Day 17 homework
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Assume the following very approximate scaling relations for the temperature, density, and age of the universe, as a function of redshift z.

◆ T kinetic = 2.7 (1+z) K
◆ n(H) ≈ 2.51×10⁻⁷ (1+z)³ cm ⁻³
◆ Age ≈ 1.3×10¹⁰ (1+z)\^-1.5  yr

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Find the H + , H 0 , n e density vs z. Neglect all elements other than H, and assume that H + is the only source of free electrons. Use the Saha equation for the kinetic temperature given. Make a plot showing the H + /H and H 0 /H ionization fractions as a function of redshift, for 0.1 < z < 10 6 . Make both axes a base 10 log.

I use the approximate total hydrogen number density as a function of z.

    n(H) ≈ 2.51×10⁻⁷ (1+z)³ cm ⁻³

    

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Find the brems opacity at 0.5 microns vs z. Plot this opacity vs redshift with both axes logs.
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Find the photon mean free path vs z and plot this. Add the visible diameter of the universe, c × Age, to this plot.
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At what redshift does the photon mean free path equal the visible diameter of the universe? The universe is transparent with the mean free path is larger than the diameter, and a foggy photosphere when the mean free path is less.
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What is the temperature at this redshift?
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Assuming Wein’s Law, λₚₑₐₖ ∝ T⁻¹ , what would be the temperature of this blackbody today? How does this compare with the observed CMB temperature of 2.7K?