you specified in preamble, without any reference to function a(t), parameters perhaps to its present value a(t). You are not expected to evaluate this except [Hint One method is to use integral. a(t) x a(t) variable of integration, just as we did when we derived first of as for t shown in formula sheets.] expressions ( points) Astronomers often describe distances in terms of redshifts, so it useful to find redshift of event horizon. That is, if a light ray that is at r rmax arrived at Earth today, what would be its redshift zeh originated event horizon)? You are not asked to find an explicit expression for (eh but instead an equation that could be solved numerically to determine zeh, For this part you can treat rmax as given, so it does not matter if you zeh. done parts and. You will get half credit for a correct answer that have function a(t), and full credit for a correct answer that involves only involves integrals depending only on parameters specified in preamble, explicit possibly a(t). and QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 2 SOLUTIONS PROBLEM DID YOU DO THE READING? (25 points) following parts are each worth 5 points. (CMB basic facts) Which one of following statements about CMB is not correct After dipole distortion of CMB is subtracted away, mean temperature (i) averaging over sky is ht i 2725K. (ii) After dipole distortion of CMB is subtracted away, root mean square temperature fluctuation is 3. dipole distortion is a simple Doppler shift, caused by net motion of observer relative to a frame of reference in which CMB is isotropic. In ir groundbreaking paper, Wilson and Penzias reported measurement of an excess temperature of about 3.5 K that was isotropic, unpolar- and free from seasonal variations. In a companion paper written by ized, Peebles, Roll and Wilkinson, authors interpreted radiation Dicke, to be a relic of an early, hot, dense, and opaque state of universe. After subtracting dipole contribution, temperature Explanation is about 5. fluctuation (CMB experiments) current mean energy per CMB photon, about 6 ev, is comparable to energy of vibration or rotation for a small such as H2O. Thus microwaves with wavelengths shorter than molecule ο 3 cm are strongly absorbed by water molecules in atmosphere. To CMB at <3cm, which one of following methods is not a measure solution to this problem? feasible (i) Measure CMB from high-altitude balloons, e.g. MAXIMA. (ii) Measure CMB from South Pole, e.g. DASI. Measure CMB from North Pole, e.g. BOOMERANG. Measure CMB from a satellite above atmosphere of Earth, e.g. WMAP and PLANCK. COBE, North Pole is at sea level. In contrast, South Pole Explanation nearly 3 kilometers above sea level. BOOMERANG is a balloon-borne is experiment launched from Antarctica. D ffit 2 E 2 T

13 8.286 QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 25 (Temperature fluctuations) creation of temperature fluctuations in CMB variations in gravitational potential is known as Sachs-Wolfe effect. by Which one of following statements is not correct concerning this effect? A CMB photon is redshifted when climbing out of a gravitational potential (i) and is blueshifted when falling down a potential hill. well, At time of last scattering, nonbaryonic dark matter dominated (ii) density, and hence gravitational potential, of universe. energy large-scale fluctuations in CMB temperatures arise from gravitational effect of primordial density fluctuations in distribution of nonbaryonic dark matter. peaks in plot of temperature fluctuation T vs. multipole l are to variations in density of nonbaryonic dark matter, while due contributions from baryons alone would not show such peaks. se peaks are due to acoustic oscillations in photonbaryon Explanation fluid. (Dark matter candidates) Which one of following is not a candidate of (d) dark matter? nonbaryonic (i) massive neutrinos (ii) axions matter made of top quarks (a type of quarks with heavy mass of about GeV). 7 WIMPs (Weakly Interacting Massive Particles) (v) primordial black holes Matter made of top quarks is so unstable that it is seen only Explanation as a product in high energy particle collisions. fleetingly (Signatures of dark matter) By what methods can signatures of dark matter (e) detected? List two methods. (Grading 3 points for one correct answer, be points for two correct answers. If you give more than two answers, your 5 will be based on number of right answers minus number of wrong score answers, with a lower bound of zero.) Answers Galaxy rotation curves. (I.e., measurements of orbital speed of stars (i) spiral galaxies as a function of radius R show that se curves remain in at radii far beyond visible stellar disk. If most of matter were flat in disk, n se velocities should fall off as p R.) contained QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 26 Use virial orem to estimate mass of a galaxy cluster. (For (ii) virial analysis shows that only 2% of mass of Coma example, cluster consists of stars, and only % consists of hot intracluster gas. Gravitational lensing. (For example, mass of a cluster can be estimated distortion of shapes of galaxies behind cluster.) from CMB temperature fluctuations. (I.e., analysis of intensity of as a function of multipole number shows that tot ß, and fluctuations dark energy contributes Λ ß 7, baryonic matter contributes bary ß that and dark matter contributes dark matter ß 26.), are or possible answers as well, but se are ones discussed by re in Chapters 8 and 9. Ryden PROBLEM 2 DID YOU DO THE READING? (25 points) ( points) This question concerns some numbers related to cosmic microwave background (CMB) that one should never forget. State values of numbers, to within an order of magnitude unless orwise stated. In all se question refers to present value of se quantities. cases (i) average temperature T of CMB (to within %) K speed of Local Group with respect to CMB, expressed as a (ii) vc of speed of light. ( speed of Local Group is found fraction measuring dipole pattern of CMB temperature to determine by velocity of spacecraft with respect to CMB, and n removing motion, orbital motion of Earth about Sun, Sun spacecraft galaxy, and galaxy relative to center of mass of Local about Group.) dipole anisotropy corresponds to a peculiar velocity" (that is, velocity is not due to expansion of universe) of 63 ± 2 km s,or which in terms of speed of light, vc ß 2 3. intrinsic relative temperature fluctuations TT, after removing anisotropy corresponding to motion of observer relative to dipole CMB. 5 ratio of baryon number density to photon number density, nbarynfl. WMAP 5-year value for nbnfl (6225 ± 7), which to closest order of magnitude is 9.

