Observational Cosmology - Unit 3 Solutions
3.
(a) Mean E=3kT for blackbody radiation. Hence T=52600K when E=13.6
eV.
(b) Since T=2.7x(1+z) K, the temperature is 52600K at z=19270.
(c) At this epoch the gas will be fully ionised, and hence the Universe is very opaque (due to electron scattering of photons). The matter and radiation are therefore closely coupled and come into thermodynamic equilibrium, resulting in a blackbody radiation field.
(d) In practice, decoupling occurs at z~1000 - i.e. considerably later and at a lower temperature. There are several reasons: even when the mean energy is significantly less than 13.6 eV, photons in the high frequency tail of the blackbody curve may be able to ionise H; also when an H atom recombines it emits an energetic photon which can ionise or excite another atom; in addition, the situation is dynamic, and the state of ionisation may not keep up with the expanding Universe.