Math

Q2: Preview In cosmology, the size of the universe is controlled by a positive function of time a(t), called the scale factor. The bigger a(t), the bigger the universe. In a flat, radiation-dominated universe (which describes relatively well the universe right after the Big Bang), one can calculate, using general relativity, that a (t) = kvt, where k is a positive constant. (a) Sketch the graph of the scale factor a(t) (fort > 0) in a flat, radiation-dominated universe. (b) Recall that the slope of the tangent line to a(t) at a point gives the instantaneous rate of change of a(t) at this point. By looking at the graph in (a), what can you say about the expansion of the universe? Justify your answer. (c) Would your conclusion be different if the scale factor was given by a(t) = 100 - (t - 10)2? What would happen to the universe in this case? Justify your answer. (Hint: sketch the graph of a(t) ... it should remind you of a geometric shape that you know very well) File