14 8.286 QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 27 angular size H, in degrees, corresponding to what was Hubble (v) ch at surface of last scattering. This answer must be within distance a factor of 3 to be correct. ο ffi (3 points) Because photons outnumber baryons by so much, exponential of photon blackbody distribution is important in ionizing hydrogen tail after ktfl falls below QH 36 ev. What is ratio ktflqh when well fraction of universe is 2? ionization (i) 5 (ii) 5 3 (v) 5 is not a number one has to commit to memory if one can remember This temperature of(re)combination in ev, or if only in K along with factor (k ß ev K ). One can n calculate that near conversion ktflqh ß ( ev K )(3 K)(36eV)ß 5. recombination, (2 points) Which of following describes Sachs-Wolfe effect? Photons from fluid which had a velocity toward us along line of sight (i) redder because of Doppler effect. appear Photons from fluid which had a velocity toward us along line of sight (ii) bluer because of Doppler effect. appear Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. redder Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. bluer Photons traveling toward us from surface of last scattering appear (v) because of absorption in intergalactic medium. redder Photons traveling toward us from surface of last scattering appear (vi) because of absorption in intergalactic medium. bluer Denser regions have a deeper (more negative) gravitational Explanation Photons which travel through a spatially varying potential ac- potential. a redshift or blueshift depending on wher y are going up or down quire potential, respectively. Photons originating in denser regions start a lower potential and must climb out, so y end up being redshifted at to ir original energies. relative (d) ( points) For each of following statements, say wher it is true or false Dark matter interacts through gravitational, weak, and electromagnetic (i) forces. T or F? QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 28 virial orem can be applied to a cluster of galaxies to find its total (ii) most of which is dark matter. T or F? mass, Neutrinos are thought to comprise a significant fraction of energy density of dark matter. T or F? Magnetic monopoles are thought to comprise a significant fraction of density of dark matter. T or F? energy Lensing observations have shown that MACHOs cannot account for (v) matter in galactic halos, but that as much as 2% of halo mass dark could be in form of MACHOs. T or F? PROBLEM 3 DID YOU DO THE READING? (35 points) (5 points) Ryden summarizes results of COBE satellite experiment for measurements of cosmic microwave background (CMB) in form of important results. first was that, in any particular direction of three spectrum of CMB is very close to that of an ideal blackbody. sky, instrument on COBE satellite could have detected deviations from FIRAS blackbody spectrum as small as fflffl ß n, n is an integer. To within ±, what is n? Answer n (5 points) second result was measurement of a dipole distortion of CMB spectrum; that is, radiation is slightly blueshifted to higher tem- in one direction, and slightly redshifted to lower temperatures in peratures opposite direction. To what physical effect was this dipole distortion at- tributed? large dipole in CMB is attributed to motion of satellite Answer to frame in which CMB is very nearly isotropic. ( entire relative Local Group is moving relative to this frame at a speed of about.2c.) (5 points) third result concerned measurement of temperature fluctuations after dipole feature mentioned above was subtracted out. Defining ffit T T ( ; ffi) ht i ( ; ffi) i ht ht i 2725 K, average value of T, y found a root mean square fluctuation, ffit * 2 2+ ; T ;

15 8.286 QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 29 to some number. To within an order of magnitude, what was that number? equal Answer ffit * T 5 (d) (5 points) Which of following describes Sachs-Wolfe effect? Photons from fluid which had a velocity toward us along line of sight (i) redder because of Doppler effect. appear Photons from fluid which had a velocity toward us along line of sight (ii) bluer because of Doppler effect. appear Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. redder Photons from overdense regions at surface of last scattering appear because y must climb out of gravitational potential well. bluer Photons traveling toward us from surface of last scattering appear (v) because