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Hydrogen recombines at a redshift of approximately z=1200, with recombination starting already at z=1600.

a. What was the ratio of the number density of hydrogen-ionizing photons (i.e. photons with energies E > 13.6eV) to the number density of hydrogen atoms at z=1600? Explain why this ratio does not have to equal unity at recombination.

b. What was the ratio of the number density of helium-ionizing photons (i.e. photons with energies E>24.6eV) to the number density of helium atoms at z=1600? Does your answer imply that helium recombines before or after hydrogen?

For this problem, assume the following: at the present day, baryons contribute a fraction Ωb=0.04 of the critical density, of which YH = 76% by mass is hydrogen, and YHe = 24% is helium. Assume a Hubble constant Ho = 7- km/s/Mpc. The temperature of the CMB today is To=2.725K.

2. Alternative Origin of Helium and CMB

It was shown that the “hot big bang” model explains the origin and mass fraction of cosmic helium, as well as the presence of the CMB. Suppose, however, that the universe instead consisted originally of hydrogen alone, and that all of the observed helium (with a mass fraction of YHe = 24%) was created by fusion inside starts. How much energy was released per unit volume in the universe as a result of this helium fusion? Suppose that all of this energy is now in the form of radiation. Compare the resulting energy density to the present-day energy density of the CMB. Could the helium fusion account for the present-day CMB based on the energy density alone? Can you think of additional problems for this being the origin of the CMB? (Note: 4He has a total binding energy of 28.3 MeV.)

3. Modified Nucleosynthesis

We estimated the maximum possible mass fraction of helium, YHe,max in the universe, by assuming that at the time of nucleosysnthesis, all available neutrons were converted into 4He nuclei. Suppose that the neutron decay time, τn = 890s, was ten times shorter, τn = 89s. What would be the value of YHe,max ? (Hint: nucleosynthesis occurred approximately 200 seconds after neutron-proton freeze-out).

4. Reionization

Recent observations by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite showed that approximately 10% of the CMB photons suffered a scattering with an electron on their way from redshift z=1100 to Earth. The explanations of this result may be that the universe is kept fully ionized by starlight at all reshifts below zr. What is the value of zr required to explain the electron scattering probability measured by WMAP? This would have to correspond to the epoch when the first stars were formed. For simplicity, assume a flat, Ω1 = 1 universe with Ωb = 0.04 and Ho = 70km/s/Mpc. Compute zr by assuming that all of the hydrogen is ionized, but all of the helium is neutral, throughout the entire interval 0<z<zr. How does your answer change if you assume instead that all of the helium is in doubly ionized (He++) form? (Note: the Thomson cross-section is σT = 6.65x 10^-25 cm2.)

If anyone can help me with any these that would be greatly appreciated.