Equation (12-7) assumes a matter-dominated universe, in which the energy density of radiation is insignificant. This situation prevails today and has to do with the different rates at which the densities of matter and radiation vary with the size of the universe, Matter density is simply inversely proportional to the volume, obeying pMIR3, where pMis the matter density now. Radiation density, however, would be proportional to 1/R4. (Not only does the volume increase, but also all wavelengths are stretched in proportion to R, lowering the energy density by the extra factor.) This density drops faster as the universe grows, but it also grows more quickly in the backward time direction. In other words, long ago, the universe would have been radiation dominated. Show that if the function used for matter density in equation (12-7) is replaced by one appropriate to radiation, but retaining the assumption that K' and Ω∆ are both 0, then the scale factor R would grow as t1/2.
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