of absorption in intergalactic medium. redder Photons traveling toward us from surface of last scattering appear (vi) because of absorption in intergalactic medium. bluer (5 points) flatness problem refers to extreme fine-tuning that is needed (e) at early times, in order for it to be as close to today aswe observe. in with assumption that today is equal to within about %, one Starting that at one second after big bang, concludes j jt sec< m ; m is an integer. To within ± 3, what is m? Answer m 8. (See derivation in Lecture Notes 8.) (5 points) total energy density of present universe consists mainly of (f) matter, dark matter, and dark energy. Give percentages of each, baryonic to best fit obtained from Planck 23 data. You will get full according if first (baryonic matter) is accurate to ±2%, and or two are credit accurate to within ±5%. Baryonic matter 5%. Dark matter 26.5%. Dark energy 68.5%. Answer Planck 23 numbers were given in Lecture Notes 7. To requested however, numbers such as Ryden's Benchmark Model would also be accuracy, satisfactory QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 3 (5 points) Within conventional hot big bang cosmology (without inflation), (g) is difficult to understand how temperature of CMB can be correlated it angular separations that are so large that points on surface of last at was separated from each or by more than a horizon distance. Ap- scattering what angle, in degrees, corresponds to a separation on surface proximately scattering of one horizon length? You will get full credit if your answer is last right to within a factor of 2. Ryden gives ffi as angle subtended by Hubble length on Answer of last scattering. For a matter-dominated universe, which would be surface good model for our universe, horizon length is twice Hubble length. a number from ffi to 5 ffi was considered acceptable. Any NUMBER DENSITIES IN THE COSMIC BACK- PROBLEM RADIATION GROUND general, number density of a particle in black-body radiation is given In by For photons, one has g Λ 2. n 38 6 erg ffi K k 9> T 27 ffi K Λ kt n 2 g ο(3) μhc ß ) erg-sec μh ; > c 2998 cm/sec n using ο(3) ' 22, one finds For neutrinos, 3 kt μhc nfl 399cm 3 g Λ ν per species cm 3 factor of 2 is to account forν and μν, and factor of 3/ arises from exclusion principle. So for three species of neutrinos one has Pauli g Λ ν 9 2

16 8.286 QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 3 Using result 3 ν T T 3 fl Problem of Problem Set 3 (2), one finds from 8 Λ T ν ν g nν g Λ fl 9 Tfl 3 nfl 3 399cm ) nν 326cm 3 (for all three species combined). PROBLEM 5 PROPERTIES OF BLACK-BODY RADIATION average energy per photon is found by dividing energy density by density. photon is a boson with two spin states, so g g Λ 2. number Using formulas on front of exam, E ß2 g 3 (kt) 3 (μhc) Λ (3) g 2 (kt) 3 (μhc) 3 ß ß kt 3 (3) were not expected to evaluate this numerically, but it is interesting to You that know E 27 kt that average energy per photon is significantly more than kt, which Note often used as a rough estimate. is method is same as above, except this time we use formula for density entropy S 2ß2 g 5 T 3 k 3 (μhc) Λ (3) g 2 (kt) 3 (μhc) 3 ß 2ß k 5 (3) QUIZ 3 REVIEW PROBLEM SOLUTIONS, FALL 26 p. 32 Numerically, this gives 362 k, k is Boltzmann constant. In this case we would have g g Λ. average energy per particle and average entropy particle depends only on ratio gg Λ, so re would be no difference from answers given in parts and. For a fermion, g is 7/8 times number of spin states, and g Λ is 3/ times (d) of spin states. So average energy per particle is number Numerically, E 35 kt. E ß2 g 3 (kt) 3 (μhc) Λ (3) g 2 (kt) 3 (μhc) 3 ß 7 ß (kt) 3 (μhc) 3 (3) ß 2 (kt) 3 (μhc) 3 7ß kt 8 (3) Mamatician General has determined Warning memorization of this number may adversely that affect your ability to remember value of ß. one takes into account both neutrinos and antineutrinos, average energy If particle is unaffected energy density and total number density per are both doubled, but ir ratio is unchanged. that energy per particle is higher for fermions than it is for bosons. Note result can be understood as a natural consequence of fact that fermions This obey exclusion principle, while bosons do not. Large numbers of must can refore collect in lowest energy levels. In fermion systems, bosons or hand, low-lying levels can accommodate at most one particle, on n additional particles are forced to higher energy levels. and