The early universe (about 800 million years after the big bang) contained atomic hydrogen gas. accretion of gas onto early super massive black holes (quasars) emitted uv light which ionized the surrounding hydrogen. consider a quasar uv source emitting n photons per second into hydrogen gas of number density n per cm cubed. each photon ionizes 1 hydrogen atom. (assume n and n to be constant in time) a assuming an infinite speed of light what is the expression for the volume vionized by the quasar after time ? b) using this expression calculate the rate of expansion dr/dt. under what conditions can dr/d be larger than c? given the constant speed of light this expression must be incorrect. in fact to account for the finite speed of light we calculate the volume ionized as the difference between the volume contained within a radius equal to the light horizon at time t, and the volume ionized during the light crossing time of the region (equal to r/c) at time t. the latter are subtracted because those photons did not encounter any hydrogen atoms yet. c) derive the correct expression for drdt d) show that this expression always has dr/dt