20 April 2012 Ay 127 E.S. Phinney & C.C. SteidelCosmology and Galaxy FormationProblem Set 3 Due in class, Friday 27 April 2012Reading: See the on-line syllabus for lecture-by-lecture readings.Collaboration policy: See the on-line collaboration p olicy.Homework Problems:1. The Cosmic Neutrino background: At temperatures high enough that all spec ies areultrarelativistic, and if all species have 0 chemical potential, by integrating over an appropriatethermal distribution,a) show that the energy densities of photons, neutrinos and electrons areργ= aT4, ρνi= ρ¯νi=716aT4, ρe+= ρe−=78aT4, (1)where a =π215k4c3¯h3, and i = e, µ, τ (and no more, according to SLAC and LEP Z0width!).Alsob) show that in physical volume V , the entropies areSi=43ρiVT. (2)c) Use these results to show that neutrinos today form a thermal background like the cosmicmicrowave background, but with Tν= (4/11)1/3TCM B. When many classic cosmologytexts were written, neutrinos were believed to be massless. Why does the fact thatneutrinos are now known to have masses of order 10−2eV not affect this result?d) Assuming neutrinos are massless, estimate the number of cosmic neutrinos passingthrough your body each second. How would this be changed if mν= 1eV or 10−2eV?e) Has a cosmic background neutrino ever interacted with a nucleus in your body? Inanyone’s?2. WIMP dark matter:a) Show that the number density today of stable relics of a particle species X of mass mXwhich falls out of equilibrium when it is nonrelativistic (this is called “cold dark matter”)is proportional to m−1xσ−1a, where σais the annihilation cross-section (at the energiescharacteristic of the freeze-out time).b) Thus show that their contribution to Ω0depends only on σa, and show that (for chemicalpotential µX= 0)ΩX'7 × 10−27cm3s−1hσavi× (slowly varying logarithms) .1c) Before precision electroweak experiments showed that there is room only for 3 neutrinotypes, it used to be popular to let X be a fourth neutrino species. Show that thiswould contribute significantly to Ω only if mx∼ 1 GeV. (Hints for those with a weakparticle-physics background: with ¯h = c = 1, the weak coupling constant is GF=1.2 × 10−5GeV−2, 1 GeV−2= 0.4 × 10−27cm2, and σa∼ G2Fm2X; those able to doso may justify and perhaps improve on the expression for σa). [Note for informationonly: It is today popular to let X be the lightest supersymmetric particle, perhapsa neutralino (a linear combination of the supersymmetric partners to the photon, Z0and/or Higgs boson), which can annihilate into all the usual particles we know about,plus Higgs bosons. Unfortunately, the cross-sections for these depend on many unknownparameters, though experimental limits suggest that these could give annihilation ratesand thus dark matter densities of the right values.]3. Thomson Scattering Optical DepthIn the standard cosmological scenario, all the electrons and protons in the Universe combineto form hydrogen at a redshift z ∼ 1000– the epoch of recombination. However, at someredshift zreion, stars begin to form and emit radiation that ionizes all the hydrogen in theUniverse. If so, then cosmic microwave background (CMB) photons may Thomson scatterfrom the free electrons en route from the surface of last scattering. Calculate the optical depthτreionfor Thomson scattering of CMB photons as a function of the reionization redshift zreionfor Ωm= 0.3 and ΩΛ= 0.7. Derive an analytic approximation for the redshift zreion>> Ω−1m.Write your answer in terms of the baryon density Ωbh2= 0.023 (where h = H0/100 km s−1Mpc−1) and in terms of the helium mass fraction Y ' 0.23. At what zreiondoes τreion=